Theorem evlfcl 16862

Description: The evaluation functor is a bifunctor (a two-argument functor) with the first parameter taking values in the set of functors C --> D, and the second parameter in D. (Contributed by Mario Carneiro, 12-Jan-2017.)

Hypotheses

Ref	Expression
evlfcl.e	|- E = ( C evalF D )
evlfcl.q	|- Q = ( C FuncCat D )
evlfcl.c	|- ( ph -> C e. Cat )
evlfcl.d	|- ( ph -> D e. Cat )

Assertion

Ref	Expression
evlfcl	|- ( ph -> E e. ( ( Q X.c C ) Func D ) )

Proof of Theorem evlfcl

Dummy variables f a g h m n u v x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		evlfcl.e	. . . . 5 |- E = ( C evalF D )
2		evlfcl.c	. . . . 5 |- ( ph -> C e. Cat )
3		evlfcl.d	. . . . 5 |- ( ph -> D e. Cat )
4		eqid 2622	. . . . 5 |- ( Base ` C ) = ( Base ` C )
5		eqid 2622	. . . . 5 |- ( Hom ` C ) = ( Hom ` C )
6		eqid 2622	. . . . 5 |- (comp ` D ) = (comp ` D )
7		eqid 2622	. . . . 5 |- ( C Nat D ) = ( C Nat D )
8	1, 2, 3, 4, 5, 6, 7	evlfval 16857	. . . 4 |- ( ph -> E = <. ( f e. ( C Func D ) , x e. ( Base ` C ) |-> ( ( 1st ` f ) ` x ) ) , ( x e. ( ( C Func D ) X. ( Base ` C ) ) , y e. ( ( C Func D ) X. ( Base ` C ) ) |-> [_ ( 1st ` x ) / m ]_ [_ ( 1st ` y ) / n ]_ ( a e. ( m ( C Nat D ) n ) , g e. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) |-> ( ( a ` ( 2nd ` y ) ) ( <. ( ( 1st ` m ) ` ( 2nd ` x ) ) , ( ( 1st ` m ) ` ( 2nd ` y ) ) >. (comp ` D ) ( ( 1st ` n ) ` ( 2nd ` y ) ) ) ( ( ( 2nd ` x ) ( 2nd ` m ) ( 2nd ` y ) ) ` g ) ) ) ) >. )
9		ovex 6678	. . . . . 6 |- ( C Func D ) e. _V
10		fvex 6201	. . . . . 6 |- ( Base ` C ) e. _V
11	9, 10	mpt2ex 7247	. . . . 5 |- ( f e. ( C Func D ) , x e. ( Base ` C ) |-> ( ( 1st ` f ) ` x ) ) e. _V
12	9, 10	xpex 6962	. . . . . 6 |- ( ( C Func D ) X. ( Base ` C ) ) e. _V
13	12, 12	mpt2ex 7247	. . . . 5 |- ( x e. ( ( C Func D ) X. ( Base ` C ) ) , y e. ( ( C Func D ) X. ( Base ` C ) ) |-> [_ ( 1st ` x ) / m ]_ [_ ( 1st ` y ) / n ]_ ( a e. ( m ( C Nat D ) n ) , g e. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) |-> ( ( a ` ( 2nd ` y ) ) ( <. ( ( 1st ` m ) ` ( 2nd ` x ) ) , ( ( 1st ` m ) ` ( 2nd ` y ) ) >. (comp ` D ) ( ( 1st ` n ) ` ( 2nd ` y ) ) ) ( ( ( 2nd ` x ) ( 2nd ` m ) ( 2nd ` y ) ) ` g ) ) ) ) e. _V
14	11, 13	opelvv 5166	. . . 4 |- <. ( f e. ( C Func D ) , x e. ( Base ` C ) |-> ( ( 1st ` f ) ` x ) ) , ( x e. ( ( C Func D ) X. ( Base ` C ) ) , y e. ( ( C Func D ) X. ( Base ` C ) ) |-> [_ ( 1st ` x ) / m ]_ [_ ( 1st ` y ) / n ]_ ( a e. ( m ( C Nat D ) n ) , g e. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) |-> ( ( a ` ( 2nd ` y ) ) ( <. ( ( 1st ` m ) ` ( 2nd ` x ) ) , ( ( 1st ` m ) ` ( 2nd ` y ) ) >. (comp ` D ) ( ( 1st ` n ) ` ( 2nd ` y ) ) ) ( ( ( 2nd ` x ) ( 2nd ` m ) ( 2nd ` y ) ) ` g ) ) ) ) >. e. ( _V X. _V )
15	8, 14	syl6eqel 2709	. . 3 |- ( ph -> E e. ( _V X. _V ) )
16		1st2nd2 7205	. . 3 |- ( E e. ( _V X. _V ) -> E = <. ( 1st ` E ) , ( 2nd ` E ) >. )
17	15, 16	syl 17	. 2 |- ( ph -> E = <. ( 1st ` E ) , ( 2nd ` E ) >. )
18		eqid 2622	. . . . 5 |- ( Q X.c C ) = ( Q X.c C )
19		evlfcl.q	. . . . . 6 |- Q = ( C FuncCat D )
20	19	fucbas 16620	. . . . 5 |- ( C Func D ) = ( Base ` Q )
21	18, 20, 4	xpcbas 16818	. . . 4 |- ( ( C Func D ) X. ( Base ` C ) ) = ( Base ` ( Q X.c C ) )
22		eqid 2622	. . . 4 |- ( Base ` D ) = ( Base ` D )
23		eqid 2622	. . . 4 |- ( Hom ` ( Q X.c C ) ) = ( Hom ` ( Q X.c C ) )
24		eqid 2622	. . . 4 |- ( Hom ` D ) = ( Hom ` D )
25		eqid 2622	. . . 4 |- ( Id ` ( Q X.c C ) ) = ( Id ` ( Q X.c C ) )
26		eqid 2622	. . . 4 |- ( Id ` D ) = ( Id ` D )
27		eqid 2622	. . . 4 |- (comp ` ( Q X.c C ) ) = (comp ` ( Q X.c C ) )
28	19, 2, 3	fuccat 16630	. . . . 5 |- ( ph -> Q e. Cat )
29	18, 28, 2	xpccat 16830	. . . 4 |- ( ph -> ( Q X.c C ) e. Cat )
30		relfunc 16522	. . . . . . . . . . 11 |- Rel ( C Func D )
31		simpr 477	. . . . . . . . . . 11 |- ( ( ph /\ f e. ( C Func D ) ) -> f e. ( C Func D ) )
32		1st2ndbr 7217	. . . . . . . . . . 11 |- ( ( Rel ( C Func D ) /\ f e. ( C Func D ) ) -> ( 1st ` f ) ( C Func D ) ( 2nd ` f ) )
33	30, 31, 32	sylancr 695	. . . . . . . . . 10 |- ( ( ph /\ f e. ( C Func D ) ) -> ( 1st ` f ) ( C Func D ) ( 2nd ` f ) )
34	4, 22, 33	funcf1 16526	. . . . . . . . 9 |- ( ( ph /\ f e. ( C Func D ) ) -> ( 1st ` f ) : ( Base ` C ) --> ( Base ` D ) )
35	34	ffvelrnda 6359	. . . . . . . 8 |- ( ( ( ph /\ f e. ( C Func D ) ) /\ x e. ( Base ` C ) ) -> ( ( 1st ` f ) ` x ) e. ( Base ` D ) )
36	35	ralrimiva 2966	. . . . . . 7 |- ( ( ph /\ f e. ( C Func D ) ) -> A. x e. ( Base ` C ) ( ( 1st ` f ) ` x ) e. ( Base ` D ) )
37	36	ralrimiva 2966	. . . . . 6 |- ( ph -> A. f e. ( C Func D ) A. x e. ( Base ` C ) ( ( 1st ` f ) ` x ) e. ( Base ` D ) )
38		eqid 2622	. . . . . . 7 |- ( f e. ( C Func D ) , x e. ( Base ` C ) |-> ( ( 1st ` f ) ` x ) ) = ( f e. ( C Func D ) , x e. ( Base ` C ) |-> ( ( 1st ` f ) ` x ) )
39	38	fmpt2 7237	. . . . . 6 |- ( A. f e. ( C Func D ) A. x e. ( Base ` C ) ( ( 1st ` f ) ` x ) e. ( Base ` D ) <-> ( f e. ( C Func D ) , x e. ( Base ` C ) |-> ( ( 1st ` f ) ` x ) ) : ( ( C Func D ) X. ( Base ` C ) ) --> ( Base ` D ) )
40	37, 39	sylib 208	. . . . 5 |- ( ph -> ( f e. ( C Func D ) , x e. ( Base ` C ) |-> ( ( 1st ` f ) ` x ) ) : ( ( C Func D ) X. ( Base ` C ) ) --> ( Base ` D ) )
41	11, 13	op1std 7178	. . . . . . 7 |- ( E = <. ( f e. ( C Func D ) , x e. ( Base ` C ) |-> ( ( 1st ` f ) ` x ) ) , ( x e. ( ( C Func D ) X. ( Base ` C ) ) , y e. ( ( C Func D ) X. ( Base ` C ) ) |-> [_ ( 1st ` x ) / m ]_ [_ ( 1st ` y ) / n ]_ ( a e. ( m ( C Nat D ) n ) , g e. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) |-> ( ( a ` ( 2nd ` y ) ) ( <. ( ( 1st ` m ) ` ( 2nd ` x ) ) , ( ( 1st ` m ) ` ( 2nd ` y ) ) >. (comp ` D ) ( ( 1st ` n ) ` ( 2nd ` y ) ) ) ( ( ( 2nd ` x ) ( 2nd ` m ) ( 2nd ` y ) ) ` g ) ) ) ) >. -> ( 1st ` E ) = ( f e. ( C Func D ) , x e. ( Base ` C ) |-> ( ( 1st ` f ) ` x ) ) )
42	8, 41	syl 17	. . . . . 6 |- ( ph -> ( 1st ` E ) = ( f e. ( C Func D ) , x e. ( Base ` C ) |-> ( ( 1st ` f ) ` x ) ) )
43	42	feq1d 6030	. . . . 5 |- ( ph -> ( ( 1st ` E ) : ( ( C Func D ) X. ( Base ` C ) ) --> ( Base ` D ) <-> ( f e. ( C Func D ) , x e. ( Base ` C ) |-> ( ( 1st ` f ) ` x ) ) : ( ( C Func D ) X. ( Base ` C ) ) --> ( Base ` D ) ) )
44	40, 43	mpbird 247	. . . 4 |- ( ph -> ( 1st ` E ) : ( ( C Func D ) X. ( Base ` C ) ) --> ( Base ` D ) )
45		eqid 2622	. . . . . 6 |- ( x e. ( ( C Func D ) X. ( Base ` C ) ) , y e. ( ( C Func D ) X. ( Base ` C ) ) |-> [_ ( 1st ` x ) / m ]_ [_ ( 1st ` y ) / n ]_ ( a e. ( m ( C Nat D ) n ) , g e. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) |-> ( ( a ` ( 2nd ` y ) ) ( <. ( ( 1st ` m ) ` ( 2nd ` x ) ) , ( ( 1st ` m ) ` ( 2nd ` y ) ) >. (comp ` D ) ( ( 1st ` n ) ` ( 2nd ` y ) ) ) ( ( ( 2nd ` x ) ( 2nd ` m ) ( 2nd ` y ) ) ` g ) ) ) ) = ( x e. ( ( C Func D ) X. ( Base ` C ) ) , y e. ( ( C Func D ) X. ( Base ` C ) ) |-> [_ ( 1st ` x ) / m ]_ [_ ( 1st ` y ) / n ]_ ( a e. ( m ( C Nat D ) n ) , g e. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) |-> ( ( a ` ( 2nd ` y ) ) ( <. ( ( 1st ` m ) ` ( 2nd ` x ) ) , ( ( 1st ` m ) ` ( 2nd ` y ) ) >. (comp ` D ) ( ( 1st ` n ) ` ( 2nd ` y ) ) ) ( ( ( 2nd ` x ) ( 2nd ` m ) ( 2nd ` y ) ) ` g ) ) ) )
46		ovex 6678	. . . . . . . . 9 |- ( m ( C Nat D ) n ) e. _V
47		ovex 6678	. . . . . . . . 9 |- ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) e. _V
48	46, 47	mpt2ex 7247	. . . . . . . 8 |- ( a e. ( m ( C Nat D ) n ) , g e. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) |-> ( ( a ` ( 2nd ` y ) ) ( <. ( ( 1st ` m ) ` ( 2nd ` x ) ) , ( ( 1st ` m ) ` ( 2nd ` y ) ) >. (comp ` D ) ( ( 1st ` n ) ` ( 2nd ` y ) ) ) ( ( ( 2nd ` x ) ( 2nd ` m ) ( 2nd ` y ) ) ` g ) ) ) e. _V
49	48	csbex 4793	. . . . . . 7 |- [_ ( 1st ` y ) / n ]_ ( a e. ( m ( C Nat D ) n ) , g e. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) |-> ( ( a ` ( 2nd ` y ) ) ( <. ( ( 1st ` m ) ` ( 2nd ` x ) ) , ( ( 1st ` m ) ` ( 2nd ` y ) ) >. (comp ` D ) ( ( 1st ` n ) ` ( 2nd ` y ) ) ) ( ( ( 2nd ` x ) ( 2nd ` m ) ( 2nd ` y ) ) ` g ) ) ) e. _V
50	49	csbex 4793	. . . . . 6 |- [_ ( 1st ` x ) / m ]_ [_ ( 1st ` y ) / n ]_ ( a e. ( m ( C Nat D ) n ) , g e. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) |-> ( ( a ` ( 2nd ` y ) ) ( <. ( ( 1st ` m ) ` ( 2nd ` x ) ) , ( ( 1st ` m ) ` ( 2nd ` y ) ) >. (comp ` D ) ( ( 1st ` n ) ` ( 2nd ` y ) ) ) ( ( ( 2nd ` x ) ( 2nd ` m ) ( 2nd ` y ) ) ` g ) ) ) e. _V
51	45, 50	fnmpt2i 7239	. . . . 5 |- ( x e. ( ( C Func D ) X. ( Base ` C ) ) , y e. ( ( C Func D ) X. ( Base ` C ) ) |-> [_ ( 1st ` x ) / m ]_ [_ ( 1st ` y ) / n ]_ ( a e. ( m ( C Nat D ) n ) , g e. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) |-> ( ( a ` ( 2nd ` y ) ) ( <. ( ( 1st ` m ) ` ( 2nd ` x ) ) , ( ( 1st ` m ) ` ( 2nd ` y ) ) >. (comp ` D ) ( ( 1st ` n ) ` ( 2nd ` y ) ) ) ( ( ( 2nd ` x ) ( 2nd ` m ) ( 2nd ` y ) ) ` g ) ) ) ) Fn ( ( ( C Func D ) X. ( Base ` C ) ) X. ( ( C Func D ) X. ( Base ` C ) ) )
52	11, 13	op2ndd 7179	. . . . . . 7 |- ( E = <. ( f e. ( C Func D ) , x e. ( Base ` C ) |-> ( ( 1st ` f ) ` x ) ) , ( x e. ( ( C Func D ) X. ( Base ` C ) ) , y e. ( ( C Func D ) X. ( Base ` C ) ) |-> [_ ( 1st ` x ) / m ]_ [_ ( 1st ` y ) / n ]_ ( a e. ( m ( C Nat D ) n ) , g e. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) |-> ( ( a ` ( 2nd ` y ) ) ( <. ( ( 1st ` m ) ` ( 2nd ` x ) ) , ( ( 1st ` m ) ` ( 2nd ` y ) ) >. (comp ` D ) ( ( 1st ` n ) ` ( 2nd ` y ) ) ) ( ( ( 2nd ` x ) ( 2nd ` m ) ( 2nd ` y ) ) ` g ) ) ) ) >. -> ( 2nd ` E ) = ( x e. ( ( C Func D ) X. ( Base ` C ) ) , y e. ( ( C Func D ) X. ( Base ` C ) ) |-> [_ ( 1st ` x ) / m ]_ [_ ( 1st ` y ) / n ]_ ( a e. ( m ( C Nat D ) n ) , g e. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) |-> ( ( a ` ( 2nd ` y ) ) ( <. ( ( 1st ` m ) ` ( 2nd ` x ) ) , ( ( 1st ` m ) ` ( 2nd ` y ) ) >. (comp ` D ) ( ( 1st ` n ) ` ( 2nd ` y ) ) ) ( ( ( 2nd ` x ) ( 2nd ` m ) ( 2nd ` y ) ) ` g ) ) ) ) )
53	8, 52	syl 17	. . . . . 6 |- ( ph -> ( 2nd ` E ) = ( x e. ( ( C Func D ) X. ( Base ` C ) ) , y e. ( ( C Func D ) X. ( Base ` C ) ) |-> [_ ( 1st ` x ) / m ]_ [_ ( 1st ` y ) / n ]_ ( a e. ( m ( C Nat D ) n ) , g e. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) |-> ( ( a ` ( 2nd ` y ) ) ( <. ( ( 1st ` m ) ` ( 2nd ` x ) ) , ( ( 1st ` m ) ` ( 2nd ` y ) ) >. (comp ` D ) ( ( 1st ` n ) ` ( 2nd ` y ) ) ) ( ( ( 2nd ` x ) ( 2nd ` m ) ( 2nd ` y ) ) ` g ) ) ) ) )
54	53	fneq1d 5981	. . . . 5 |- ( ph -> ( ( 2nd ` E ) Fn ( ( ( C Func D ) X. ( Base ` C ) ) X. ( ( C Func D ) X. ( Base ` C ) ) ) <-> ( x e. ( ( C Func D ) X. ( Base ` C ) ) , y e. ( ( C Func D ) X. ( Base ` C ) ) |-> [_ ( 1st ` x ) / m ]_ [_ ( 1st ` y ) / n ]_ ( a e. ( m ( C Nat D ) n ) , g e. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) |-> ( ( a ` ( 2nd ` y ) ) ( <. ( ( 1st ` m ) ` ( 2nd ` x ) ) , ( ( 1st ` m ) ` ( 2nd ` y ) ) >. (comp ` D ) ( ( 1st ` n ) ` ( 2nd ` y ) ) ) ( ( ( 2nd ` x ) ( 2nd ` m ) ( 2nd ` y ) ) ` g ) ) ) ) Fn ( ( ( C Func D ) X. ( Base ` C ) ) X. ( ( C Func D ) X. ( Base ` C ) ) ) ) )
55	51, 54	mpbiri 248	. . . 4 |- ( ph -> ( 2nd ` E ) Fn ( ( ( C Func D ) X. ( Base ` C ) ) X. ( ( C Func D ) X. ( Base ` C ) ) ) )
56	3	ad2antrr 762	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) /\ ( g e. ( C Func D ) /\ v e. ( Base ` C ) ) ) -> D e. Cat )
57	56	adantr 481	. . . . . . . . . . . . . . 15 |- ( ( ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) /\ ( g e. ( C Func D ) /\ v e. ( Base ` C ) ) ) /\ ( a e. ( f ( C Nat D ) g ) /\ h e. ( u ( Hom ` C ) v ) ) ) -> D e. Cat )
58		simplrl 800	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) /\ ( g e. ( C Func D ) /\ v e. ( Base ` C ) ) ) -> f e. ( C Func D ) )
59	30, 58, 32	sylancr 695	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) /\ ( g e. ( C Func D ) /\ v e. ( Base ` C ) ) ) -> ( 1st ` f ) ( C Func D ) ( 2nd ` f ) )
60	4, 22, 59	funcf1 16526	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) /\ ( g e. ( C Func D ) /\ v e. ( Base ` C ) ) ) -> ( 1st ` f ) : ( Base ` C ) --> ( Base ` D ) )
61	60	adantr 481	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) /\ ( g e. ( C Func D ) /\ v e. ( Base ` C ) ) ) /\ ( a e. ( f ( C Nat D ) g ) /\ h e. ( u ( Hom ` C ) v ) ) ) -> ( 1st ` f ) : ( Base ` C ) --> ( Base ` D ) )
62		simplrr 801	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) /\ ( g e. ( C Func D ) /\ v e. ( Base ` C ) ) ) -> u e. ( Base ` C ) )
63	62	adantr 481	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) /\ ( g e. ( C Func D ) /\ v e. ( Base ` C ) ) ) /\ ( a e. ( f ( C Nat D ) g ) /\ h e. ( u ( Hom ` C ) v ) ) ) -> u e. ( Base ` C ) )
64	61, 63	ffvelrnd 6360	. . . . . . . . . . . . . . 15 |- ( ( ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) /\ ( g e. ( C Func D ) /\ v e. ( Base ` C ) ) ) /\ ( a e. ( f ( C Nat D ) g ) /\ h e. ( u ( Hom ` C ) v ) ) ) -> ( ( 1st ` f ) ` u ) e. ( Base ` D ) )
65		simplrr 801	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) /\ ( g e. ( C Func D ) /\ v e. ( Base ` C ) ) ) /\ ( a e. ( f ( C Nat D ) g ) /\ h e. ( u ( Hom ` C ) v ) ) ) -> v e. ( Base ` C ) )
66	61, 65	ffvelrnd 6360	. . . . . . . . . . . . . . 15 |- ( ( ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) /\ ( g e. ( C Func D ) /\ v e. ( Base ` C ) ) ) /\ ( a e. ( f ( C Nat D ) g ) /\ h e. ( u ( Hom ` C ) v ) ) ) -> ( ( 1st ` f ) ` v ) e. ( Base ` D ) )
67		simprl 794	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) /\ ( g e. ( C Func D ) /\ v e. ( Base ` C ) ) ) -> g e. ( C Func D ) )
68		1st2ndbr 7217	. . . . . . . . . . . . . . . . . . 19 |- ( ( Rel ( C Func D ) /\ g e. ( C Func D ) ) -> ( 1st ` g ) ( C Func D ) ( 2nd ` g ) )
69	30, 67, 68	sylancr 695	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) /\ ( g e. ( C Func D ) /\ v e. ( Base ` C ) ) ) -> ( 1st ` g ) ( C Func D ) ( 2nd ` g ) )
70	4, 22, 69	funcf1 16526	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) /\ ( g e. ( C Func D ) /\ v e. ( Base ` C ) ) ) -> ( 1st ` g ) : ( Base ` C ) --> ( Base ` D ) )
71	70	adantr 481	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) /\ ( g e. ( C Func D ) /\ v e. ( Base ` C ) ) ) /\ ( a e. ( f ( C Nat D ) g ) /\ h e. ( u ( Hom ` C ) v ) ) ) -> ( 1st ` g ) : ( Base ` C ) --> ( Base ` D ) )
72	71, 65	ffvelrnd 6360	. . . . . . . . . . . . . . 15 |- ( ( ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) /\ ( g e. ( C Func D ) /\ v e. ( Base ` C ) ) ) /\ ( a e. ( f ( C Nat D ) g ) /\ h e. ( u ( Hom ` C ) v ) ) ) -> ( ( 1st ` g ) ` v ) e. ( Base ` D ) )
73		simprr 796	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) /\ ( g e. ( C Func D ) /\ v e. ( Base ` C ) ) ) -> v e. ( Base ` C ) )
74	4, 5, 24, 59, 62, 73	funcf2 16528	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) /\ ( g e. ( C Func D ) /\ v e. ( Base ` C ) ) ) -> ( u ( 2nd ` f ) v ) : ( u ( Hom ` C ) v ) --> ( ( ( 1st ` f ) ` u ) ( Hom ` D ) ( ( 1st ` f ) ` v ) ) )
75	74	adantr 481	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) /\ ( g e. ( C Func D ) /\ v e. ( Base ` C ) ) ) /\ ( a e. ( f ( C Nat D ) g ) /\ h e. ( u ( Hom ` C ) v ) ) ) -> ( u ( 2nd ` f ) v ) : ( u ( Hom ` C ) v ) --> ( ( ( 1st ` f ) ` u ) ( Hom ` D ) ( ( 1st ` f ) ` v ) ) )
76		simprr 796	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) /\ ( g e. ( C Func D ) /\ v e. ( Base ` C ) ) ) /\ ( a e. ( f ( C Nat D ) g ) /\ h e. ( u ( Hom ` C ) v ) ) ) -> h e. ( u ( Hom ` C ) v ) )
77	75, 76	ffvelrnd 6360	. . . . . . . . . . . . . . 15 |- ( ( ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) /\ ( g e. ( C Func D ) /\ v e. ( Base ` C ) ) ) /\ ( a e. ( f ( C Nat D ) g ) /\ h e. ( u ( Hom ` C ) v ) ) ) -> ( ( u ( 2nd ` f ) v ) ` h ) e. ( ( ( 1st ` f ) ` u ) ( Hom ` D ) ( ( 1st ` f ) ` v ) ) )
78		simprl 794	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) /\ ( g e. ( C Func D ) /\ v e. ( Base ` C ) ) ) /\ ( a e. ( f ( C Nat D ) g ) /\ h e. ( u ( Hom ` C ) v ) ) ) -> a e. ( f ( C Nat D ) g ) )
79	7, 78	nat1st2nd 16611	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) /\ ( g e. ( C Func D ) /\ v e. ( Base ` C ) ) ) /\ ( a e. ( f ( C Nat D ) g ) /\ h e. ( u ( Hom ` C ) v ) ) ) -> a e. ( <. ( 1st ` f ) , ( 2nd ` f ) >. ( C Nat D ) <. ( 1st ` g ) , ( 2nd ` g ) >. ) )
80	7, 79, 4, 24, 65	natcl 16613	. . . . . . . . . . . . . . 15 |- ( ( ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) /\ ( g e. ( C Func D ) /\ v e. ( Base ` C ) ) ) /\ ( a e. ( f ( C Nat D ) g ) /\ h e. ( u ( Hom ` C ) v ) ) ) -> ( a ` v ) e. ( ( ( 1st ` f ) ` v ) ( Hom ` D ) ( ( 1st ` g ) ` v ) ) )
81	22, 24, 6, 57, 64, 66, 72, 77, 80	catcocl 16346	. . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) /\ ( g e. ( C Func D ) /\ v e. ( Base ` C ) ) ) /\ ( a e. ( f ( C Nat D ) g ) /\ h e. ( u ( Hom ` C ) v ) ) ) -> ( ( a ` v ) ( <. ( ( 1st ` f ) ` u ) , ( ( 1st ` f ) ` v ) >. (comp ` D ) ( ( 1st ` g ) ` v ) ) ( ( u ( 2nd ` f ) v ) ` h ) ) e. ( ( ( 1st ` f ) ` u ) ( Hom ` D ) ( ( 1st ` g ) ` v ) ) )
82	81	ralrimivva 2971	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) /\ ( g e. ( C Func D ) /\ v e. ( Base ` C ) ) ) -> A. a e. ( f ( C Nat D ) g ) A. h e. ( u ( Hom ` C ) v ) ( ( a ` v ) ( <. ( ( 1st ` f ) ` u ) , ( ( 1st ` f ) ` v ) >. (comp ` D ) ( ( 1st ` g ) ` v ) ) ( ( u ( 2nd ` f ) v ) ` h ) ) e. ( ( ( 1st ` f ) ` u ) ( Hom ` D ) ( ( 1st ` g ) ` v ) ) )
83		eqid 2622	. . . . . . . . . . . . . 14 |- ( a e. ( f ( C Nat D ) g ) , h e. ( u ( Hom ` C ) v ) |-> ( ( a ` v ) ( <. ( ( 1st ` f ) ` u ) , ( ( 1st ` f ) ` v ) >. (comp ` D ) ( ( 1st ` g ) ` v ) ) ( ( u ( 2nd ` f ) v ) ` h ) ) ) = ( a e. ( f ( C Nat D ) g ) , h e. ( u ( Hom ` C ) v ) |-> ( ( a ` v ) ( <. ( ( 1st ` f ) ` u ) , ( ( 1st ` f ) ` v ) >. (comp ` D ) ( ( 1st ` g ) ` v ) ) ( ( u ( 2nd ` f ) v ) ` h ) ) )
84	83	fmpt2 7237	. . . . . . . . . . . . 13 |- ( A. a e. ( f ( C Nat D ) g ) A. h e. ( u ( Hom ` C ) v ) ( ( a ` v ) ( <. ( ( 1st ` f ) ` u ) , ( ( 1st ` f ) ` v ) >. (comp ` D ) ( ( 1st ` g ) ` v ) ) ( ( u ( 2nd ` f ) v ) ` h ) ) e. ( ( ( 1st ` f ) ` u ) ( Hom ` D ) ( ( 1st ` g ) ` v ) ) <-> ( a e. ( f ( C Nat D ) g ) , h e. ( u ( Hom ` C ) v ) |-> ( ( a ` v ) ( <. ( ( 1st ` f ) ` u ) , ( ( 1st ` f ) ` v ) >. (comp ` D ) ( ( 1st ` g ) ` v ) ) ( ( u ( 2nd ` f ) v ) ` h ) ) ) : ( ( f ( C Nat D ) g ) X. ( u ( Hom ` C ) v ) ) --> ( ( ( 1st ` f ) ` u ) ( Hom ` D ) ( ( 1st ` g ) ` v ) ) )
85	82, 84	sylib 208	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) /\ ( g e. ( C Func D ) /\ v e. ( Base ` C ) ) ) -> ( a e. ( f ( C Nat D ) g ) , h e. ( u ( Hom ` C ) v ) |-> ( ( a ` v ) ( <. ( ( 1st ` f ) ` u ) , ( ( 1st ` f ) ` v ) >. (comp ` D ) ( ( 1st ` g ) ` v ) ) ( ( u ( 2nd ` f ) v ) ` h ) ) ) : ( ( f ( C Nat D ) g ) X. ( u ( Hom ` C ) v ) ) --> ( ( ( 1st ` f ) ` u ) ( Hom ` D ) ( ( 1st ` g ) ` v ) ) )
86	2	ad2antrr 762	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) /\ ( g e. ( C Func D ) /\ v e. ( Base ` C ) ) ) -> C e. Cat )
87		eqid 2622	. . . . . . . . . . . . . 14 |- ( <. f , u >. ( 2nd ` E ) <. g , v >. ) = ( <. f , u >. ( 2nd ` E ) <. g , v >. )
88	1, 86, 56, 4, 5, 6, 7, 58, 67, 62, 73, 87	evlf2 16858	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) /\ ( g e. ( C Func D ) /\ v e. ( Base ` C ) ) ) -> ( <. f , u >. ( 2nd ` E ) <. g , v >. ) = ( a e. ( f ( C Nat D ) g ) , h e. ( u ( Hom ` C ) v ) |-> ( ( a ` v ) ( <. ( ( 1st ` f ) ` u ) , ( ( 1st ` f ) ` v ) >. (comp ` D ) ( ( 1st ` g ) ` v ) ) ( ( u ( 2nd ` f ) v ) ` h ) ) ) )
89	88	feq1d 6030	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) /\ ( g e. ( C Func D ) /\ v e. ( Base ` C ) ) ) -> ( ( <. f , u >. ( 2nd ` E ) <. g , v >. ) : ( ( f ( C Nat D ) g ) X. ( u ( Hom ` C ) v ) ) --> ( ( ( 1st ` f ) ` u ) ( Hom ` D ) ( ( 1st ` g ) ` v ) ) <-> ( a e. ( f ( C Nat D ) g ) , h e. ( u ( Hom ` C ) v ) |-> ( ( a ` v ) ( <. ( ( 1st ` f ) ` u ) , ( ( 1st ` f ) ` v ) >. (comp ` D ) ( ( 1st ` g ) ` v ) ) ( ( u ( 2nd ` f ) v ) ` h ) ) ) : ( ( f ( C Nat D ) g ) X. ( u ( Hom ` C ) v ) ) --> ( ( ( 1st ` f ) ` u ) ( Hom ` D ) ( ( 1st ` g ) ` v ) ) ) )
90	85, 89	mpbird 247	. . . . . . . . . . 11 |- ( ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) /\ ( g e. ( C Func D ) /\ v e. ( Base ` C ) ) ) -> ( <. f , u >. ( 2nd ` E ) <. g , v >. ) : ( ( f ( C Nat D ) g ) X. ( u ( Hom ` C ) v ) ) --> ( ( ( 1st ` f ) ` u ) ( Hom ` D ) ( ( 1st ` g ) ` v ) ) )
91	19, 7	fuchom 16621	. . . . . . . . . . . . 13 |- ( C Nat D ) = ( Hom ` Q )
92	18, 20, 4, 91, 5, 58, 62, 67, 73, 23	xpchom2 16826	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) /\ ( g e. ( C Func D ) /\ v e. ( Base ` C ) ) ) -> ( <. f , u >. ( Hom ` ( Q X.c C ) ) <. g , v >. ) = ( ( f ( C Nat D ) g ) X. ( u ( Hom ` C ) v ) ) )
93	1, 86, 56, 4, 58, 62	evlf1 16860	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) /\ ( g e. ( C Func D ) /\ v e. ( Base ` C ) ) ) -> ( f ( 1st ` E ) u ) = ( ( 1st ` f ) ` u ) )
94	1, 86, 56, 4, 67, 73	evlf1 16860	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) /\ ( g e. ( C Func D ) /\ v e. ( Base ` C ) ) ) -> ( g ( 1st ` E ) v ) = ( ( 1st ` g ) ` v ) )
95	93, 94	oveq12d 6668	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) /\ ( g e. ( C Func D ) /\ v e. ( Base ` C ) ) ) -> ( ( f ( 1st ` E ) u ) ( Hom ` D ) ( g ( 1st ` E ) v ) ) = ( ( ( 1st ` f ) ` u ) ( Hom ` D ) ( ( 1st ` g ) ` v ) ) )
96	92, 95	feq23d 6040	. . . . . . . . . . 11 |- ( ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) /\ ( g e. ( C Func D ) /\ v e. ( Base ` C ) ) ) -> ( ( <. f , u >. ( 2nd ` E ) <. g , v >. ) : ( <. f , u >. ( Hom ` ( Q X.c C ) ) <. g , v >. ) --> ( ( f ( 1st ` E ) u ) ( Hom ` D ) ( g ( 1st ` E ) v ) ) <-> ( <. f , u >. ( 2nd ` E ) <. g , v >. ) : ( ( f ( C Nat D ) g ) X. ( u ( Hom ` C ) v ) ) --> ( ( ( 1st ` f ) ` u ) ( Hom ` D ) ( ( 1st ` g ) ` v ) ) ) )
97	90, 96	mpbird 247	. . . . . . . . . 10 |- ( ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) /\ ( g e. ( C Func D ) /\ v e. ( Base ` C ) ) ) -> ( <. f , u >. ( 2nd ` E ) <. g , v >. ) : ( <. f , u >. ( Hom ` ( Q X.c C ) ) <. g , v >. ) --> ( ( f ( 1st ` E ) u ) ( Hom ` D ) ( g ( 1st ` E ) v ) ) )
98	97	ralrimivva 2971	. . . . . . . . 9 |- ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) -> A. g e. ( C Func D ) A. v e. ( Base ` C ) ( <. f , u >. ( 2nd ` E ) <. g , v >. ) : ( <. f , u >. ( Hom ` ( Q X.c C ) ) <. g , v >. ) --> ( ( f ( 1st ` E ) u ) ( Hom ` D ) ( g ( 1st ` E ) v ) ) )
99	98	ralrimivva 2971	. . . . . . . 8 |- ( ph -> A. f e. ( C Func D ) A. u e. ( Base ` C ) A. g e. ( C Func D ) A. v e. ( Base ` C ) ( <. f , u >. ( 2nd ` E ) <. g , v >. ) : ( <. f , u >. ( Hom ` ( Q X.c C ) ) <. g , v >. ) --> ( ( f ( 1st ` E ) u ) ( Hom ` D ) ( g ( 1st ` E ) v ) ) )
100		oveq2 6658	. . . . . . . . . . . 12 |- ( y = <. g , v >. -> ( x ( 2nd ` E ) y ) = ( x ( 2nd ` E ) <. g , v >. ) )
101		oveq2 6658	. . . . . . . . . . . 12 |- ( y = <. g , v >. -> ( x ( Hom ` ( Q X.c C ) ) y ) = ( x ( Hom ` ( Q X.c C ) ) <. g , v >. ) )
102		fveq2 6191	. . . . . . . . . . . . . 14 |- ( y = <. g , v >. -> ( ( 1st ` E ) ` y ) = ( ( 1st ` E ) ` <. g , v >. ) )
103		df-ov 6653	. . . . . . . . . . . . . 14 |- ( g ( 1st ` E ) v ) = ( ( 1st ` E ) ` <. g , v >. )
104	102, 103	syl6eqr 2674	. . . . . . . . . . . . 13 |- ( y = <. g , v >. -> ( ( 1st ` E ) ` y ) = ( g ( 1st ` E ) v ) )
105	104	oveq2d 6666	. . . . . . . . . . . 12 |- ( y = <. g , v >. -> ( ( ( 1st ` E ) ` x ) ( Hom ` D ) ( ( 1st ` E ) ` y ) ) = ( ( ( 1st ` E ) ` x ) ( Hom ` D ) ( g ( 1st ` E ) v ) ) )
106	100, 101, 105	feq123d 6034	. . . . . . . . . . 11 |- ( y = <. g , v >. -> ( ( x ( 2nd ` E ) y ) : ( x ( Hom ` ( Q X.c C ) ) y ) --> ( ( ( 1st ` E ) ` x ) ( Hom ` D ) ( ( 1st ` E ) ` y ) ) <-> ( x ( 2nd ` E ) <. g , v >. ) : ( x ( Hom ` ( Q X.c C ) ) <. g , v >. ) --> ( ( ( 1st ` E ) ` x ) ( Hom ` D ) ( g ( 1st ` E ) v ) ) ) )
107	106	ralxp 5263	. . . . . . . . . 10 |- ( A. y e. ( ( C Func D ) X. ( Base ` C ) ) ( x ( 2nd ` E ) y ) : ( x ( Hom ` ( Q X.c C ) ) y ) --> ( ( ( 1st ` E ) ` x ) ( Hom ` D ) ( ( 1st ` E ) ` y ) ) <-> A. g e. ( C Func D ) A. v e. ( Base ` C ) ( x ( 2nd ` E ) <. g , v >. ) : ( x ( Hom ` ( Q X.c C ) ) <. g , v >. ) --> ( ( ( 1st ` E ) ` x ) ( Hom ` D ) ( g ( 1st ` E ) v ) ) )
108		oveq1 6657	. . . . . . . . . . . 12 |- ( x = <. f , u >. -> ( x ( 2nd ` E ) <. g , v >. ) = ( <. f , u >. ( 2nd ` E ) <. g , v >. ) )
109		oveq1 6657	. . . . . . . . . . . 12 |- ( x = <. f , u >. -> ( x ( Hom ` ( Q X.c C ) ) <. g , v >. ) = ( <. f , u >. ( Hom ` ( Q X.c C ) ) <. g , v >. ) )
110		fveq2 6191	. . . . . . . . . . . . . 14 |- ( x = <. f , u >. -> ( ( 1st ` E ) ` x ) = ( ( 1st ` E ) ` <. f , u >. ) )
111		df-ov 6653	. . . . . . . . . . . . . 14 |- ( f ( 1st ` E ) u ) = ( ( 1st ` E ) ` <. f , u >. )
112	110, 111	syl6eqr 2674	. . . . . . . . . . . . 13 |- ( x = <. f , u >. -> ( ( 1st ` E ) ` x ) = ( f ( 1st ` E ) u ) )
113	112	oveq1d 6665	. . . . . . . . . . . 12 |- ( x = <. f , u >. -> ( ( ( 1st ` E ) ` x ) ( Hom ` D ) ( g ( 1st ` E ) v ) ) = ( ( f ( 1st ` E ) u ) ( Hom ` D ) ( g ( 1st ` E ) v ) ) )
114	108, 109, 113	feq123d 6034	. . . . . . . . . . 11 |- ( x = <. f , u >. -> ( ( x ( 2nd ` E ) <. g , v >. ) : ( x ( Hom ` ( Q X.c C ) ) <. g , v >. ) --> ( ( ( 1st ` E ) ` x ) ( Hom ` D ) ( g ( 1st ` E ) v ) ) <-> ( <. f , u >. ( 2nd ` E ) <. g , v >. ) : ( <. f , u >. ( Hom ` ( Q X.c C ) ) <. g , v >. ) --> ( ( f ( 1st ` E ) u ) ( Hom ` D ) ( g ( 1st ` E ) v ) ) ) )
115	114	2ralbidv 2989	. . . . . . . . . 10 |- ( x = <. f , u >. -> ( A. g e. ( C Func D ) A. v e. ( Base ` C ) ( x ( 2nd ` E ) <. g , v >. ) : ( x ( Hom ` ( Q X.c C ) ) <. g , v >. ) --> ( ( ( 1st ` E ) ` x ) ( Hom ` D ) ( g ( 1st ` E ) v ) ) <-> A. g e. ( C Func D ) A. v e. ( Base ` C ) ( <. f , u >. ( 2nd ` E ) <. g , v >. ) : ( <. f , u >. ( Hom ` ( Q X.c C ) ) <. g , v >. ) --> ( ( f ( 1st ` E ) u ) ( Hom ` D ) ( g ( 1st ` E ) v ) ) ) )
116	107, 115	syl5bb 272	. . . . . . . . 9 |- ( x = <. f , u >. -> ( A. y e. ( ( C Func D ) X. ( Base ` C ) ) ( x ( 2nd ` E ) y ) : ( x ( Hom ` ( Q X.c C ) ) y ) --> ( ( ( 1st ` E ) ` x ) ( Hom ` D ) ( ( 1st ` E ) ` y ) ) <-> A. g e. ( C Func D ) A. v e. ( Base ` C ) ( <. f , u >. ( 2nd ` E ) <. g , v >. ) : ( <. f , u >. ( Hom ` ( Q X.c C ) ) <. g , v >. ) --> ( ( f ( 1st ` E ) u ) ( Hom ` D ) ( g ( 1st ` E ) v ) ) ) )
117	116	ralxp 5263	. . . . . . . 8 |- ( A. x e. ( ( C Func D ) X. ( Base ` C ) ) A. y e. ( ( C Func D ) X. ( Base ` C ) ) ( x ( 2nd ` E ) y ) : ( x ( Hom ` ( Q X.c C ) ) y ) --> ( ( ( 1st ` E ) ` x ) ( Hom ` D ) ( ( 1st ` E ) ` y ) ) <-> A. f e. ( C Func D ) A. u e. ( Base ` C ) A. g e. ( C Func D ) A. v e. ( Base ` C ) ( <. f , u >. ( 2nd ` E ) <. g , v >. ) : ( <. f , u >. ( Hom ` ( Q X.c C ) ) <. g , v >. ) --> ( ( f ( 1st ` E ) u ) ( Hom ` D ) ( g ( 1st ` E ) v ) ) )
118	99, 117	sylibr 224	. . . . . . 7 |- ( ph -> A. x e. ( ( C Func D ) X. ( Base ` C ) ) A. y e. ( ( C Func D ) X. ( Base ` C ) ) ( x ( 2nd ` E ) y ) : ( x ( Hom ` ( Q X.c C ) ) y ) --> ( ( ( 1st ` E ) ` x ) ( Hom ` D ) ( ( 1st ` E ) ` y ) ) )
119	118	r19.21bi 2932	. . . . . 6 |- ( ( ph /\ x e. ( ( C Func D ) X. ( Base ` C ) ) ) -> A. y e. ( ( C Func D ) X. ( Base ` C ) ) ( x ( 2nd ` E ) y ) : ( x ( Hom ` ( Q X.c C ) ) y ) --> ( ( ( 1st ` E ) ` x ) ( Hom ` D ) ( ( 1st ` E ) ` y ) ) )
120	119	r19.21bi 2932	. . . . 5 |- ( ( ( ph /\ x e. ( ( C Func D ) X. ( Base ` C ) ) ) /\ y e. ( ( C Func D ) X. ( Base ` C ) ) ) -> ( x ( 2nd ` E ) y ) : ( x ( Hom ` ( Q X.c C ) ) y ) --> ( ( ( 1st ` E ) ` x ) ( Hom ` D ) ( ( 1st ` E ) ` y ) ) )
121	120	anasss 679	. . . 4 |- ( ( ph /\ ( x e. ( ( C Func D ) X. ( Base ` C ) ) /\ y e. ( ( C Func D ) X. ( Base ` C ) ) ) ) -> ( x ( 2nd ` E ) y ) : ( x ( Hom ` ( Q X.c C ) ) y ) --> ( ( ( 1st ` E ) ` x ) ( Hom ` D ) ( ( 1st ` E ) ` y ) ) )
122	28	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) -> Q e. Cat )
123	2	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) -> C e. Cat )
124		eqid 2622	. . . . . . . . . . 11 |- ( Id ` Q ) = ( Id ` Q )
125		eqid 2622	. . . . . . . . . . 11 |- ( Id ` C ) = ( Id ` C )
126		simprl 794	. . . . . . . . . . 11 |- ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) -> f e. ( C Func D ) )
127		simprr 796	. . . . . . . . . . 11 |- ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) -> u e. ( Base ` C ) )
128	18, 122, 123, 20, 4, 124, 125, 25, 126, 127	xpcid 16829	. . . . . . . . . 10 |- ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) -> ( ( Id ` ( Q X.c C ) ) ` <. f , u >. ) = <. ( ( Id ` Q ) ` f ) , ( ( Id ` C ) ` u ) >. )
129	128	fveq2d 6195	. . . . . . . . 9 |- ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) -> ( ( <. f , u >. ( 2nd ` E ) <. f , u >. ) ` ( ( Id ` ( Q X.c C ) ) ` <. f , u >. ) ) = ( ( <. f , u >. ( 2nd ` E ) <. f , u >. ) ` <. ( ( Id ` Q ) ` f ) , ( ( Id ` C ) ` u ) >. ) )
130		df-ov 6653	. . . . . . . . 9 |- ( ( ( Id ` Q ) ` f ) ( <. f , u >. ( 2nd ` E ) <. f , u >. ) ( ( Id ` C ) ` u ) ) = ( ( <. f , u >. ( 2nd ` E ) <. f , u >. ) ` <. ( ( Id ` Q ) ` f ) , ( ( Id ` C ) ` u ) >. )
131	129, 130	syl6eqr 2674	. . . . . . . 8 |- ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) -> ( ( <. f , u >. ( 2nd ` E ) <. f , u >. ) ` ( ( Id ` ( Q X.c C ) ) ` <. f , u >. ) ) = ( ( ( Id ` Q ) ` f ) ( <. f , u >. ( 2nd ` E ) <. f , u >. ) ( ( Id ` C ) ` u ) ) )
132	3	adantr 481	. . . . . . . . 9 |- ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) -> D e. Cat )
133		eqid 2622	. . . . . . . . 9 |- ( <. f , u >. ( 2nd ` E ) <. f , u >. ) = ( <. f , u >. ( 2nd ` E ) <. f , u >. )
134	20, 91, 124, 122, 126	catidcl 16343	. . . . . . . . 9 |- ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) -> ( ( Id ` Q ) ` f ) e. ( f ( C Nat D ) f ) )
135	4, 5, 125, 123, 127	catidcl 16343	. . . . . . . . 9 |- ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) -> ( ( Id ` C ) ` u ) e. ( u ( Hom ` C ) u ) )
136	1, 123, 132, 4, 5, 6, 7, 126, 126, 127, 127, 133, 134, 135	evlf2val 16859	. . . . . . . 8 |- ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) -> ( ( ( Id ` Q ) ` f ) ( <. f , u >. ( 2nd ` E ) <. f , u >. ) ( ( Id ` C ) ` u ) ) = ( ( ( ( Id ` Q ) ` f ) ` u ) ( <. ( ( 1st ` f ) ` u ) , ( ( 1st ` f ) ` u ) >. (comp ` D ) ( ( 1st ` f ) ` u ) ) ( ( u ( 2nd ` f ) u ) ` ( ( Id ` C ) ` u ) ) ) )
137	30, 126, 32	sylancr 695	. . . . . . . . . . . 12 |- ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) -> ( 1st ` f ) ( C Func D ) ( 2nd ` f ) )
138	4, 22, 137	funcf1 16526	. . . . . . . . . . 11 |- ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) -> ( 1st ` f ) : ( Base ` C ) --> ( Base ` D ) )
139	138, 127	ffvelrnd 6360	. . . . . . . . . 10 |- ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) -> ( ( 1st ` f ) ` u ) e. ( Base ` D ) )
140	22, 24, 26, 132, 139	catidcl 16343	. . . . . . . . . 10 |- ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) -> ( ( Id ` D ) ` ( ( 1st ` f ) ` u ) ) e. ( ( ( 1st ` f ) ` u ) ( Hom ` D ) ( ( 1st ` f ) ` u ) ) )
141	22, 24, 26, 132, 139, 6, 139, 140	catlid 16344	. . . . . . . . 9 |- ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) -> ( ( ( Id ` D ) ` ( ( 1st ` f ) ` u ) ) ( <. ( ( 1st ` f ) ` u ) , ( ( 1st ` f ) ` u ) >. (comp ` D ) ( ( 1st ` f ) ` u ) ) ( ( Id ` D ) ` ( ( 1st ` f ) ` u ) ) ) = ( ( Id ` D ) ` ( ( 1st ` f ) ` u ) ) )
142	19, 124, 26, 126	fucid 16631	. . . . . . . . . . . 12 |- ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) -> ( ( Id ` Q ) ` f ) = ( ( Id ` D ) o. ( 1st ` f ) ) )
143	142	fveq1d 6193	. . . . . . . . . . 11 |- ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) -> ( ( ( Id ` Q ) ` f ) ` u ) = ( ( ( Id ` D ) o. ( 1st ` f ) ) ` u ) )
144		fvco3 6275	. . . . . . . . . . . 12 |- ( ( ( 1st ` f ) : ( Base ` C ) --> ( Base ` D ) /\ u e. ( Base ` C ) ) -> ( ( ( Id ` D ) o. ( 1st ` f ) ) ` u ) = ( ( Id ` D ) ` ( ( 1st ` f ) ` u ) ) )
145	138, 127, 144	syl2anc 693	. . . . . . . . . . 11 |- ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) -> ( ( ( Id ` D ) o. ( 1st ` f ) ) ` u ) = ( ( Id ` D ) ` ( ( 1st ` f ) ` u ) ) )
146	143, 145	eqtrd 2656	. . . . . . . . . 10 |- ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) -> ( ( ( Id ` Q ) ` f ) ` u ) = ( ( Id ` D ) ` ( ( 1st ` f ) ` u ) ) )
147	4, 125, 26, 137, 127	funcid 16530	. . . . . . . . . 10 |- ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) -> ( ( u ( 2nd ` f ) u ) ` ( ( Id ` C ) ` u ) ) = ( ( Id ` D ) ` ( ( 1st ` f ) ` u ) ) )
148	146, 147	oveq12d 6668	. . . . . . . . 9 |- ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) -> ( ( ( ( Id ` Q ) ` f ) ` u ) ( <. ( ( 1st ` f ) ` u ) , ( ( 1st ` f ) ` u ) >. (comp ` D ) ( ( 1st ` f ) ` u ) ) ( ( u ( 2nd ` f ) u ) ` ( ( Id ` C ) ` u ) ) ) = ( ( ( Id ` D ) ` ( ( 1st ` f ) ` u ) ) ( <. ( ( 1st ` f ) ` u ) , ( ( 1st ` f ) ` u ) >. (comp ` D ) ( ( 1st ` f ) ` u ) ) ( ( Id ` D ) ` ( ( 1st ` f ) ` u ) ) ) )
149	1, 123, 132, 4, 126, 127	evlf1 16860	. . . . . . . . . 10 |- ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) -> ( f ( 1st ` E ) u ) = ( ( 1st ` f ) ` u ) )
150	149	fveq2d 6195	. . . . . . . . 9 |- ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) -> ( ( Id ` D ) ` ( f ( 1st ` E ) u ) ) = ( ( Id ` D ) ` ( ( 1st ` f ) ` u ) ) )
151	141, 148, 150	3eqtr4d 2666	. . . . . . . 8 |- ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) -> ( ( ( ( Id ` Q ) ` f ) ` u ) ( <. ( ( 1st ` f ) ` u ) , ( ( 1st ` f ) ` u ) >. (comp ` D ) ( ( 1st ` f ) ` u ) ) ( ( u ( 2nd ` f ) u ) ` ( ( Id ` C ) ` u ) ) ) = ( ( Id ` D ) ` ( f ( 1st ` E ) u ) ) )
152	131, 136, 151	3eqtrd 2660	. . . . . . 7 |- ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) -> ( ( <. f , u >. ( 2nd ` E ) <. f , u >. ) ` ( ( Id ` ( Q X.c C ) ) ` <. f , u >. ) ) = ( ( Id ` D ) ` ( f ( 1st ` E ) u ) ) )
153	152	ralrimivva 2971	. . . . . 6 |- ( ph -> A. f e. ( C Func D ) A. u e. ( Base ` C ) ( ( <. f , u >. ( 2nd ` E ) <. f , u >. ) ` ( ( Id ` ( Q X.c C ) ) ` <. f , u >. ) ) = ( ( Id ` D ) ` ( f ( 1st ` E ) u ) ) )
154		id 22	. . . . . . . . . 10 |- ( x = <. f , u >. -> x = <. f , u >. )
155	154, 154	oveq12d 6668	. . . . . . . . 9 |- ( x = <. f , u >. -> ( x ( 2nd ` E ) x ) = ( <. f , u >. ( 2nd ` E ) <. f , u >. ) )
156		fveq2 6191	. . . . . . . . 9 |- ( x = <. f , u >. -> ( ( Id ` ( Q X.c C ) ) ` x ) = ( ( Id ` ( Q X.c C ) ) ` <. f , u >. ) )
157	155, 156	fveq12d 6197	. . . . . . . 8 |- ( x = <. f , u >. -> ( ( x ( 2nd ` E ) x ) ` ( ( Id ` ( Q X.c C ) ) ` x ) ) = ( ( <. f , u >. ( 2nd ` E ) <. f , u >. ) ` ( ( Id ` ( Q X.c C ) ) ` <. f , u >. ) ) )
158	112	fveq2d 6195	. . . . . . . 8 |- ( x = <. f , u >. -> ( ( Id ` D ) ` ( ( 1st ` E ) ` x ) ) = ( ( Id ` D ) ` ( f ( 1st ` E ) u ) ) )
159	157, 158	eqeq12d 2637	. . . . . . 7 |- ( x = <. f , u >. -> ( ( ( x ( 2nd ` E ) x ) ` ( ( Id ` ( Q X.c C ) ) ` x ) ) = ( ( Id ` D ) ` ( ( 1st ` E ) ` x ) ) <-> ( ( <. f , u >. ( 2nd ` E ) <. f , u >. ) ` ( ( Id ` ( Q X.c C ) ) ` <. f , u >. ) ) = ( ( Id ` D ) ` ( f ( 1st ` E ) u ) ) ) )
160	159	ralxp 5263	. . . . . 6 |- ( A. x e. ( ( C Func D ) X. ( Base ` C ) ) ( ( x ( 2nd ` E ) x ) ` ( ( Id ` ( Q X.c C ) ) ` x ) ) = ( ( Id ` D ) ` ( ( 1st ` E ) ` x ) ) <-> A. f e. ( C Func D ) A. u e. ( Base ` C ) ( ( <. f , u >. ( 2nd ` E ) <. f , u >. ) ` ( ( Id ` ( Q X.c C ) ) ` <. f , u >. ) ) = ( ( Id ` D ) ` ( f ( 1st ` E ) u ) ) )
161	153, 160	sylibr 224	. . . . 5 |- ( ph -> A. x e. ( ( C Func D ) X. ( Base ` C ) ) ( ( x ( 2nd ` E ) x ) ` ( ( Id ` ( Q X.c C ) ) ` x ) ) = ( ( Id ` D ) ` ( ( 1st ` E ) ` x ) ) )
162	161	r19.21bi 2932	. . . 4 |- ( ( ph /\ x e. ( ( C Func D ) X. ( Base ` C ) ) ) -> ( ( x ( 2nd ` E ) x ) ` ( ( Id ` ( Q X.c C ) ) ` x ) ) = ( ( Id ` D ) ` ( ( 1st ` E ) ` x ) ) )
163	2	3ad2ant1 1082	. . . . . 6 |- ( ( ph /\ ( x e. ( ( C Func D ) X. ( Base ` C ) ) /\ y e. ( ( C Func D ) X. ( Base ` C ) ) /\ z e. ( ( C Func D ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( Q X.c C ) ) y ) /\ g e. ( y ( Hom ` ( Q X.c C ) ) z ) ) ) -> C e. Cat )
164	3	3ad2ant1 1082	. . . . . 6 |- ( ( ph /\ ( x e. ( ( C Func D ) X. ( Base ` C ) ) /\ y e. ( ( C Func D ) X. ( Base ` C ) ) /\ z e. ( ( C Func D ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( Q X.c C ) ) y ) /\ g e. ( y ( Hom ` ( Q X.c C ) ) z ) ) ) -> D e. Cat )
165		simp21 1094	. . . . . . . . 9 |- ( ( ph /\ ( x e. ( ( C Func D ) X. ( Base ` C ) ) /\ y e. ( ( C Func D ) X. ( Base ` C ) ) /\ z e. ( ( C Func D ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( Q X.c C ) ) y ) /\ g e. ( y ( Hom ` ( Q X.c C ) ) z ) ) ) -> x e. ( ( C Func D ) X. ( Base ` C ) ) )
166		1st2nd2 7205	. . . . . . . . 9 |- ( x e. ( ( C Func D ) X. ( Base ` C ) ) -> x = <. ( 1st ` x ) , ( 2nd ` x ) >. )
167	165, 166	syl 17	. . . . . . . 8 |- ( ( ph /\ ( x e. ( ( C Func D ) X. ( Base ` C ) ) /\ y e. ( ( C Func D ) X. ( Base ` C ) ) /\ z e. ( ( C Func D ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( Q X.c C ) ) y ) /\ g e. ( y ( Hom ` ( Q X.c C ) ) z ) ) ) -> x = <. ( 1st ` x ) , ( 2nd ` x ) >. )
168	167, 165	eqeltrrd 2702	. . . . . . 7 |- ( ( ph /\ ( x e. ( ( C Func D ) X. ( Base ` C ) ) /\ y e. ( ( C Func D ) X. ( Base ` C ) ) /\ z e. ( ( C Func D ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( Q X.c C ) ) y ) /\ g e. ( y ( Hom ` ( Q X.c C ) ) z ) ) ) -> <. ( 1st ` x ) , ( 2nd ` x ) >. e. ( ( C Func D ) X. ( Base ` C ) ) )
169		opelxp 5146	. . . . . . 7 |- ( <. ( 1st ` x ) , ( 2nd ` x ) >. e. ( ( C Func D ) X. ( Base ` C ) ) <-> ( ( 1st ` x ) e. ( C Func D ) /\ ( 2nd ` x ) e. ( Base ` C ) ) )
170	168, 169	sylib 208	. . . . . 6 |- ( ( ph /\ ( x e. ( ( C Func D ) X. ( Base ` C ) ) /\ y e. ( ( C Func D ) X. ( Base ` C ) ) /\ z e. ( ( C Func D ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( Q X.c C ) ) y ) /\ g e. ( y ( Hom ` ( Q X.c C ) ) z ) ) ) -> ( ( 1st ` x ) e. ( C Func D ) /\ ( 2nd ` x ) e. ( Base ` C ) ) )
171		simp22 1095	. . . . . . . . 9 |- ( ( ph /\ ( x e. ( ( C Func D ) X. ( Base ` C ) ) /\ y e. ( ( C Func D ) X. ( Base ` C ) ) /\ z e. ( ( C Func D ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( Q X.c C ) ) y ) /\ g e. ( y ( Hom ` ( Q X.c C ) ) z ) ) ) -> y e. ( ( C Func D ) X. ( Base ` C ) ) )
172		1st2nd2 7205	. . . . . . . . 9 |- ( y e. ( ( C Func D ) X. ( Base ` C ) ) -> y = <. ( 1st ` y ) , ( 2nd ` y ) >. )
173	171, 172	syl 17	. . . . . . . 8 |- ( ( ph /\ ( x e. ( ( C Func D ) X. ( Base ` C ) ) /\ y e. ( ( C Func D ) X. ( Base ` C ) ) /\ z e. ( ( C Func D ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( Q X.c C ) ) y ) /\ g e. ( y ( Hom ` ( Q X.c C ) ) z ) ) ) -> y = <. ( 1st ` y ) , ( 2nd ` y ) >. )
174	173, 171	eqeltrrd 2702	. . . . . . 7 |- ( ( ph /\ ( x e. ( ( C Func D ) X. ( Base ` C ) ) /\ y e. ( ( C Func D ) X. ( Base ` C ) ) /\ z e. ( ( C Func D ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( Q X.c C ) ) y ) /\ g e. ( y ( Hom ` ( Q X.c C ) ) z ) ) ) -> <. ( 1st ` y ) , ( 2nd ` y ) >. e. ( ( C Func D ) X. ( Base ` C ) ) )
175		opelxp 5146	. . . . . . 7 |- ( <. ( 1st ` y ) , ( 2nd ` y ) >. e. ( ( C Func D ) X. ( Base ` C ) ) <-> ( ( 1st ` y ) e. ( C Func D ) /\ ( 2nd ` y ) e. ( Base ` C ) ) )
176	174, 175	sylib 208	. . . . . 6 |- ( ( ph /\ ( x e. ( ( C Func D ) X. ( Base ` C ) ) /\ y e. ( ( C Func D ) X. ( Base ` C ) ) /\ z e. ( ( C Func D ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( Q X.c C ) ) y ) /\ g e. ( y ( Hom ` ( Q X.c C ) ) z ) ) ) -> ( ( 1st ` y ) e. ( C Func D ) /\ ( 2nd ` y ) e. ( Base ` C ) ) )
177		simp23 1096	. . . . . . . . 9 |- ( ( ph /\ ( x e. ( ( C Func D ) X. ( Base ` C ) ) /\ y e. ( ( C Func D ) X. ( Base ` C ) ) /\ z e. ( ( C Func D ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( Q X.c C ) ) y ) /\ g e. ( y ( Hom ` ( Q X.c C ) ) z ) ) ) -> z e. ( ( C Func D ) X. ( Base ` C ) ) )
178		1st2nd2 7205	. . . . . . . . 9 |- ( z e. ( ( C Func D ) X. ( Base ` C ) ) -> z = <. ( 1st ` z ) , ( 2nd ` z ) >. )
179	177, 178	syl 17	. . . . . . . 8 |- ( ( ph /\ ( x e. ( ( C Func D ) X. ( Base ` C ) ) /\ y e. ( ( C Func D ) X. ( Base ` C ) ) /\ z e. ( ( C Func D ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( Q X.c C ) ) y ) /\ g e. ( y ( Hom ` ( Q X.c C ) ) z ) ) ) -> z = <. ( 1st ` z ) , ( 2nd ` z ) >. )
180	179, 177	eqeltrrd 2702	. . . . . . 7 |- ( ( ph /\ ( x e. ( ( C Func D ) X. ( Base ` C ) ) /\ y e. ( ( C Func D ) X. ( Base ` C ) ) /\ z e. ( ( C Func D ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( Q X.c C ) ) y ) /\ g e. ( y ( Hom ` ( Q X.c C ) ) z ) ) ) -> <. ( 1st ` z ) , ( 2nd ` z ) >. e. ( ( C Func D ) X. ( Base ` C ) ) )
181		opelxp 5146	. . . . . . 7 |- ( <. ( 1st ` z ) , ( 2nd ` z ) >. e. ( ( C Func D ) X. ( Base ` C ) ) <-> ( ( 1st ` z ) e. ( C Func D ) /\ ( 2nd ` z ) e. ( Base ` C ) ) )
182	180, 181	sylib 208	. . . . . 6 |- ( ( ph /\ ( x e. ( ( C Func D ) X. ( Base ` C ) ) /\ y e. ( ( C Func D ) X. ( Base ` C ) ) /\ z e. ( ( C Func D ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( Q X.c C ) ) y ) /\ g e. ( y ( Hom ` ( Q X.c C ) ) z ) ) ) -> ( ( 1st ` z ) e. ( C Func D ) /\ ( 2nd ` z ) e. ( Base ` C ) ) )
183		simp3l 1089	. . . . . . . . . 10 |- ( ( ph /\ ( x e. ( ( C Func D ) X. ( Base ` C ) ) /\ y e. ( ( C Func D ) X. ( Base ` C ) ) /\ z e. ( ( C Func D ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( Q X.c C ) ) y ) /\ g e. ( y ( Hom ` ( Q X.c C ) ) z ) ) ) -> f e. ( x ( Hom ` ( Q X.c C ) ) y ) )
184	18, 21, 91, 5, 23, 165, 171	xpchom 16820	. . . . . . . . . 10 |- ( ( ph /\ ( x e. ( ( C Func D ) X. ( Base ` C ) ) /\ y e. ( ( C Func D ) X. ( Base ` C ) ) /\ z e. ( ( C Func D ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( Q X.c C ) ) y ) /\ g e. ( y ( Hom ` ( Q X.c C ) ) z ) ) ) -> ( x ( Hom ` ( Q X.c C ) ) y ) = ( ( ( 1st ` x ) ( C Nat D ) ( 1st ` y ) ) X. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) ) )
185	183, 184	eleqtrd 2703	. . . . . . . . 9 |- ( ( ph /\ ( x e. ( ( C Func D ) X. ( Base ` C ) ) /\ y e. ( ( C Func D ) X. ( Base ` C ) ) /\ z e. ( ( C Func D ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( Q X.c C ) ) y ) /\ g e. ( y ( Hom ` ( Q X.c C ) ) z ) ) ) -> f e. ( ( ( 1st ` x ) ( C Nat D ) ( 1st ` y ) ) X. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) ) )
186		1st2nd2 7205	. . . . . . . . 9 |- ( f e. ( ( ( 1st ` x ) ( C Nat D ) ( 1st ` y ) ) X. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) ) -> f = <. ( 1st ` f ) , ( 2nd ` f ) >. )
187	185, 186	syl 17	. . . . . . . 8 |- ( ( ph /\ ( x e. ( ( C Func D ) X. ( Base ` C ) ) /\ y e. ( ( C Func D ) X. ( Base ` C ) ) /\ z e. ( ( C Func D ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( Q X.c C ) ) y ) /\ g e. ( y ( Hom ` ( Q X.c C ) ) z ) ) ) -> f = <. ( 1st ` f ) , ( 2nd ` f ) >. )
188	187, 185	eqeltrrd 2702	. . . . . . 7 |- ( ( ph /\ ( x e. ( ( C Func D ) X. ( Base ` C ) ) /\ y e. ( ( C Func D ) X. ( Base ` C ) ) /\ z e. ( ( C Func D ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( Q X.c C ) ) y ) /\ g e. ( y ( Hom ` ( Q X.c C ) ) z ) ) ) -> <. ( 1st ` f ) , ( 2nd ` f ) >. e. ( ( ( 1st ` x ) ( C Nat D ) ( 1st ` y ) ) X. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) ) )
189		opelxp 5146	. . . . . . 7 |- ( <. ( 1st ` f ) , ( 2nd ` f ) >. e. ( ( ( 1st ` x ) ( C Nat D ) ( 1st ` y ) ) X. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) ) <-> ( ( 1st ` f ) e. ( ( 1st ` x ) ( C Nat D ) ( 1st ` y ) ) /\ ( 2nd ` f ) e. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) ) )
190	188, 189	sylib 208	. . . . . 6 |- ( ( ph /\ ( x e. ( ( C Func D ) X. ( Base ` C ) ) /\ y e. ( ( C Func D ) X. ( Base ` C ) ) /\ z e. ( ( C Func D ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( Q X.c C ) ) y ) /\ g e. ( y ( Hom ` ( Q X.c C ) ) z ) ) ) -> ( ( 1st ` f ) e. ( ( 1st ` x ) ( C Nat D ) ( 1st ` y ) ) /\ ( 2nd ` f ) e. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) ) )
191		simp3r 1090	. . . . . . . . . 10 |- ( ( ph /\ ( x e. ( ( C Func D ) X. ( Base ` C ) ) /\ y e. ( ( C Func D ) X. ( Base ` C ) ) /\ z e. ( ( C Func D ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( Q X.c C ) ) y ) /\ g e. ( y ( Hom ` ( Q X.c C ) ) z ) ) ) -> g e. ( y ( Hom ` ( Q X.c C ) ) z ) )
192	18, 21, 91, 5, 23, 171, 177	xpchom 16820	. . . . . . . . . 10 |- ( ( ph /\ ( x e. ( ( C Func D ) X. ( Base ` C ) ) /\ y e. ( ( C Func D ) X. ( Base ` C ) ) /\ z e. ( ( C Func D ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( Q X.c C ) ) y ) /\ g e. ( y ( Hom ` ( Q X.c C ) ) z ) ) ) -> ( y ( Hom ` ( Q X.c C ) ) z ) = ( ( ( 1st ` y ) ( C Nat D ) ( 1st ` z ) ) X. ( ( 2nd ` y ) ( Hom ` C ) ( 2nd ` z ) ) ) )
193	191, 192	eleqtrd 2703	. . . . . . . . 9 |- ( ( ph /\ ( x e. ( ( C Func D ) X. ( Base ` C ) ) /\ y e. ( ( C Func D ) X. ( Base ` C ) ) /\ z e. ( ( C Func D ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( Q X.c C ) ) y ) /\ g e. ( y ( Hom ` ( Q X.c C ) ) z ) ) ) -> g e. ( ( ( 1st ` y ) ( C Nat D ) ( 1st ` z ) ) X. ( ( 2nd ` y ) ( Hom ` C ) ( 2nd ` z ) ) ) )
194		1st2nd2 7205	. . . . . . . . 9 |- ( g e. ( ( ( 1st ` y ) ( C Nat D ) ( 1st ` z ) ) X. ( ( 2nd ` y ) ( Hom ` C ) ( 2nd ` z ) ) ) -> g = <. ( 1st ` g ) , ( 2nd ` g ) >. )
195	193, 194	syl 17	. . . . . . . 8 |- ( ( ph /\ ( x e. ( ( C Func D ) X. ( Base ` C ) ) /\ y e. ( ( C Func D ) X. ( Base ` C ) ) /\ z e. ( ( C Func D ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( Q X.c C ) ) y ) /\ g e. ( y ( Hom ` ( Q X.c C ) ) z ) ) ) -> g = <. ( 1st ` g ) , ( 2nd ` g ) >. )
196	195, 193	eqeltrrd 2702	. . . . . . 7 |- ( ( ph /\ ( x e. ( ( C Func D ) X. ( Base ` C ) ) /\ y e. ( ( C Func D ) X. ( Base ` C ) ) /\ z e. ( ( C Func D ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( Q X.c C ) ) y ) /\ g e. ( y ( Hom ` ( Q X.c C ) ) z ) ) ) -> <. ( 1st ` g ) , ( 2nd ` g ) >. e. ( ( ( 1st ` y ) ( C Nat D ) ( 1st ` z ) ) X. ( ( 2nd ` y ) ( Hom ` C ) ( 2nd ` z ) ) ) )
197		opelxp 5146	. . . . . . 7 |- ( <. ( 1st ` g ) , ( 2nd ` g ) >. e. ( ( ( 1st ` y ) ( C Nat D ) ( 1st ` z ) ) X. ( ( 2nd ` y ) ( Hom ` C ) ( 2nd ` z ) ) ) <-> ( ( 1st ` g ) e. ( ( 1st ` y ) ( C Nat D ) ( 1st ` z ) ) /\ ( 2nd ` g ) e. ( ( 2nd ` y ) ( Hom ` C ) ( 2nd ` z ) ) ) )
198	196, 197	sylib 208	. . . . . 6 |- ( ( ph /\ ( x e. ( ( C Func D ) X. ( Base ` C ) ) /\ y e. ( ( C Func D ) X. ( Base ` C ) ) /\ z e. ( ( C Func D ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( Q X.c C ) ) y ) /\ g e. ( y ( Hom ` ( Q X.c C ) ) z ) ) ) -> ( ( 1st ` g ) e. ( ( 1st ` y ) ( C Nat D ) ( 1st ` z ) ) /\ ( 2nd ` g ) e. ( ( 2nd ` y ) ( Hom ` C ) ( 2nd ` z ) ) ) )
199	1, 19, 163, 164, 7, 170, 176, 182, 190, 198	evlfcllem 16861	. . . . 5 |- ( ( ph /\ ( x e. ( ( C Func D ) X. ( Base ` C ) ) /\ y e. ( ( C Func D ) X. ( Base ` C ) ) /\ z e. ( ( C Func D ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( Q X.c C ) ) y ) /\ g e. ( y ( Hom ` ( Q X.c C ) ) z ) ) ) -> ( ( <. ( 1st ` x ) , ( 2nd ` x ) >. ( 2nd ` E ) <. ( 1st ` z ) , ( 2nd ` z ) >. ) ` ( <. ( 1st ` g ) , ( 2nd ` g ) >. ( <. <. ( 1st ` x ) , ( 2nd ` x ) >. , <. ( 1st ` y ) , ( 2nd ` y ) >. >. (comp ` ( Q X.c C ) ) <. ( 1st ` z ) , ( 2nd ` z ) >. ) <. ( 1st ` f ) , ( 2nd ` f ) >. ) ) = ( ( ( <. ( 1st ` y ) , ( 2nd ` y ) >. ( 2nd ` E ) <. ( 1st ` z ) , ( 2nd ` z ) >. ) ` <. ( 1st ` g ) , ( 2nd ` g ) >. ) ( <. ( ( 1st ` E ) ` <. ( 1st ` x ) , ( 2nd ` x ) >. ) , ( ( 1st ` E ) ` <. ( 1st ` y ) , ( 2nd ` y ) >. ) >. (comp ` D ) ( ( 1st ` E ) ` <. ( 1st ` z ) , ( 2nd ` z ) >. ) ) ( ( <. ( 1st ` x ) , ( 2nd ` x ) >. ( 2nd ` E ) <. ( 1st ` y ) , ( 2nd ` y ) >. ) ` <. ( 1st ` f ) , ( 2nd ` f ) >. ) ) )
200	167, 179	oveq12d 6668	. . . . . 6 |- ( ( ph /\ ( x e. ( ( C Func D ) X. ( Base ` C ) ) /\ y e. ( ( C Func D ) X. ( Base ` C ) ) /\ z e. ( ( C Func D ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( Q X.c C ) ) y ) /\ g e. ( y ( Hom ` ( Q X.c C ) ) z ) ) ) -> ( x ( 2nd ` E ) z ) = ( <. ( 1st ` x ) , ( 2nd ` x ) >. ( 2nd ` E ) <. ( 1st ` z ) , ( 2nd ` z ) >. ) )
201	167, 173	opeq12d 4410	. . . . . . . 8 |- ( ( ph /\ ( x e. ( ( C Func D ) X. ( Base ` C ) ) /\ y e. ( ( C Func D ) X. ( Base ` C ) ) /\ z e. ( ( C Func D ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( Q X.c C ) ) y ) /\ g e. ( y ( Hom ` ( Q X.c C ) ) z ) ) ) -> <. x , y >. = <. <. ( 1st ` x ) , ( 2nd ` x ) >. , <. ( 1st ` y ) , ( 2nd ` y ) >. >. )
202	201, 179	oveq12d 6668	. . . . . . 7 |- ( ( ph /\ ( x e. ( ( C Func D ) X. ( Base ` C ) ) /\ y e. ( ( C Func D ) X. ( Base ` C ) ) /\ z e. ( ( C Func D ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( Q X.c C ) ) y ) /\ g e. ( y ( Hom ` ( Q X.c C ) ) z ) ) ) -> ( <. x , y >. (comp ` ( Q X.c C ) ) z ) = ( <. <. ( 1st ` x ) , ( 2nd ` x ) >. , <. ( 1st ` y ) , ( 2nd ` y ) >. >. (comp ` ( Q X.c C ) ) <. ( 1st ` z ) , ( 2nd ` z ) >. ) )
203	202, 195, 187	oveq123d 6671	. . . . . 6 |- ( ( ph /\ ( x e. ( ( C Func D ) X. ( Base ` C ) ) /\ y e. ( ( C Func D ) X. ( Base ` C ) ) /\ z e. ( ( C Func D ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( Q X.c C ) ) y ) /\ g e. ( y ( Hom ` ( Q X.c C ) ) z ) ) ) -> ( g ( <. x , y >. (comp ` ( Q X.c C ) ) z ) f ) = ( <. ( 1st ` g ) , ( 2nd ` g ) >. ( <. <. ( 1st ` x ) , ( 2nd ` x ) >. , <. ( 1st ` y ) , ( 2nd ` y ) >. >. (comp ` ( Q X.c C ) ) <. ( 1st ` z ) , ( 2nd ` z ) >. ) <. ( 1st ` f ) , ( 2nd ` f ) >. ) )
204	200, 203	fveq12d 6197	. . . . 5 |- ( ( ph /\ ( x e. ( ( C Func D ) X. ( Base ` C ) ) /\ y e. ( ( C Func D ) X. ( Base ` C ) ) /\ z e. ( ( C Func D ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( Q X.c C ) ) y ) /\ g e. ( y ( Hom ` ( Q X.c C ) ) z ) ) ) -> ( ( x ( 2nd ` E ) z ) ` ( g ( <. x , y >. (comp ` ( Q X.c C ) ) z ) f ) ) = ( ( <. ( 1st ` x ) , ( 2nd ` x ) >. ( 2nd ` E ) <. ( 1st ` z ) , ( 2nd ` z ) >. ) ` ( <. ( 1st ` g ) , ( 2nd ` g ) >. ( <. <. ( 1st ` x ) , ( 2nd ` x ) >. , <. ( 1st ` y ) , ( 2nd ` y ) >. >. (comp ` ( Q X.c C ) ) <. ( 1st ` z ) , ( 2nd ` z ) >. ) <. ( 1st ` f ) , ( 2nd ` f ) >. ) ) )
205	167	fveq2d 6195	. . . . . . . 8 |- ( ( ph /\ ( x e. ( ( C Func D ) X. ( Base ` C ) ) /\ y e. ( ( C Func D ) X. ( Base ` C ) ) /\ z e. ( ( C Func D ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( Q X.c C ) ) y ) /\ g e. ( y ( Hom ` ( Q X.c C ) ) z ) ) ) -> ( ( 1st ` E ) ` x ) = ( ( 1st ` E ) ` <. ( 1st ` x ) , ( 2nd ` x ) >. ) )
206	173	fveq2d 6195	. . . . . . . 8 |- ( ( ph /\ ( x e. ( ( C Func D ) X. ( Base ` C ) ) /\ y e. ( ( C Func D ) X. ( Base ` C ) ) /\ z e. ( ( C Func D ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( Q X.c C ) ) y ) /\ g e. ( y ( Hom ` ( Q X.c C ) ) z ) ) ) -> ( ( 1st ` E ) ` y ) = ( ( 1st ` E ) ` <. ( 1st ` y ) , ( 2nd ` y ) >. ) )
207	205, 206	opeq12d 4410	. . . . . . 7 |- ( ( ph /\ ( x e. ( ( C Func D ) X. ( Base ` C ) ) /\ y e. ( ( C Func D ) X. ( Base ` C ) ) /\ z e. ( ( C Func D ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( Q X.c C ) ) y ) /\ g e. ( y ( Hom ` ( Q X.c C ) ) z ) ) ) -> <. ( ( 1st ` E ) ` x ) , ( ( 1st ` E ) ` y ) >. = <. ( ( 1st ` E ) ` <. ( 1st ` x ) , ( 2nd ` x ) >. ) , ( ( 1st ` E ) ` <. ( 1st ` y ) , ( 2nd ` y ) >. ) >. )
208	179	fveq2d 6195	. . . . . . 7 |- ( ( ph /\ ( x e. ( ( C Func D ) X. ( Base ` C ) ) /\ y e. ( ( C Func D ) X. ( Base ` C ) ) /\ z e. ( ( C Func D ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( Q X.c C ) ) y ) /\ g e. ( y ( Hom ` ( Q X.c C ) ) z ) ) ) -> ( ( 1st ` E ) ` z ) = ( ( 1st ` E ) ` <. ( 1st ` z ) , ( 2nd ` z ) >. ) )
209	207, 208	oveq12d 6668	. . . . . 6 |- ( ( ph /\ ( x e. ( ( C Func D ) X. ( Base ` C ) ) /\ y e. ( ( C Func D ) X. ( Base ` C ) ) /\ z e. ( ( C Func D ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( Q X.c C ) ) y ) /\ g e. ( y ( Hom ` ( Q X.c C ) ) z ) ) ) -> ( <. ( ( 1st ` E ) ` x ) , ( ( 1st ` E ) ` y ) >. (comp ` D ) ( ( 1st ` E ) ` z ) ) = ( <. ( ( 1st ` E ) ` <. ( 1st ` x ) , ( 2nd ` x ) >. ) , ( ( 1st ` E ) ` <. ( 1st ` y ) , ( 2nd ` y ) >. ) >. (comp ` D ) ( ( 1st ` E ) ` <. ( 1st ` z ) , ( 2nd ` z ) >. ) ) )
210	173, 179	oveq12d 6668	. . . . . . 7 |- ( ( ph /\ ( x e. ( ( C Func D ) X. ( Base ` C ) ) /\ y e. ( ( C Func D ) X. ( Base ` C ) ) /\ z e. ( ( C Func D ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( Q X.c C ) ) y ) /\ g e. ( y ( Hom ` ( Q X.c C ) ) z ) ) ) -> ( y ( 2nd ` E ) z ) = ( <. ( 1st ` y ) , ( 2nd ` y ) >. ( 2nd ` E ) <. ( 1st ` z ) , ( 2nd ` z ) >. ) )
211	210, 195	fveq12d 6197	. . . . . 6 |- ( ( ph /\ ( x e. ( ( C Func D ) X. ( Base ` C ) ) /\ y e. ( ( C Func D ) X. ( Base ` C ) ) /\ z e. ( ( C Func D ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( Q X.c C ) ) y ) /\ g e. ( y ( Hom ` ( Q X.c C ) ) z ) ) ) -> ( ( y ( 2nd ` E ) z ) ` g ) = ( ( <. ( 1st ` y ) , ( 2nd ` y ) >. ( 2nd ` E ) <. ( 1st ` z ) , ( 2nd ` z ) >. ) ` <. ( 1st ` g ) , ( 2nd ` g ) >. ) )
212	167, 173	oveq12d 6668	. . . . . . 7 |- ( ( ph /\ ( x e. ( ( C Func D ) X. ( Base ` C ) ) /\ y e. ( ( C Func D ) X. ( Base ` C ) ) /\ z e. ( ( C Func D ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( Q X.c C ) ) y ) /\ g e. ( y ( Hom ` ( Q X.c C ) ) z ) ) ) -> ( x ( 2nd ` E ) y ) = ( <. ( 1st ` x ) , ( 2nd ` x ) >. ( 2nd ` E ) <. ( 1st ` y ) , ( 2nd ` y ) >. ) )
213	212, 187	fveq12d 6197	. . . . . 6 |- ( ( ph /\ ( x e. ( ( C Func D ) X. ( Base ` C ) ) /\ y e. ( ( C Func D ) X. ( Base ` C ) ) /\ z e. ( ( C Func D ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( Q X.c C ) ) y ) /\ g e. ( y ( Hom ` ( Q X.c C ) ) z ) ) ) -> ( ( x ( 2nd ` E ) y ) ` f ) = ( ( <. ( 1st ` x ) , ( 2nd ` x ) >. ( 2nd ` E ) <. ( 1st ` y ) , ( 2nd ` y ) >. ) ` <. ( 1st ` f ) , ( 2nd ` f ) >. ) )
214	209, 211, 213	oveq123d 6671	. . . . 5 |- ( ( ph /\ ( x e. ( ( C Func D ) X. ( Base ` C ) ) /\ y e. ( ( C Func D ) X. ( Base ` C ) ) /\ z e. ( ( C Func D ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( Q X.c C ) ) y ) /\ g e. ( y ( Hom ` ( Q X.c C ) ) z ) ) ) -> ( ( ( y ( 2nd ` E ) z ) ` g ) ( <. ( ( 1st ` E ) ` x ) , ( ( 1st ` E ) ` y ) >. (comp ` D ) ( ( 1st ` E ) ` z ) ) ( ( x ( 2nd ` E ) y ) ` f ) ) = ( ( ( <. ( 1st ` y ) , ( 2nd ` y ) >. ( 2nd ` E ) <. ( 1st ` z ) , ( 2nd ` z ) >. ) ` <. ( 1st ` g ) , ( 2nd ` g ) >. ) ( <. ( ( 1st ` E ) ` <. ( 1st ` x ) , ( 2nd ` x ) >. ) , ( ( 1st ` E ) ` <. ( 1st ` y ) , ( 2nd ` y ) >. ) >. (comp ` D ) ( ( 1st ` E ) ` <. ( 1st ` z ) , ( 2nd ` z ) >. ) ) ( ( <. ( 1st ` x ) , ( 2nd ` x ) >. ( 2nd ` E ) <. ( 1st ` y ) , ( 2nd ` y ) >. ) ` <. ( 1st ` f ) , ( 2nd ` f ) >. ) ) )
215	199, 204, 214	3eqtr4d 2666	. . . 4 |- ( ( ph /\ ( x e. ( ( C Func D ) X. ( Base ` C ) ) /\ y e. ( ( C Func D ) X. ( Base ` C ) ) /\ z e. ( ( C Func D ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( Q X.c C ) ) y ) /\ g e. ( y ( Hom ` ( Q X.c C ) ) z ) ) ) -> ( ( x ( 2nd ` E ) z ) ` ( g ( <. x , y >. (comp ` ( Q X.c C ) ) z ) f ) ) = ( ( ( y ( 2nd ` E ) z ) ` g ) ( <. ( ( 1st ` E ) ` x ) , ( ( 1st ` E ) ` y ) >. (comp ` D ) ( ( 1st ` E ) ` z ) ) ( ( x ( 2nd ` E ) y ) ` f ) ) )
216	21, 22, 23, 24, 25, 26, 27, 6, 29, 3, 44, 55, 121, 162, 215	isfuncd 16525	. . 3 |- ( ph -> ( 1st ` E ) ( ( Q X.c C ) Func D ) ( 2nd ` E ) )
217		df-br 4654	. . 3 |- ( ( 1st ` E ) ( ( Q X.c C ) Func D ) ( 2nd ` E ) <-> <. ( 1st ` E ) , ( 2nd ` E ) >. e. ( ( Q X.c C ) Func D ) )
218	216, 217	sylib 208	. 2 |- ( ph -> <. ( 1st ` E ) , ( 2nd ` E ) >. e. ( ( Q X.c C ) Func D ) )
219	17, 218	eqeltrd 2701	1 |- ( ph -> E e. ( ( Q X.c C ) Func D ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   [_csb 3533   <.cop 4183   class class class wbr 4653   X. cxp 5112   o. ccom 5118   Rel wrel 5119   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   1stc1st 7166   2ndc2nd 7167   Basecbs 15857   Hom chom 15952  compcco 15953   Catccat 16325   Idccid 16326   Func cfunc 16514   Nat cnat 16601   FuncCat cfuc 16602   X._c cxpc 16808   eval_F cevlf 16849

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-fz 12327  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-hom 15966  df-cco 15967  df-cat 16329  df-cid 16330  df-func 16518  df-nat 16603  df-fuc 16604  df-xpc 16812  df-evlf 16853

This theorem is referenced by:  uncfcl  16875  uncf1  16876  uncf2  16877  yonedalem1  16912