Theorem curf1cl 16868

Description: The partially evaluated curry functor is a functor. (Contributed by Mario Carneiro, 13-Jan-2017.)

Hypotheses

Ref	Expression
curfval.g	|- G = ( <. C , D >. curryF F )
curfval.a	|- A = ( Base ` C )
curfval.c	|- ( ph -> C e. Cat )
curfval.d	|- ( ph -> D e. Cat )
curfval.f	|- ( ph -> F e. ( ( C X.c D ) Func E ) )
curfval.b	|- B = ( Base ` D )
curf1.x	|- ( ph -> X e. A )
curf1.k	|- K = ( ( 1st ` G ) ` X )

Assertion

Ref	Expression
curf1cl	|- ( ph -> K e. ( D Func E ) )

Proof of Theorem curf1cl

Dummy variables g y z h w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		curfval.g	. . . 4 |- G = ( <. C , D >. curryF F )
2		curfval.a	. . . 4 |- A = ( Base ` C )
3		curfval.c	. . . 4 |- ( ph -> C e. Cat )
4		curfval.d	. . . 4 |- ( ph -> D e. Cat )
5		curfval.f	. . . 4 |- ( ph -> F e. ( ( C X.c D ) Func E ) )
6		curfval.b	. . . 4 |- B = ( Base ` D )
7		curf1.x	. . . 4 |- ( ph -> X e. A )
8		curf1.k	. . . 4 |- K = ( ( 1st ` G ) ` X )
9		eqid 2622	. . . 4 |- ( Hom ` D ) = ( Hom ` D )
10		eqid 2622	. . . 4 |- ( Id ` C ) = ( Id ` C )
11	1, 2, 3, 4, 5, 6, 7, 8, 9, 10	curf1 16865	. . 3 |- ( ph -> K = <. ( y e. B |-> ( X ( 1st ` F ) y ) ) , ( y e. B , z e. B |-> ( g e. ( y ( Hom ` D ) z ) |-> ( ( ( Id ` C ) ` X ) ( <. X , y >. ( 2nd ` F ) <. X , z >. ) g ) ) ) >. )
12		fvex 6201	. . . . . . . 8 |- ( Base ` D ) e. _V
13	6, 12	eqeltri 2697	. . . . . . 7 |- B e. _V
14	13	mptex 6486	. . . . . 6 |- ( y e. B |-> ( X ( 1st ` F ) y ) ) e. _V
15	13, 13	mpt2ex 7247	. . . . . 6 |- ( y e. B , z e. B |-> ( g e. ( y ( Hom ` D ) z ) |-> ( ( ( Id ` C ) ` X ) ( <. X , y >. ( 2nd ` F ) <. X , z >. ) g ) ) ) e. _V
16	14, 15	op1std 7178	. . . . 5 |- ( K = <. ( y e. B |-> ( X ( 1st ` F ) y ) ) , ( y e. B , z e. B |-> ( g e. ( y ( Hom ` D ) z ) |-> ( ( ( Id ` C ) ` X ) ( <. X , y >. ( 2nd ` F ) <. X , z >. ) g ) ) ) >. -> ( 1st ` K ) = ( y e. B |-> ( X ( 1st ` F ) y ) ) )
17	11, 16	syl 17	. . . 4 |- ( ph -> ( 1st ` K ) = ( y e. B |-> ( X ( 1st ` F ) y ) ) )
18	14, 15	op2ndd 7179	. . . . 5 |- ( K = <. ( y e. B |-> ( X ( 1st ` F ) y ) ) , ( y e. B , z e. B |-> ( g e. ( y ( Hom ` D ) z ) |-> ( ( ( Id ` C ) ` X ) ( <. X , y >. ( 2nd ` F ) <. X , z >. ) g ) ) ) >. -> ( 2nd ` K ) = ( y e. B , z e. B |-> ( g e. ( y ( Hom ` D ) z ) |-> ( ( ( Id ` C ) ` X ) ( <. X , y >. ( 2nd ` F ) <. X , z >. ) g ) ) ) )
19	11, 18	syl 17	. . . 4 |- ( ph -> ( 2nd ` K ) = ( y e. B , z e. B |-> ( g e. ( y ( Hom ` D ) z ) |-> ( ( ( Id ` C ) ` X ) ( <. X , y >. ( 2nd ` F ) <. X , z >. ) g ) ) ) )
20	17, 19	opeq12d 4410	. . 3 |- ( ph -> <. ( 1st ` K ) , ( 2nd ` K ) >. = <. ( y e. B |-> ( X ( 1st ` F ) y ) ) , ( y e. B , z e. B |-> ( g e. ( y ( Hom ` D ) z ) |-> ( ( ( Id ` C ) ` X ) ( <. X , y >. ( 2nd ` F ) <. X , z >. ) g ) ) ) >. )
21	11, 20	eqtr4d 2659	. 2 |- ( ph -> K = <. ( 1st ` K ) , ( 2nd ` K ) >. )
22		eqid 2622	. . . 4 |- ( Base ` E ) = ( Base ` E )
23		eqid 2622	. . . 4 |- ( Hom ` E ) = ( Hom ` E )
24		eqid 2622	. . . 4 |- ( Id ` D ) = ( Id ` D )
25		eqid 2622	. . . 4 |- ( Id ` E ) = ( Id ` E )
26		eqid 2622	. . . 4 |- (comp ` D ) = (comp ` D )
27		eqid 2622	. . . 4 |- (comp ` E ) = (comp ` E )
28		funcrcl 16523	. . . . . 6 |- ( F e. ( ( C X.c D ) Func E ) -> ( ( C X.c D ) e. Cat /\ E e. Cat ) )
29	5, 28	syl 17	. . . . 5 |- ( ph -> ( ( C X.c D ) e. Cat /\ E e. Cat ) )
30	29	simprd 479	. . . 4 |- ( ph -> E e. Cat )
31		eqid 2622	. . . . . . . . . 10 |- ( C X.c D ) = ( C X.c D )
32	31, 2, 6	xpcbas 16818	. . . . . . . . 9 |- ( A X. B ) = ( Base ` ( C X.c D ) )
33		relfunc 16522	. . . . . . . . . 10 |- Rel ( ( C X.c D ) Func E )
34		1st2ndbr 7217	. . . . . . . . . 10 |- ( ( Rel ( ( C X.c D ) Func E ) /\ F e. ( ( C X.c D ) Func E ) ) -> ( 1st ` F ) ( ( C X.c D ) Func E ) ( 2nd ` F ) )
35	33, 5, 34	sylancr 695	. . . . . . . . 9 |- ( ph -> ( 1st ` F ) ( ( C X.c D ) Func E ) ( 2nd ` F ) )
36	32, 22, 35	funcf1 16526	. . . . . . . 8 |- ( ph -> ( 1st ` F ) : ( A X. B ) --> ( Base ` E ) )
37	36	adantr 481	. . . . . . 7 |- ( ( ph /\ y e. B ) -> ( 1st ` F ) : ( A X. B ) --> ( Base ` E ) )
38	7	adantr 481	. . . . . . 7 |- ( ( ph /\ y e. B ) -> X e. A )
39		simpr 477	. . . . . . 7 |- ( ( ph /\ y e. B ) -> y e. B )
40	37, 38, 39	fovrnd 6806	. . . . . 6 |- ( ( ph /\ y e. B ) -> ( X ( 1st ` F ) y ) e. ( Base ` E ) )
41		eqid 2622	. . . . . 6 |- ( y e. B |-> ( X ( 1st ` F ) y ) ) = ( y e. B |-> ( X ( 1st ` F ) y ) )
42	40, 41	fmptd 6385	. . . . 5 |- ( ph -> ( y e. B |-> ( X ( 1st ` F ) y ) ) : B --> ( Base ` E ) )
43	17	feq1d 6030	. . . . 5 |- ( ph -> ( ( 1st ` K ) : B --> ( Base ` E ) <-> ( y e. B |-> ( X ( 1st ` F ) y ) ) : B --> ( Base ` E ) ) )
44	42, 43	mpbird 247	. . . 4 |- ( ph -> ( 1st ` K ) : B --> ( Base ` E ) )
45		eqid 2622	. . . . . 6 |- ( y e. B , z e. B |-> ( g e. ( y ( Hom ` D ) z ) |-> ( ( ( Id ` C ) ` X ) ( <. X , y >. ( 2nd ` F ) <. X , z >. ) g ) ) ) = ( y e. B , z e. B |-> ( g e. ( y ( Hom ` D ) z ) |-> ( ( ( Id ` C ) ` X ) ( <. X , y >. ( 2nd ` F ) <. X , z >. ) g ) ) )
46		ovex 6678	. . . . . . 7 |- ( y ( Hom ` D ) z ) e. _V
47	46	mptex 6486	. . . . . 6 |- ( g e. ( y ( Hom ` D ) z ) |-> ( ( ( Id ` C ) ` X ) ( <. X , y >. ( 2nd ` F ) <. X , z >. ) g ) ) e. _V
48	45, 47	fnmpt2i 7239	. . . . 5 |- ( y e. B , z e. B |-> ( g e. ( y ( Hom ` D ) z ) |-> ( ( ( Id ` C ) ` X ) ( <. X , y >. ( 2nd ` F ) <. X , z >. ) g ) ) ) Fn ( B X. B )
49	19	fneq1d 5981	. . . . 5 |- ( ph -> ( ( 2nd ` K ) Fn ( B X. B ) <-> ( y e. B , z e. B |-> ( g e. ( y ( Hom ` D ) z ) |-> ( ( ( Id ` C ) ` X ) ( <. X , y >. ( 2nd ` F ) <. X , z >. ) g ) ) ) Fn ( B X. B ) ) )
50	48, 49	mpbiri 248	. . . 4 |- ( ph -> ( 2nd ` K ) Fn ( B X. B ) )
51		eqid 2622	. . . . . . . . 9 |- ( Hom ` ( C X.c D ) ) = ( Hom ` ( C X.c D ) )
52	35	ad2antrr 762	. . . . . . . . 9 |- ( ( ( ph /\ ( y e. B /\ z e. B ) ) /\ g e. ( y ( Hom ` D ) z ) ) -> ( 1st ` F ) ( ( C X.c D ) Func E ) ( 2nd ` F ) )
53	7	ad2antrr 762	. . . . . . . . . 10 |- ( ( ( ph /\ ( y e. B /\ z e. B ) ) /\ g e. ( y ( Hom ` D ) z ) ) -> X e. A )
54		simplrl 800	. . . . . . . . . 10 |- ( ( ( ph /\ ( y e. B /\ z e. B ) ) /\ g e. ( y ( Hom ` D ) z ) ) -> y e. B )
55		opelxpi 5148	. . . . . . . . . 10 |- ( ( X e. A /\ y e. B ) -> <. X , y >. e. ( A X. B ) )
56	53, 54, 55	syl2anc 693	. . . . . . . . 9 |- ( ( ( ph /\ ( y e. B /\ z e. B ) ) /\ g e. ( y ( Hom ` D ) z ) ) -> <. X , y >. e. ( A X. B ) )
57		simplrr 801	. . . . . . . . . 10 |- ( ( ( ph /\ ( y e. B /\ z e. B ) ) /\ g e. ( y ( Hom ` D ) z ) ) -> z e. B )
58		opelxpi 5148	. . . . . . . . . 10 |- ( ( X e. A /\ z e. B ) -> <. X , z >. e. ( A X. B ) )
59	53, 57, 58	syl2anc 693	. . . . . . . . 9 |- ( ( ( ph /\ ( y e. B /\ z e. B ) ) /\ g e. ( y ( Hom ` D ) z ) ) -> <. X , z >. e. ( A X. B ) )
60	32, 51, 23, 52, 56, 59	funcf2 16528	. . . . . . . 8 |- ( ( ( ph /\ ( y e. B /\ z e. B ) ) /\ g e. ( y ( Hom ` D ) z ) ) -> ( <. X , y >. ( 2nd ` F ) <. X , z >. ) : ( <. X , y >. ( Hom ` ( C X.c D ) ) <. X , z >. ) --> ( ( ( 1st ` F ) ` <. X , y >. ) ( Hom ` E ) ( ( 1st ` F ) ` <. X , z >. ) ) )
61		eqid 2622	. . . . . . . . . 10 |- ( Hom ` C ) = ( Hom ` C )
62	31, 32, 61, 9, 51, 56, 59	xpchom 16820	. . . . . . . . 9 |- ( ( ( ph /\ ( y e. B /\ z e. B ) ) /\ g e. ( y ( Hom ` D ) z ) ) -> ( <. X , y >. ( Hom ` ( C X.c D ) ) <. X , z >. ) = ( ( ( 1st ` <. X , y >. ) ( Hom ` C ) ( 1st ` <. X , z >. ) ) X. ( ( 2nd ` <. X , y >. ) ( Hom ` D ) ( 2nd ` <. X , z >. ) ) ) )
63	3	ad2antrr 762	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( y e. B /\ z e. B ) ) /\ g e. ( y ( Hom ` D ) z ) ) -> C e. Cat )
64	4	ad2antrr 762	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( y e. B /\ z e. B ) ) /\ g e. ( y ( Hom ` D ) z ) ) -> D e. Cat )
65	5	ad2antrr 762	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( y e. B /\ z e. B ) ) /\ g e. ( y ( Hom ` D ) z ) ) -> F e. ( ( C X.c D ) Func E ) )
66	1, 2, 63, 64, 65, 6, 53, 8, 54	curf11 16866	. . . . . . . . . . 11 |- ( ( ( ph /\ ( y e. B /\ z e. B ) ) /\ g e. ( y ( Hom ` D ) z ) ) -> ( ( 1st ` K ) ` y ) = ( X ( 1st ` F ) y ) )
67		df-ov 6653	. . . . . . . . . . 11 |- ( X ( 1st ` F ) y ) = ( ( 1st ` F ) ` <. X , y >. )
68	66, 67	syl6req 2673	. . . . . . . . . 10 |- ( ( ( ph /\ ( y e. B /\ z e. B ) ) /\ g e. ( y ( Hom ` D ) z ) ) -> ( ( 1st ` F ) ` <. X , y >. ) = ( ( 1st ` K ) ` y ) )
69	1, 2, 63, 64, 65, 6, 53, 8, 57	curf11 16866	. . . . . . . . . . 11 |- ( ( ( ph /\ ( y e. B /\ z e. B ) ) /\ g e. ( y ( Hom ` D ) z ) ) -> ( ( 1st ` K ) ` z ) = ( X ( 1st ` F ) z ) )
70		df-ov 6653	. . . . . . . . . . 11 |- ( X ( 1st ` F ) z ) = ( ( 1st ` F ) ` <. X , z >. )
71	69, 70	syl6req 2673	. . . . . . . . . 10 |- ( ( ( ph /\ ( y e. B /\ z e. B ) ) /\ g e. ( y ( Hom ` D ) z ) ) -> ( ( 1st ` F ) ` <. X , z >. ) = ( ( 1st ` K ) ` z ) )
72	68, 71	oveq12d 6668	. . . . . . . . 9 |- ( ( ( ph /\ ( y e. B /\ z e. B ) ) /\ g e. ( y ( Hom ` D ) z ) ) -> ( ( ( 1st ` F ) ` <. X , y >. ) ( Hom ` E ) ( ( 1st ` F ) ` <. X , z >. ) ) = ( ( ( 1st ` K ) ` y ) ( Hom ` E ) ( ( 1st ` K ) ` z ) ) )
73	62, 72	feq23d 6040	. . . . . . . 8 |- ( ( ( ph /\ ( y e. B /\ z e. B ) ) /\ g e. ( y ( Hom ` D ) z ) ) -> ( ( <. X , y >. ( 2nd ` F ) <. X , z >. ) : ( <. X , y >. ( Hom ` ( C X.c D ) ) <. X , z >. ) --> ( ( ( 1st ` F ) ` <. X , y >. ) ( Hom ` E ) ( ( 1st ` F ) ` <. X , z >. ) ) <-> ( <. X , y >. ( 2nd ` F ) <. X , z >. ) : ( ( ( 1st ` <. X , y >. ) ( Hom ` C ) ( 1st ` <. X , z >. ) ) X. ( ( 2nd ` <. X , y >. ) ( Hom ` D ) ( 2nd ` <. X , z >. ) ) ) --> ( ( ( 1st ` K ) ` y ) ( Hom ` E ) ( ( 1st ` K ) ` z ) ) ) )
74	60, 73	mpbid 222	. . . . . . 7 |- ( ( ( ph /\ ( y e. B /\ z e. B ) ) /\ g e. ( y ( Hom ` D ) z ) ) -> ( <. X , y >. ( 2nd ` F ) <. X , z >. ) : ( ( ( 1st ` <. X , y >. ) ( Hom ` C ) ( 1st ` <. X , z >. ) ) X. ( ( 2nd ` <. X , y >. ) ( Hom ` D ) ( 2nd ` <. X , z >. ) ) ) --> ( ( ( 1st ` K ) ` y ) ( Hom ` E ) ( ( 1st ` K ) ` z ) ) )
75	2, 61, 10, 63, 53	catidcl 16343	. . . . . . . 8 |- ( ( ( ph /\ ( y e. B /\ z e. B ) ) /\ g e. ( y ( Hom ` D ) z ) ) -> ( ( Id ` C ) ` X ) e. ( X ( Hom ` C ) X ) )
76		op1stg 7180	. . . . . . . . . 10 |- ( ( X e. A /\ y e. B ) -> ( 1st ` <. X , y >. ) = X )
77	53, 54, 76	syl2anc 693	. . . . . . . . 9 |- ( ( ( ph /\ ( y e. B /\ z e. B ) ) /\ g e. ( y ( Hom ` D ) z ) ) -> ( 1st ` <. X , y >. ) = X )
78		op1stg 7180	. . . . . . . . . 10 |- ( ( X e. A /\ z e. B ) -> ( 1st ` <. X , z >. ) = X )
79	53, 57, 78	syl2anc 693	. . . . . . . . 9 |- ( ( ( ph /\ ( y e. B /\ z e. B ) ) /\ g e. ( y ( Hom ` D ) z ) ) -> ( 1st ` <. X , z >. ) = X )
80	77, 79	oveq12d 6668	. . . . . . . 8 |- ( ( ( ph /\ ( y e. B /\ z e. B ) ) /\ g e. ( y ( Hom ` D ) z ) ) -> ( ( 1st ` <. X , y >. ) ( Hom ` C ) ( 1st ` <. X , z >. ) ) = ( X ( Hom ` C ) X ) )
81	75, 80	eleqtrrd 2704	. . . . . . 7 |- ( ( ( ph /\ ( y e. B /\ z e. B ) ) /\ g e. ( y ( Hom ` D ) z ) ) -> ( ( Id ` C ) ` X ) e. ( ( 1st ` <. X , y >. ) ( Hom ` C ) ( 1st ` <. X , z >. ) ) )
82		simpr 477	. . . . . . . 8 |- ( ( ( ph /\ ( y e. B /\ z e. B ) ) /\ g e. ( y ( Hom ` D ) z ) ) -> g e. ( y ( Hom ` D ) z ) )
83		op2ndg 7181	. . . . . . . . . 10 |- ( ( X e. A /\ y e. B ) -> ( 2nd ` <. X , y >. ) = y )
84	53, 54, 83	syl2anc 693	. . . . . . . . 9 |- ( ( ( ph /\ ( y e. B /\ z e. B ) ) /\ g e. ( y ( Hom ` D ) z ) ) -> ( 2nd ` <. X , y >. ) = y )
85		op2ndg 7181	. . . . . . . . . 10 |- ( ( X e. A /\ z e. B ) -> ( 2nd ` <. X , z >. ) = z )
86	53, 57, 85	syl2anc 693	. . . . . . . . 9 |- ( ( ( ph /\ ( y e. B /\ z e. B ) ) /\ g e. ( y ( Hom ` D ) z ) ) -> ( 2nd ` <. X , z >. ) = z )
87	84, 86	oveq12d 6668	. . . . . . . 8 |- ( ( ( ph /\ ( y e. B /\ z e. B ) ) /\ g e. ( y ( Hom ` D ) z ) ) -> ( ( 2nd ` <. X , y >. ) ( Hom ` D ) ( 2nd ` <. X , z >. ) ) = ( y ( Hom ` D ) z ) )
88	82, 87	eleqtrrd 2704	. . . . . . 7 |- ( ( ( ph /\ ( y e. B /\ z e. B ) ) /\ g e. ( y ( Hom ` D ) z ) ) -> g e. ( ( 2nd ` <. X , y >. ) ( Hom ` D ) ( 2nd ` <. X , z >. ) ) )
89	74, 81, 88	fovrnd 6806	. . . . . 6 |- ( ( ( ph /\ ( y e. B /\ z e. B ) ) /\ g e. ( y ( Hom ` D ) z ) ) -> ( ( ( Id ` C ) ` X ) ( <. X , y >. ( 2nd ` F ) <. X , z >. ) g ) e. ( ( ( 1st ` K ) ` y ) ( Hom ` E ) ( ( 1st ` K ) ` z ) ) )
90		eqid 2622	. . . . . 6 |- ( g e. ( y ( Hom ` D ) z ) |-> ( ( ( Id ` C ) ` X ) ( <. X , y >. ( 2nd ` F ) <. X , z >. ) g ) ) = ( g e. ( y ( Hom ` D ) z ) |-> ( ( ( Id ` C ) ` X ) ( <. X , y >. ( 2nd ` F ) <. X , z >. ) g ) )
91	89, 90	fmptd 6385	. . . . 5 |- ( ( ph /\ ( y e. B /\ z e. B ) ) -> ( g e. ( y ( Hom ` D ) z ) |-> ( ( ( Id ` C ) ` X ) ( <. X , y >. ( 2nd ` F ) <. X , z >. ) g ) ) : ( y ( Hom ` D ) z ) --> ( ( ( 1st ` K ) ` y ) ( Hom ` E ) ( ( 1st ` K ) ` z ) ) )
92	19	oveqd 6667	. . . . . . 7 |- ( ph -> ( y ( 2nd ` K ) z ) = ( y ( y e. B , z e. B |-> ( g e. ( y ( Hom ` D ) z ) |-> ( ( ( Id ` C ) ` X ) ( <. X , y >. ( 2nd ` F ) <. X , z >. ) g ) ) ) z ) )
93	45	ovmpt4g 6783	. . . . . . . 8 |- ( ( y e. B /\ z e. B /\ ( g e. ( y ( Hom ` D ) z ) |-> ( ( ( Id ` C ) ` X ) ( <. X , y >. ( 2nd ` F ) <. X , z >. ) g ) ) e. _V ) -> ( y ( y e. B , z e. B |-> ( g e. ( y ( Hom ` D ) z ) |-> ( ( ( Id ` C ) ` X ) ( <. X , y >. ( 2nd ` F ) <. X , z >. ) g ) ) ) z ) = ( g e. ( y ( Hom ` D ) z ) |-> ( ( ( Id ` C ) ` X ) ( <. X , y >. ( 2nd ` F ) <. X , z >. ) g ) ) )
94	47, 93	mp3an3 1413	. . . . . . 7 |- ( ( y e. B /\ z e. B ) -> ( y ( y e. B , z e. B |-> ( g e. ( y ( Hom ` D ) z ) |-> ( ( ( Id ` C ) ` X ) ( <. X , y >. ( 2nd ` F ) <. X , z >. ) g ) ) ) z ) = ( g e. ( y ( Hom ` D ) z ) |-> ( ( ( Id ` C ) ` X ) ( <. X , y >. ( 2nd ` F ) <. X , z >. ) g ) ) )
95	92, 94	sylan9eq 2676	. . . . . 6 |- ( ( ph /\ ( y e. B /\ z e. B ) ) -> ( y ( 2nd ` K ) z ) = ( g e. ( y ( Hom ` D ) z ) |-> ( ( ( Id ` C ) ` X ) ( <. X , y >. ( 2nd ` F ) <. X , z >. ) g ) ) )
96	95	feq1d 6030	. . . . 5 |- ( ( ph /\ ( y e. B /\ z e. B ) ) -> ( ( y ( 2nd ` K ) z ) : ( y ( Hom ` D ) z ) --> ( ( ( 1st ` K ) ` y ) ( Hom ` E ) ( ( 1st ` K ) ` z ) ) <-> ( g e. ( y ( Hom ` D ) z ) |-> ( ( ( Id ` C ) ` X ) ( <. X , y >. ( 2nd ` F ) <. X , z >. ) g ) ) : ( y ( Hom ` D ) z ) --> ( ( ( 1st ` K ) ` y ) ( Hom ` E ) ( ( 1st ` K ) ` z ) ) ) )
97	91, 96	mpbird 247	. . . 4 |- ( ( ph /\ ( y e. B /\ z e. B ) ) -> ( y ( 2nd ` K ) z ) : ( y ( Hom ` D ) z ) --> ( ( ( 1st ` K ) ` y ) ( Hom ` E ) ( ( 1st ` K ) ` z ) ) )
98	3	adantr 481	. . . . . . . . 9 |- ( ( ph /\ y e. B ) -> C e. Cat )
99	4	adantr 481	. . . . . . . . 9 |- ( ( ph /\ y e. B ) -> D e. Cat )
100		eqid 2622	. . . . . . . . 9 |- ( Id ` ( C X.c D ) ) = ( Id ` ( C X.c D ) )
101	31, 98, 99, 2, 6, 10, 24, 100, 38, 39	xpcid 16829	. . . . . . . 8 |- ( ( ph /\ y e. B ) -> ( ( Id ` ( C X.c D ) ) ` <. X , y >. ) = <. ( ( Id ` C ) ` X ) , ( ( Id ` D ) ` y ) >. )
102	101	fveq2d 6195	. . . . . . 7 |- ( ( ph /\ y e. B ) -> ( ( <. X , y >. ( 2nd ` F ) <. X , y >. ) ` ( ( Id ` ( C X.c D ) ) ` <. X , y >. ) ) = ( ( <. X , y >. ( 2nd ` F ) <. X , y >. ) ` <. ( ( Id ` C ) ` X ) , ( ( Id ` D ) ` y ) >. ) )
103		df-ov 6653	. . . . . . 7 |- ( ( ( Id ` C ) ` X ) ( <. X , y >. ( 2nd ` F ) <. X , y >. ) ( ( Id ` D ) ` y ) ) = ( ( <. X , y >. ( 2nd ` F ) <. X , y >. ) ` <. ( ( Id ` C ) ` X ) , ( ( Id ` D ) ` y ) >. )
104	102, 103	syl6eqr 2674	. . . . . 6 |- ( ( ph /\ y e. B ) -> ( ( <. X , y >. ( 2nd ` F ) <. X , y >. ) ` ( ( Id ` ( C X.c D ) ) ` <. X , y >. ) ) = ( ( ( Id ` C ) ` X ) ( <. X , y >. ( 2nd ` F ) <. X , y >. ) ( ( Id ` D ) ` y ) ) )
105	35	adantr 481	. . . . . . 7 |- ( ( ph /\ y e. B ) -> ( 1st ` F ) ( ( C X.c D ) Func E ) ( 2nd ` F ) )
106	7, 55	sylan 488	. . . . . . 7 |- ( ( ph /\ y e. B ) -> <. X , y >. e. ( A X. B ) )
107	32, 100, 25, 105, 106	funcid 16530	. . . . . 6 |- ( ( ph /\ y e. B ) -> ( ( <. X , y >. ( 2nd ` F ) <. X , y >. ) ` ( ( Id ` ( C X.c D ) ) ` <. X , y >. ) ) = ( ( Id ` E ) ` ( ( 1st ` F ) ` <. X , y >. ) ) )
108	104, 107	eqtr3d 2658	. . . . 5 |- ( ( ph /\ y e. B ) -> ( ( ( Id ` C ) ` X ) ( <. X , y >. ( 2nd ` F ) <. X , y >. ) ( ( Id ` D ) ` y ) ) = ( ( Id ` E ) ` ( ( 1st ` F ) ` <. X , y >. ) ) )
109	5	adantr 481	. . . . . 6 |- ( ( ph /\ y e. B ) -> F e. ( ( C X.c D ) Func E ) )
110	6, 9, 24, 99, 39	catidcl 16343	. . . . . 6 |- ( ( ph /\ y e. B ) -> ( ( Id ` D ) ` y ) e. ( y ( Hom ` D ) y ) )
111	1, 2, 98, 99, 109, 6, 38, 8, 39, 9, 10, 39, 110	curf12 16867	. . . . 5 |- ( ( ph /\ y e. B ) -> ( ( y ( 2nd ` K ) y ) ` ( ( Id ` D ) ` y ) ) = ( ( ( Id ` C ) ` X ) ( <. X , y >. ( 2nd ` F ) <. X , y >. ) ( ( Id ` D ) ` y ) ) )
112	1, 2, 98, 99, 109, 6, 38, 8, 39	curf11 16866	. . . . . . 7 |- ( ( ph /\ y e. B ) -> ( ( 1st ` K ) ` y ) = ( X ( 1st ` F ) y ) )
113	112, 67	syl6eq 2672	. . . . . 6 |- ( ( ph /\ y e. B ) -> ( ( 1st ` K ) ` y ) = ( ( 1st ` F ) ` <. X , y >. ) )
114	113	fveq2d 6195	. . . . 5 |- ( ( ph /\ y e. B ) -> ( ( Id ` E ) ` ( ( 1st ` K ) ` y ) ) = ( ( Id ` E ) ` ( ( 1st ` F ) ` <. X , y >. ) ) )
115	108, 111, 114	3eqtr4d 2666	. . . 4 |- ( ( ph /\ y e. B ) -> ( ( y ( 2nd ` K ) y ) ` ( ( Id ` D ) ` y ) ) = ( ( Id ` E ) ` ( ( 1st ` K ) ` y ) ) )
116	7	3ad2ant1 1082	. . . . . . . . . 10 |- ( ( ph /\ ( y e. B /\ z e. B /\ w e. B ) /\ ( g e. ( y ( Hom ` D ) z ) /\ h e. ( z ( Hom ` D ) w ) ) ) -> X e. A )
117		simp21 1094	. . . . . . . . . 10 |- ( ( ph /\ ( y e. B /\ z e. B /\ w e. B ) /\ ( g e. ( y ( Hom ` D ) z ) /\ h e. ( z ( Hom ` D ) w ) ) ) -> y e. B )
118		simp22 1095	. . . . . . . . . 10 |- ( ( ph /\ ( y e. B /\ z e. B /\ w e. B ) /\ ( g e. ( y ( Hom ` D ) z ) /\ h e. ( z ( Hom ` D ) w ) ) ) -> z e. B )
119		eqid 2622	. . . . . . . . . 10 |- (comp ` C ) = (comp ` C )
120		eqid 2622	. . . . . . . . . 10 |- (comp ` ( C X.c D ) ) = (comp ` ( C X.c D ) )
121		simp23 1096	. . . . . . . . . 10 |- ( ( ph /\ ( y e. B /\ z e. B /\ w e. B ) /\ ( g e. ( y ( Hom ` D ) z ) /\ h e. ( z ( Hom ` D ) w ) ) ) -> w e. B )
122	3	3ad2ant1 1082	. . . . . . . . . . 11 |- ( ( ph /\ ( y e. B /\ z e. B /\ w e. B ) /\ ( g e. ( y ( Hom ` D ) z ) /\ h e. ( z ( Hom ` D ) w ) ) ) -> C e. Cat )
123	2, 61, 10, 122, 116	catidcl 16343	. . . . . . . . . 10 |- ( ( ph /\ ( y e. B /\ z e. B /\ w e. B ) /\ ( g e. ( y ( Hom ` D ) z ) /\ h e. ( z ( Hom ` D ) w ) ) ) -> ( ( Id ` C ) ` X ) e. ( X ( Hom ` C ) X ) )
124		simp3l 1089	. . . . . . . . . 10 |- ( ( ph /\ ( y e. B /\ z e. B /\ w e. B ) /\ ( g e. ( y ( Hom ` D ) z ) /\ h e. ( z ( Hom ` D ) w ) ) ) -> g e. ( y ( Hom ` D ) z ) )
125		simp3r 1090	. . . . . . . . . 10 |- ( ( ph /\ ( y e. B /\ z e. B /\ w e. B ) /\ ( g e. ( y ( Hom ` D ) z ) /\ h e. ( z ( Hom ` D ) w ) ) ) -> h e. ( z ( Hom ` D ) w ) )
126	31, 2, 6, 61, 9, 116, 117, 116, 118, 119, 26, 120, 116, 121, 123, 124, 123, 125	xpcco2 16827	. . . . . . . . 9 |- ( ( ph /\ ( y e. B /\ z e. B /\ w e. B ) /\ ( g e. ( y ( Hom ` D ) z ) /\ h e. ( z ( Hom ` D ) w ) ) ) -> ( <. ( ( Id ` C ) ` X ) , h >. ( <. <. X , y >. , <. X , z >. >. (comp ` ( C X.c D ) ) <. X , w >. ) <. ( ( Id ` C ) ` X ) , g >. ) = <. ( ( ( Id ` C ) ` X ) ( <. X , X >. (comp ` C ) X ) ( ( Id ` C ) ` X ) ) , ( h ( <. y , z >. (comp ` D ) w ) g ) >. )
127	2, 61, 10, 122, 116, 119, 116, 123	catlid 16344	. . . . . . . . . 10 |- ( ( ph /\ ( y e. B /\ z e. B /\ w e. B ) /\ ( g e. ( y ( Hom ` D ) z ) /\ h e. ( z ( Hom ` D ) w ) ) ) -> ( ( ( Id ` C ) ` X ) ( <. X , X >. (comp ` C ) X ) ( ( Id ` C ) ` X ) ) = ( ( Id ` C ) ` X ) )
128	127	opeq1d 4408	. . . . . . . . 9 |- ( ( ph /\ ( y e. B /\ z e. B /\ w e. B ) /\ ( g e. ( y ( Hom ` D ) z ) /\ h e. ( z ( Hom ` D ) w ) ) ) -> <. ( ( ( Id ` C ) ` X ) ( <. X , X >. (comp ` C ) X ) ( ( Id ` C ) ` X ) ) , ( h ( <. y , z >. (comp ` D ) w ) g ) >. = <. ( ( Id ` C ) ` X ) , ( h ( <. y , z >. (comp ` D ) w ) g ) >. )
129	126, 128	eqtrd 2656	. . . . . . . 8 |- ( ( ph /\ ( y e. B /\ z e. B /\ w e. B ) /\ ( g e. ( y ( Hom ` D ) z ) /\ h e. ( z ( Hom ` D ) w ) ) ) -> ( <. ( ( Id ` C ) ` X ) , h >. ( <. <. X , y >. , <. X , z >. >. (comp ` ( C X.c D ) ) <. X , w >. ) <. ( ( Id ` C ) ` X ) , g >. ) = <. ( ( Id ` C ) ` X ) , ( h ( <. y , z >. (comp ` D ) w ) g ) >. )
130	129	fveq2d 6195	. . . . . . 7 |- ( ( ph /\ ( y e. B /\ z e. B /\ w e. B ) /\ ( g e. ( y ( Hom ` D ) z ) /\ h e. ( z ( Hom ` D ) w ) ) ) -> ( ( <. X , y >. ( 2nd ` F ) <. X , w >. ) ` ( <. ( ( Id ` C ) ` X ) , h >. ( <. <. X , y >. , <. X , z >. >. (comp ` ( C X.c D ) ) <. X , w >. ) <. ( ( Id ` C ) ` X ) , g >. ) ) = ( ( <. X , y >. ( 2nd ` F ) <. X , w >. ) ` <. ( ( Id ` C ) ` X ) , ( h ( <. y , z >. (comp ` D ) w ) g ) >. ) )
131		df-ov 6653	. . . . . . 7 |- ( ( ( Id ` C ) ` X ) ( <. X , y >. ( 2nd ` F ) <. X , w >. ) ( h ( <. y , z >. (comp ` D ) w ) g ) ) = ( ( <. X , y >. ( 2nd ` F ) <. X , w >. ) ` <. ( ( Id ` C ) ` X ) , ( h ( <. y , z >. (comp ` D ) w ) g ) >. )
132	130, 131	syl6eqr 2674	. . . . . 6 |- ( ( ph /\ ( y e. B /\ z e. B /\ w e. B ) /\ ( g e. ( y ( Hom ` D ) z ) /\ h e. ( z ( Hom ` D ) w ) ) ) -> ( ( <. X , y >. ( 2nd ` F ) <. X , w >. ) ` ( <. ( ( Id ` C ) ` X ) , h >. ( <. <. X , y >. , <. X , z >. >. (comp ` ( C X.c D ) ) <. X , w >. ) <. ( ( Id ` C ) ` X ) , g >. ) ) = ( ( ( Id ` C ) ` X ) ( <. X , y >. ( 2nd ` F ) <. X , w >. ) ( h ( <. y , z >. (comp ` D ) w ) g ) ) )
133	35	3ad2ant1 1082	. . . . . . 7 |- ( ( ph /\ ( y e. B /\ z e. B /\ w e. B ) /\ ( g e. ( y ( Hom ` D ) z ) /\ h e. ( z ( Hom ` D ) w ) ) ) -> ( 1st ` F ) ( ( C X.c D ) Func E ) ( 2nd ` F ) )
134	116, 117, 55	syl2anc 693	. . . . . . 7 |- ( ( ph /\ ( y e. B /\ z e. B /\ w e. B ) /\ ( g e. ( y ( Hom ` D ) z ) /\ h e. ( z ( Hom ` D ) w ) ) ) -> <. X , y >. e. ( A X. B ) )
135	116, 118, 58	syl2anc 693	. . . . . . 7 |- ( ( ph /\ ( y e. B /\ z e. B /\ w e. B ) /\ ( g e. ( y ( Hom ` D ) z ) /\ h e. ( z ( Hom ` D ) w ) ) ) -> <. X , z >. e. ( A X. B ) )
136		opelxpi 5148	. . . . . . . 8 |- ( ( X e. A /\ w e. B ) -> <. X , w >. e. ( A X. B ) )
137	116, 121, 136	syl2anc 693	. . . . . . 7 |- ( ( ph /\ ( y e. B /\ z e. B /\ w e. B ) /\ ( g e. ( y ( Hom ` D ) z ) /\ h e. ( z ( Hom ` D ) w ) ) ) -> <. X , w >. e. ( A X. B ) )
138		opelxpi 5148	. . . . . . . . 9 |- ( ( ( ( Id ` C ) ` X ) e. ( X ( Hom ` C ) X ) /\ g e. ( y ( Hom ` D ) z ) ) -> <. ( ( Id ` C ) ` X ) , g >. e. ( ( X ( Hom ` C ) X ) X. ( y ( Hom ` D ) z ) ) )
139	123, 124, 138	syl2anc 693	. . . . . . . 8 |- ( ( ph /\ ( y e. B /\ z e. B /\ w e. B ) /\ ( g e. ( y ( Hom ` D ) z ) /\ h e. ( z ( Hom ` D ) w ) ) ) -> <. ( ( Id ` C ) ` X ) , g >. e. ( ( X ( Hom ` C ) X ) X. ( y ( Hom ` D ) z ) ) )
140	31, 2, 6, 61, 9, 116, 117, 116, 118, 51	xpchom2 16826	. . . . . . . 8 |- ( ( ph /\ ( y e. B /\ z e. B /\ w e. B ) /\ ( g e. ( y ( Hom ` D ) z ) /\ h e. ( z ( Hom ` D ) w ) ) ) -> ( <. X , y >. ( Hom ` ( C X.c D ) ) <. X , z >. ) = ( ( X ( Hom ` C ) X ) X. ( y ( Hom ` D ) z ) ) )
141	139, 140	eleqtrrd 2704	. . . . . . 7 |- ( ( ph /\ ( y e. B /\ z e. B /\ w e. B ) /\ ( g e. ( y ( Hom ` D ) z ) /\ h e. ( z ( Hom ` D ) w ) ) ) -> <. ( ( Id ` C ) ` X ) , g >. e. ( <. X , y >. ( Hom ` ( C X.c D ) ) <. X , z >. ) )
142		opelxpi 5148	. . . . . . . . 9 |- ( ( ( ( Id ` C ) ` X ) e. ( X ( Hom ` C ) X ) /\ h e. ( z ( Hom ` D ) w ) ) -> <. ( ( Id ` C ) ` X ) , h >. e. ( ( X ( Hom ` C ) X ) X. ( z ( Hom ` D ) w ) ) )
143	123, 125, 142	syl2anc 693	. . . . . . . 8 |- ( ( ph /\ ( y e. B /\ z e. B /\ w e. B ) /\ ( g e. ( y ( Hom ` D ) z ) /\ h e. ( z ( Hom ` D ) w ) ) ) -> <. ( ( Id ` C ) ` X ) , h >. e. ( ( X ( Hom ` C ) X ) X. ( z ( Hom ` D ) w ) ) )
144	31, 2, 6, 61, 9, 116, 118, 116, 121, 51	xpchom2 16826	. . . . . . . 8 |- ( ( ph /\ ( y e. B /\ z e. B /\ w e. B ) /\ ( g e. ( y ( Hom ` D ) z ) /\ h e. ( z ( Hom ` D ) w ) ) ) -> ( <. X , z >. ( Hom ` ( C X.c D ) ) <. X , w >. ) = ( ( X ( Hom ` C ) X ) X. ( z ( Hom ` D ) w ) ) )
145	143, 144	eleqtrrd 2704	. . . . . . 7 |- ( ( ph /\ ( y e. B /\ z e. B /\ w e. B ) /\ ( g e. ( y ( Hom ` D ) z ) /\ h e. ( z ( Hom ` D ) w ) ) ) -> <. ( ( Id ` C ) ` X ) , h >. e. ( <. X , z >. ( Hom ` ( C X.c D ) ) <. X , w >. ) )
146	32, 51, 120, 27, 133, 134, 135, 137, 141, 145	funcco 16531	. . . . . 6 |- ( ( ph /\ ( y e. B /\ z e. B /\ w e. B ) /\ ( g e. ( y ( Hom ` D ) z ) /\ h e. ( z ( Hom ` D ) w ) ) ) -> ( ( <. X , y >. ( 2nd ` F ) <. X , w >. ) ` ( <. ( ( Id ` C ) ` X ) , h >. ( <. <. X , y >. , <. X , z >. >. (comp ` ( C X.c D ) ) <. X , w >. ) <. ( ( Id ` C ) ` X ) , g >. ) ) = ( ( ( <. X , z >. ( 2nd ` F ) <. X , w >. ) ` <. ( ( Id ` C ) ` X ) , h >. ) ( <. ( ( 1st ` F ) ` <. X , y >. ) , ( ( 1st ` F ) ` <. X , z >. ) >. (comp ` E ) ( ( 1st ` F ) ` <. X , w >. ) ) ( ( <. X , y >. ( 2nd ` F ) <. X , z >. ) ` <. ( ( Id ` C ) ` X ) , g >. ) ) )
147	132, 146	eqtr3d 2658	. . . . 5 |- ( ( ph /\ ( y e. B /\ z e. B /\ w e. B ) /\ ( g e. ( y ( Hom ` D ) z ) /\ h e. ( z ( Hom ` D ) w ) ) ) -> ( ( ( Id ` C ) ` X ) ( <. X , y >. ( 2nd ` F ) <. X , w >. ) ( h ( <. y , z >. (comp ` D ) w ) g ) ) = ( ( ( <. X , z >. ( 2nd ` F ) <. X , w >. ) ` <. ( ( Id ` C ) ` X ) , h >. ) ( <. ( ( 1st ` F ) ` <. X , y >. ) , ( ( 1st ` F ) ` <. X , z >. ) >. (comp ` E ) ( ( 1st ` F ) ` <. X , w >. ) ) ( ( <. X , y >. ( 2nd ` F ) <. X , z >. ) ` <. ( ( Id ` C ) ` X ) , g >. ) ) )
148	4	3ad2ant1 1082	. . . . . 6 |- ( ( ph /\ ( y e. B /\ z e. B /\ w e. B ) /\ ( g e. ( y ( Hom ` D ) z ) /\ h e. ( z ( Hom ` D ) w ) ) ) -> D e. Cat )
149	5	3ad2ant1 1082	. . . . . 6 |- ( ( ph /\ ( y e. B /\ z e. B /\ w e. B ) /\ ( g e. ( y ( Hom ` D ) z ) /\ h e. ( z ( Hom ` D ) w ) ) ) -> F e. ( ( C X.c D ) Func E ) )
150	6, 9, 26, 148, 117, 118, 121, 124, 125	catcocl 16346	. . . . . 6 |- ( ( ph /\ ( y e. B /\ z e. B /\ w e. B ) /\ ( g e. ( y ( Hom ` D ) z ) /\ h e. ( z ( Hom ` D ) w ) ) ) -> ( h ( <. y , z >. (comp ` D ) w ) g ) e. ( y ( Hom ` D ) w ) )
151	1, 2, 122, 148, 149, 6, 116, 8, 117, 9, 10, 121, 150	curf12 16867	. . . . 5 |- ( ( ph /\ ( y e. B /\ z e. B /\ w e. B ) /\ ( g e. ( y ( Hom ` D ) z ) /\ h e. ( z ( Hom ` D ) w ) ) ) -> ( ( y ( 2nd ` K ) w ) ` ( h ( <. y , z >. (comp ` D ) w ) g ) ) = ( ( ( Id ` C ) ` X ) ( <. X , y >. ( 2nd ` F ) <. X , w >. ) ( h ( <. y , z >. (comp ` D ) w ) g ) ) )
152	1, 2, 122, 148, 149, 6, 116, 8, 117	curf11 16866	. . . . . . . . 9 |- ( ( ph /\ ( y e. B /\ z e. B /\ w e. B ) /\ ( g e. ( y ( Hom ` D ) z ) /\ h e. ( z ( Hom ` D ) w ) ) ) -> ( ( 1st ` K ) ` y ) = ( X ( 1st ` F ) y ) )
153	152, 67	syl6eq 2672	. . . . . . . 8 |- ( ( ph /\ ( y e. B /\ z e. B /\ w e. B ) /\ ( g e. ( y ( Hom ` D ) z ) /\ h e. ( z ( Hom ` D ) w ) ) ) -> ( ( 1st ` K ) ` y ) = ( ( 1st ` F ) ` <. X , y >. ) )
154	1, 2, 122, 148, 149, 6, 116, 8, 118	curf11 16866	. . . . . . . . 9 |- ( ( ph /\ ( y e. B /\ z e. B /\ w e. B ) /\ ( g e. ( y ( Hom ` D ) z ) /\ h e. ( z ( Hom ` D ) w ) ) ) -> ( ( 1st ` K ) ` z ) = ( X ( 1st ` F ) z ) )
155	154, 70	syl6eq 2672	. . . . . . . 8 |- ( ( ph /\ ( y e. B /\ z e. B /\ w e. B ) /\ ( g e. ( y ( Hom ` D ) z ) /\ h e. ( z ( Hom ` D ) w ) ) ) -> ( ( 1st ` K ) ` z ) = ( ( 1st ` F ) ` <. X , z >. ) )
156	153, 155	opeq12d 4410	. . . . . . 7 |- ( ( ph /\ ( y e. B /\ z e. B /\ w e. B ) /\ ( g e. ( y ( Hom ` D ) z ) /\ h e. ( z ( Hom ` D ) w ) ) ) -> <. ( ( 1st ` K ) ` y ) , ( ( 1st ` K ) ` z ) >. = <. ( ( 1st ` F ) ` <. X , y >. ) , ( ( 1st ` F ) ` <. X , z >. ) >. )
157	1, 2, 122, 148, 149, 6, 116, 8, 121	curf11 16866	. . . . . . . 8 |- ( ( ph /\ ( y e. B /\ z e. B /\ w e. B ) /\ ( g e. ( y ( Hom ` D ) z ) /\ h e. ( z ( Hom ` D ) w ) ) ) -> ( ( 1st ` K ) ` w ) = ( X ( 1st ` F ) w ) )
158		df-ov 6653	. . . . . . . 8 |- ( X ( 1st ` F ) w ) = ( ( 1st ` F ) ` <. X , w >. )
159	157, 158	syl6eq 2672	. . . . . . 7 |- ( ( ph /\ ( y e. B /\ z e. B /\ w e. B ) /\ ( g e. ( y ( Hom ` D ) z ) /\ h e. ( z ( Hom ` D ) w ) ) ) -> ( ( 1st ` K ) ` w ) = ( ( 1st ` F ) ` <. X , w >. ) )
160	156, 159	oveq12d 6668	. . . . . 6 |- ( ( ph /\ ( y e. B /\ z e. B /\ w e. B ) /\ ( g e. ( y ( Hom ` D ) z ) /\ h e. ( z ( Hom ` D ) w ) ) ) -> ( <. ( ( 1st ` K ) ` y ) , ( ( 1st ` K ) ` z ) >. (comp ` E ) ( ( 1st ` K ) ` w ) ) = ( <. ( ( 1st ` F ) ` <. X , y >. ) , ( ( 1st ` F ) ` <. X , z >. ) >. (comp ` E ) ( ( 1st ` F ) ` <. X , w >. ) ) )
161	1, 2, 122, 148, 149, 6, 116, 8, 118, 9, 10, 121, 125	curf12 16867	. . . . . . 7 |- ( ( ph /\ ( y e. B /\ z e. B /\ w e. B ) /\ ( g e. ( y ( Hom ` D ) z ) /\ h e. ( z ( Hom ` D ) w ) ) ) -> ( ( z ( 2nd ` K ) w ) ` h ) = ( ( ( Id ` C ) ` X ) ( <. X , z >. ( 2nd ` F ) <. X , w >. ) h ) )
162		df-ov 6653	. . . . . . 7 |- ( ( ( Id ` C ) ` X ) ( <. X , z >. ( 2nd ` F ) <. X , w >. ) h ) = ( ( <. X , z >. ( 2nd ` F ) <. X , w >. ) ` <. ( ( Id ` C ) ` X ) , h >. )
163	161, 162	syl6eq 2672	. . . . . 6 |- ( ( ph /\ ( y e. B /\ z e. B /\ w e. B ) /\ ( g e. ( y ( Hom ` D ) z ) /\ h e. ( z ( Hom ` D ) w ) ) ) -> ( ( z ( 2nd ` K ) w ) ` h ) = ( ( <. X , z >. ( 2nd ` F ) <. X , w >. ) ` <. ( ( Id ` C ) ` X ) , h >. ) )
164	1, 2, 122, 148, 149, 6, 116, 8, 117, 9, 10, 118, 124	curf12 16867	. . . . . . 7 |- ( ( ph /\ ( y e. B /\ z e. B /\ w e. B ) /\ ( g e. ( y ( Hom ` D ) z ) /\ h e. ( z ( Hom ` D ) w ) ) ) -> ( ( y ( 2nd ` K ) z ) ` g ) = ( ( ( Id ` C ) ` X ) ( <. X , y >. ( 2nd ` F ) <. X , z >. ) g ) )
165		df-ov 6653	. . . . . . 7 |- ( ( ( Id ` C ) ` X ) ( <. X , y >. ( 2nd ` F ) <. X , z >. ) g ) = ( ( <. X , y >. ( 2nd ` F ) <. X , z >. ) ` <. ( ( Id ` C ) ` X ) , g >. )
166	164, 165	syl6eq 2672	. . . . . 6 |- ( ( ph /\ ( y e. B /\ z e. B /\ w e. B ) /\ ( g e. ( y ( Hom ` D ) z ) /\ h e. ( z ( Hom ` D ) w ) ) ) -> ( ( y ( 2nd ` K ) z ) ` g ) = ( ( <. X , y >. ( 2nd ` F ) <. X , z >. ) ` <. ( ( Id ` C ) ` X ) , g >. ) )
167	160, 163, 166	oveq123d 6671	. . . . 5 |- ( ( ph /\ ( y e. B /\ z e. B /\ w e. B ) /\ ( g e. ( y ( Hom ` D ) z ) /\ h e. ( z ( Hom ` D ) w ) ) ) -> ( ( ( z ( 2nd ` K ) w ) ` h ) ( <. ( ( 1st ` K ) ` y ) , ( ( 1st ` K ) ` z ) >. (comp ` E ) ( ( 1st ` K ) ` w ) ) ( ( y ( 2nd ` K ) z ) ` g ) ) = ( ( ( <. X , z >. ( 2nd ` F ) <. X , w >. ) ` <. ( ( Id ` C ) ` X ) , h >. ) ( <. ( ( 1st ` F ) ` <. X , y >. ) , ( ( 1st ` F ) ` <. X , z >. ) >. (comp ` E ) ( ( 1st ` F ) ` <. X , w >. ) ) ( ( <. X , y >. ( 2nd ` F ) <. X , z >. ) ` <. ( ( Id ` C ) ` X ) , g >. ) ) )
168	147, 151, 167	3eqtr4d 2666	. . . 4 |- ( ( ph /\ ( y e. B /\ z e. B /\ w e. B ) /\ ( g e. ( y ( Hom ` D ) z ) /\ h e. ( z ( Hom ` D ) w ) ) ) -> ( ( y ( 2nd ` K ) w ) ` ( h ( <. y , z >. (comp ` D ) w ) g ) ) = ( ( ( z ( 2nd ` K ) w ) ` h ) ( <. ( ( 1st ` K ) ` y ) , ( ( 1st ` K ) ` z ) >. (comp ` E ) ( ( 1st ` K ) ` w ) ) ( ( y ( 2nd ` K ) z ) ` g ) ) )
169	6, 22, 9, 23, 24, 25, 26, 27, 4, 30, 44, 50, 97, 115, 168	isfuncd 16525	. . 3 |- ( ph -> ( 1st ` K ) ( D Func E ) ( 2nd ` K ) )
170		df-br 4654	. . 3 |- ( ( 1st ` K ) ( D Func E ) ( 2nd ` K ) <-> <. ( 1st ` K ) , ( 2nd ` K ) >. e. ( D Func E ) )
171	169, 170	sylib 208	. 2 |- ( ph -> <. ( 1st ` K ) , ( 2nd ` K ) >. e. ( D Func E ) )
172	21, 171	eqeltrd 2701	1 |- ( ph -> K e. ( D Func E ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   _Vcvv 3200   <.cop 4183   class class class wbr 4653   |-> cmpt 4729   X. cxp 5112   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   X._c cxpc 16808   curry_F ccurf 16850

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-xpc 16812  df-curf 16854

This theorem is referenced by:  curf2cl  16871  curfcl  16872