Theorem prfcl 16843

Description: The pairing of functors F : C --> D and G : C --> D is a functor <. F , G >. : C --> ( D X. E ). (Contributed by Mario Carneiro, 12-Jan-2017.)

Hypotheses

Ref	Expression
prfcl.p	|- P = ( F ⟨,⟩F G )
prfcl.t	|- T = ( D X.c E )
prfcl.c	|- ( ph -> F e. ( C Func D ) )
prfcl.d	|- ( ph -> G e. ( C Func E ) )

Assertion

Ref	Expression
prfcl	|- ( ph -> P e. ( C Func T ) )

Proof of Theorem prfcl

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

Step	Hyp	Ref	Expression
1		prfcl.p	. . . 4 |- P = ( F ⟨,⟩F G )
2		eqid 2622	. . . 4 |- ( Base ` C ) = ( Base ` C )
3		eqid 2622	. . . 4 |- ( Hom ` C ) = ( Hom ` C )
4		prfcl.c	. . . 4 |- ( ph -> F e. ( C Func D ) )
5		prfcl.d	. . . 4 |- ( ph -> G e. ( C Func E ) )
6	1, 2, 3, 4, 5	prfval 16839	. . 3 |- ( ph -> P = <. ( x e. ( Base ` C ) |-> <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. ) , ( x e. ( Base ` C ) , y e. ( Base ` C ) |-> ( h e. ( x ( Hom ` C ) y ) |-> <. ( ( x ( 2nd ` F ) y ) ` h ) , ( ( x ( 2nd ` G ) y ) ` h ) >. ) ) >. )
7		fvex 6201	. . . . . . 7 |- ( Base ` C ) e. _V
8	7	mptex 6486	. . . . . 6 |- ( x e. ( Base ` C ) |-> <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. ) e. _V
9	7, 7	mpt2ex 7247	. . . . . 6 |- ( x e. ( Base ` C ) , y e. ( Base ` C ) |-> ( h e. ( x ( Hom ` C ) y ) |-> <. ( ( x ( 2nd ` F ) y ) ` h ) , ( ( x ( 2nd ` G ) y ) ` h ) >. ) ) e. _V
10	8, 9	op1std 7178	. . . . 5 |- ( P = <. ( x e. ( Base ` C ) |-> <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. ) , ( x e. ( Base ` C ) , y e. ( Base ` C ) |-> ( h e. ( x ( Hom ` C ) y ) |-> <. ( ( x ( 2nd ` F ) y ) ` h ) , ( ( x ( 2nd ` G ) y ) ` h ) >. ) ) >. -> ( 1st ` P ) = ( x e. ( Base ` C ) |-> <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. ) )
11	6, 10	syl 17	. . . 4 |- ( ph -> ( 1st ` P ) = ( x e. ( Base ` C ) |-> <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. ) )
12	8, 9	op2ndd 7179	. . . . 5 |- ( P = <. ( x e. ( Base ` C ) |-> <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. ) , ( x e. ( Base ` C ) , y e. ( Base ` C ) |-> ( h e. ( x ( Hom ` C ) y ) |-> <. ( ( x ( 2nd ` F ) y ) ` h ) , ( ( x ( 2nd ` G ) y ) ` h ) >. ) ) >. -> ( 2nd ` P ) = ( x e. ( Base ` C ) , y e. ( Base ` C ) |-> ( h e. ( x ( Hom ` C ) y ) |-> <. ( ( x ( 2nd ` F ) y ) ` h ) , ( ( x ( 2nd ` G ) y ) ` h ) >. ) ) )
13	6, 12	syl 17	. . . 4 |- ( ph -> ( 2nd ` P ) = ( x e. ( Base ` C ) , y e. ( Base ` C ) |-> ( h e. ( x ( Hom ` C ) y ) |-> <. ( ( x ( 2nd ` F ) y ) ` h ) , ( ( x ( 2nd ` G ) y ) ` h ) >. ) ) )
14	11, 13	opeq12d 4410	. . 3 |- ( ph -> <. ( 1st ` P ) , ( 2nd ` P ) >. = <. ( x e. ( Base ` C ) |-> <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. ) , ( x e. ( Base ` C ) , y e. ( Base ` C ) |-> ( h e. ( x ( Hom ` C ) y ) |-> <. ( ( x ( 2nd ` F ) y ) ` h ) , ( ( x ( 2nd ` G ) y ) ` h ) >. ) ) >. )
15	6, 14	eqtr4d 2659	. 2 |- ( ph -> P = <. ( 1st ` P ) , ( 2nd ` P ) >. )
16		prfcl.t	. . . . 5 |- T = ( D X.c E )
17		eqid 2622	. . . . 5 |- ( Base ` D ) = ( Base ` D )
18		eqid 2622	. . . . 5 |- ( Base ` E ) = ( Base ` E )
19	16, 17, 18	xpcbas 16818	. . . 4 |- ( ( Base ` D ) X. ( Base ` E ) ) = ( Base ` T )
20		eqid 2622	. . . 4 |- ( Hom ` T ) = ( Hom ` T )
21		eqid 2622	. . . 4 |- ( Id ` C ) = ( Id ` C )
22		eqid 2622	. . . 4 |- ( Id ` T ) = ( Id ` T )
23		eqid 2622	. . . 4 |- (comp ` C ) = (comp ` C )
24		eqid 2622	. . . 4 |- (comp ` T ) = (comp ` T )
25		funcrcl 16523	. . . . . 6 |- ( F e. ( C Func D ) -> ( C e. Cat /\ D e. Cat ) )
26	4, 25	syl 17	. . . . 5 |- ( ph -> ( C e. Cat /\ D e. Cat ) )
27	26	simpld 475	. . . 4 |- ( ph -> C e. Cat )
28	26	simprd 479	. . . . 5 |- ( ph -> D e. Cat )
29		funcrcl 16523	. . . . . . 7 |- ( G e. ( C Func E ) -> ( C e. Cat /\ E e. Cat ) )
30	5, 29	syl 17	. . . . . 6 |- ( ph -> ( C e. Cat /\ E e. Cat ) )
31	30	simprd 479	. . . . 5 |- ( ph -> E e. Cat )
32	16, 28, 31	xpccat 16830	. . . 4 |- ( ph -> T e. Cat )
33		relfunc 16522	. . . . . . . . . 10 |- Rel ( C Func D )
34		1st2ndbr 7217	. . . . . . . . . 10 |- ( ( Rel ( C Func D ) /\ F e. ( C Func D ) ) -> ( 1st ` F ) ( C Func D ) ( 2nd ` F ) )
35	33, 4, 34	sylancr 695	. . . . . . . . 9 |- ( ph -> ( 1st ` F ) ( C Func D ) ( 2nd ` F ) )
36	2, 17, 35	funcf1 16526	. . . . . . . 8 |- ( ph -> ( 1st ` F ) : ( Base ` C ) --> ( Base ` D ) )
37	36	ffvelrnda 6359	. . . . . . 7 |- ( ( ph /\ x e. ( Base ` C ) ) -> ( ( 1st ` F ) ` x ) e. ( Base ` D ) )
38		relfunc 16522	. . . . . . . . . 10 |- Rel ( C Func E )
39		1st2ndbr 7217	. . . . . . . . . 10 |- ( ( Rel ( C Func E ) /\ G e. ( C Func E ) ) -> ( 1st ` G ) ( C Func E ) ( 2nd ` G ) )
40	38, 5, 39	sylancr 695	. . . . . . . . 9 |- ( ph -> ( 1st ` G ) ( C Func E ) ( 2nd ` G ) )
41	2, 18, 40	funcf1 16526	. . . . . . . 8 |- ( ph -> ( 1st ` G ) : ( Base ` C ) --> ( Base ` E ) )
42	41	ffvelrnda 6359	. . . . . . 7 |- ( ( ph /\ x e. ( Base ` C ) ) -> ( ( 1st ` G ) ` x ) e. ( Base ` E ) )
43		opelxpi 5148	. . . . . . 7 |- ( ( ( ( 1st ` F ) ` x ) e. ( Base ` D ) /\ ( ( 1st ` G ) ` x ) e. ( Base ` E ) ) -> <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. e. ( ( Base ` D ) X. ( Base ` E ) ) )
44	37, 42, 43	syl2anc 693	. . . . . 6 |- ( ( ph /\ x e. ( Base ` C ) ) -> <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. e. ( ( Base ` D ) X. ( Base ` E ) ) )
45		eqid 2622	. . . . . 6 |- ( x e. ( Base ` C ) |-> <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. ) = ( x e. ( Base ` C ) |-> <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. )
46	44, 45	fmptd 6385	. . . . 5 |- ( ph -> ( x e. ( Base ` C ) |-> <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. ) : ( Base ` C ) --> ( ( Base ` D ) X. ( Base ` E ) ) )
47	11	feq1d 6030	. . . . 5 |- ( ph -> ( ( 1st ` P ) : ( Base ` C ) --> ( ( Base ` D ) X. ( Base ` E ) ) <-> ( x e. ( Base ` C ) |-> <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. ) : ( Base ` C ) --> ( ( Base ` D ) X. ( Base ` E ) ) ) )
48	46, 47	mpbird 247	. . . 4 |- ( ph -> ( 1st ` P ) : ( Base ` C ) --> ( ( Base ` D ) X. ( Base ` E ) ) )
49		eqid 2622	. . . . . 6 |- ( x e. ( Base ` C ) , y e. ( Base ` C ) |-> ( h e. ( x ( Hom ` C ) y ) |-> <. ( ( x ( 2nd ` F ) y ) ` h ) , ( ( x ( 2nd ` G ) y ) ` h ) >. ) ) = ( x e. ( Base ` C ) , y e. ( Base ` C ) |-> ( h e. ( x ( Hom ` C ) y ) |-> <. ( ( x ( 2nd ` F ) y ) ` h ) , ( ( x ( 2nd ` G ) y ) ` h ) >. ) )
50		ovex 6678	. . . . . . 7 |- ( x ( Hom ` C ) y ) e. _V
51	50	mptex 6486	. . . . . 6 |- ( h e. ( x ( Hom ` C ) y ) |-> <. ( ( x ( 2nd ` F ) y ) ` h ) , ( ( x ( 2nd ` G ) y ) ` h ) >. ) e. _V
52	49, 51	fnmpt2i 7239	. . . . 5 |- ( x e. ( Base ` C ) , y e. ( Base ` C ) |-> ( h e. ( x ( Hom ` C ) y ) |-> <. ( ( x ( 2nd ` F ) y ) ` h ) , ( ( x ( 2nd ` G ) y ) ` h ) >. ) ) Fn ( ( Base ` C ) X. ( Base ` C ) )
53	13	fneq1d 5981	. . . . 5 |- ( ph -> ( ( 2nd ` P ) Fn ( ( Base ` C ) X. ( Base ` C ) ) <-> ( x e. ( Base ` C ) , y e. ( Base ` C ) |-> ( h e. ( x ( Hom ` C ) y ) |-> <. ( ( x ( 2nd ` F ) y ) ` h ) , ( ( x ( 2nd ` G ) y ) ` h ) >. ) ) Fn ( ( Base ` C ) X. ( Base ` C ) ) ) )
54	52, 53	mpbiri 248	. . . 4 |- ( ph -> ( 2nd ` P ) Fn ( ( Base ` C ) X. ( Base ` C ) ) )
55		eqid 2622	. . . . . . . . . 10 |- ( Hom ` D ) = ( Hom ` D )
56	35	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> ( 1st ` F ) ( C Func D ) ( 2nd ` F ) )
57		simprl 794	. . . . . . . . . 10 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> x e. ( Base ` C ) )
58		simprr 796	. . . . . . . . . 10 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> y e. ( Base ` C ) )
59	2, 3, 55, 56, 57, 58	funcf2 16528	. . . . . . . . 9 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> ( x ( 2nd ` F ) y ) : ( x ( Hom ` C ) y ) --> ( ( ( 1st ` F ) ` x ) ( Hom ` D ) ( ( 1st ` F ) ` y ) ) )
60	59	ffvelrnda 6359	. . . . . . . 8 |- ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ h e. ( x ( Hom ` C ) y ) ) -> ( ( x ( 2nd ` F ) y ) ` h ) e. ( ( ( 1st ` F ) ` x ) ( Hom ` D ) ( ( 1st ` F ) ` y ) ) )
61		eqid 2622	. . . . . . . . . 10 |- ( Hom ` E ) = ( Hom ` E )
62	40	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> ( 1st ` G ) ( C Func E ) ( 2nd ` G ) )
63	2, 3, 61, 62, 57, 58	funcf2 16528	. . . . . . . . 9 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> ( x ( 2nd ` G ) y ) : ( x ( Hom ` C ) y ) --> ( ( ( 1st ` G ) ` x ) ( Hom ` E ) ( ( 1st ` G ) ` y ) ) )
64	63	ffvelrnda 6359	. . . . . . . 8 |- ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ h e. ( x ( Hom ` C ) y ) ) -> ( ( x ( 2nd ` G ) y ) ` h ) e. ( ( ( 1st ` G ) ` x ) ( Hom ` E ) ( ( 1st ` G ) ` y ) ) )
65		opelxpi 5148	. . . . . . . 8 |- ( ( ( ( x ( 2nd ` F ) y ) ` h ) e. ( ( ( 1st ` F ) ` x ) ( Hom ` D ) ( ( 1st ` F ) ` y ) ) /\ ( ( x ( 2nd ` G ) y ) ` h ) e. ( ( ( 1st ` G ) ` x ) ( Hom ` E ) ( ( 1st ` G ) ` y ) ) ) -> <. ( ( x ( 2nd ` F ) y ) ` h ) , ( ( x ( 2nd ` G ) y ) ` h ) >. e. ( ( ( ( 1st ` F ) ` x ) ( Hom ` D ) ( ( 1st ` F ) ` y ) ) X. ( ( ( 1st ` G ) ` x ) ( Hom ` E ) ( ( 1st ` G ) ` y ) ) ) )
66	60, 64, 65	syl2anc 693	. . . . . . 7 |- ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ h e. ( x ( Hom ` C ) y ) ) -> <. ( ( x ( 2nd ` F ) y ) ` h ) , ( ( x ( 2nd ` G ) y ) ` h ) >. e. ( ( ( ( 1st ` F ) ` x ) ( Hom ` D ) ( ( 1st ` F ) ` y ) ) X. ( ( ( 1st ` G ) ` x ) ( Hom ` E ) ( ( 1st ` G ) ` y ) ) ) )
67	4	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> F e. ( C Func D ) )
68	5	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> G e. ( C Func E ) )
69	1, 2, 3, 67, 68, 57	prf1 16840	. . . . . . . . . 10 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> ( ( 1st ` P ) ` x ) = <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. )
70	1, 2, 3, 67, 68, 58	prf1 16840	. . . . . . . . . 10 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> ( ( 1st ` P ) ` y ) = <. ( ( 1st ` F ) ` y ) , ( ( 1st ` G ) ` y ) >. )
71	69, 70	oveq12d 6668	. . . . . . . . 9 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> ( ( ( 1st ` P ) ` x ) ( Hom ` T ) ( ( 1st ` P ) ` y ) ) = ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. ( Hom ` T ) <. ( ( 1st ` F ) ` y ) , ( ( 1st ` G ) ` y ) >. ) )
72	37	adantrr 753	. . . . . . . . . 10 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> ( ( 1st ` F ) ` x ) e. ( Base ` D ) )
73	42	adantrr 753	. . . . . . . . . 10 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> ( ( 1st ` G ) ` x ) e. ( Base ` E ) )
74	36	ffvelrnda 6359	. . . . . . . . . . 11 |- ( ( ph /\ y e. ( Base ` C ) ) -> ( ( 1st ` F ) ` y ) e. ( Base ` D ) )
75	74	adantrl 752	. . . . . . . . . 10 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> ( ( 1st ` F ) ` y ) e. ( Base ` D ) )
76	41	ffvelrnda 6359	. . . . . . . . . . 11 |- ( ( ph /\ y e. ( Base ` C ) ) -> ( ( 1st ` G ) ` y ) e. ( Base ` E ) )
77	76	adantrl 752	. . . . . . . . . 10 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> ( ( 1st ` G ) ` y ) e. ( Base ` E ) )
78	16, 17, 18, 55, 61, 72, 73, 75, 77, 20	xpchom2 16826	. . . . . . . . 9 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. ( Hom ` T ) <. ( ( 1st ` F ) ` y ) , ( ( 1st ` G ) ` y ) >. ) = ( ( ( ( 1st ` F ) ` x ) ( Hom ` D ) ( ( 1st ` F ) ` y ) ) X. ( ( ( 1st ` G ) ` x ) ( Hom ` E ) ( ( 1st ` G ) ` y ) ) ) )
79	71, 78	eqtrd 2656	. . . . . . . 8 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> ( ( ( 1st ` P ) ` x ) ( Hom ` T ) ( ( 1st ` P ) ` y ) ) = ( ( ( ( 1st ` F ) ` x ) ( Hom ` D ) ( ( 1st ` F ) ` y ) ) X. ( ( ( 1st ` G ) ` x ) ( Hom ` E ) ( ( 1st ` G ) ` y ) ) ) )
80	79	adantr 481	. . . . . . 7 |- ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ h e. ( x ( Hom ` C ) y ) ) -> ( ( ( 1st ` P ) ` x ) ( Hom ` T ) ( ( 1st ` P ) ` y ) ) = ( ( ( ( 1st ` F ) ` x ) ( Hom ` D ) ( ( 1st ` F ) ` y ) ) X. ( ( ( 1st ` G ) ` x ) ( Hom ` E ) ( ( 1st ` G ) ` y ) ) ) )
81	66, 80	eleqtrrd 2704	. . . . . 6 |- ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ h e. ( x ( Hom ` C ) y ) ) -> <. ( ( x ( 2nd ` F ) y ) ` h ) , ( ( x ( 2nd ` G ) y ) ` h ) >. e. ( ( ( 1st ` P ) ` x ) ( Hom ` T ) ( ( 1st ` P ) ` y ) ) )
82		eqid 2622	. . . . . 6 |- ( h e. ( x ( Hom ` C ) y ) |-> <. ( ( x ( 2nd ` F ) y ) ` h ) , ( ( x ( 2nd ` G ) y ) ` h ) >. ) = ( h e. ( x ( Hom ` C ) y ) |-> <. ( ( x ( 2nd ` F ) y ) ` h ) , ( ( x ( 2nd ` G ) y ) ` h ) >. )
83	81, 82	fmptd 6385	. . . . 5 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> ( h e. ( x ( Hom ` C ) y ) |-> <. ( ( x ( 2nd ` F ) y ) ` h ) , ( ( x ( 2nd ` G ) y ) ` h ) >. ) : ( x ( Hom ` C ) y ) --> ( ( ( 1st ` P ) ` x ) ( Hom ` T ) ( ( 1st ` P ) ` y ) ) )
84	13	oveqd 6667	. . . . . . 7 |- ( ph -> ( x ( 2nd ` P ) y ) = ( x ( x e. ( Base ` C ) , y e. ( Base ` C ) |-> ( h e. ( x ( Hom ` C ) y ) |-> <. ( ( x ( 2nd ` F ) y ) ` h ) , ( ( x ( 2nd ` G ) y ) ` h ) >. ) ) y ) )
85	49	ovmpt4g 6783	. . . . . . . 8 |- ( ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ ( h e. ( x ( Hom ` C ) y ) |-> <. ( ( x ( 2nd ` F ) y ) ` h ) , ( ( x ( 2nd ` G ) y ) ` h ) >. ) e. _V ) -> ( x ( x e. ( Base ` C ) , y e. ( Base ` C ) |-> ( h e. ( x ( Hom ` C ) y ) |-> <. ( ( x ( 2nd ` F ) y ) ` h ) , ( ( x ( 2nd ` G ) y ) ` h ) >. ) ) y ) = ( h e. ( x ( Hom ` C ) y ) |-> <. ( ( x ( 2nd ` F ) y ) ` h ) , ( ( x ( 2nd ` G ) y ) ` h ) >. ) )
86	51, 85	mp3an3 1413	. . . . . . 7 |- ( ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) -> ( x ( x e. ( Base ` C ) , y e. ( Base ` C ) |-> ( h e. ( x ( Hom ` C ) y ) |-> <. ( ( x ( 2nd ` F ) y ) ` h ) , ( ( x ( 2nd ` G ) y ) ` h ) >. ) ) y ) = ( h e. ( x ( Hom ` C ) y ) |-> <. ( ( x ( 2nd ` F ) y ) ` h ) , ( ( x ( 2nd ` G ) y ) ` h ) >. ) )
87	84, 86	sylan9eq 2676	. . . . . 6 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> ( x ( 2nd ` P ) y ) = ( h e. ( x ( Hom ` C ) y ) |-> <. ( ( x ( 2nd ` F ) y ) ` h ) , ( ( x ( 2nd ` G ) y ) ` h ) >. ) )
88	87	feq1d 6030	. . . . 5 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> ( ( x ( 2nd ` P ) y ) : ( x ( Hom ` C ) y ) --> ( ( ( 1st ` P ) ` x ) ( Hom ` T ) ( ( 1st ` P ) ` y ) ) <-> ( h e. ( x ( Hom ` C ) y ) |-> <. ( ( x ( 2nd ` F ) y ) ` h ) , ( ( x ( 2nd ` G ) y ) ` h ) >. ) : ( x ( Hom ` C ) y ) --> ( ( ( 1st ` P ) ` x ) ( Hom ` T ) ( ( 1st ` P ) ` y ) ) ) )
89	83, 88	mpbird 247	. . . 4 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> ( x ( 2nd ` P ) y ) : ( x ( Hom ` C ) y ) --> ( ( ( 1st ` P ) ` x ) ( Hom ` T ) ( ( 1st ` P ) ` y ) ) )
90		eqid 2622	. . . . . . 7 |- ( Id ` D ) = ( Id ` D )
91	35	adantr 481	. . . . . . 7 |- ( ( ph /\ x e. ( Base ` C ) ) -> ( 1st ` F ) ( C Func D ) ( 2nd ` F ) )
92		simpr 477	. . . . . . 7 |- ( ( ph /\ x e. ( Base ` C ) ) -> x e. ( Base ` C ) )
93	2, 21, 90, 91, 92	funcid 16530	. . . . . 6 |- ( ( ph /\ x e. ( Base ` C ) ) -> ( ( x ( 2nd ` F ) x ) ` ( ( Id ` C ) ` x ) ) = ( ( Id ` D ) ` ( ( 1st ` F ) ` x ) ) )
94		eqid 2622	. . . . . . 7 |- ( Id ` E ) = ( Id ` E )
95	40	adantr 481	. . . . . . 7 |- ( ( ph /\ x e. ( Base ` C ) ) -> ( 1st ` G ) ( C Func E ) ( 2nd ` G ) )
96	2, 21, 94, 95, 92	funcid 16530	. . . . . 6 |- ( ( ph /\ x e. ( Base ` C ) ) -> ( ( x ( 2nd ` G ) x ) ` ( ( Id ` C ) ` x ) ) = ( ( Id ` E ) ` ( ( 1st ` G ) ` x ) ) )
97	93, 96	opeq12d 4410	. . . . 5 |- ( ( ph /\ x e. ( Base ` C ) ) -> <. ( ( x ( 2nd ` F ) x ) ` ( ( Id ` C ) ` x ) ) , ( ( x ( 2nd ` G ) x ) ` ( ( Id ` C ) ` x ) ) >. = <. ( ( Id ` D ) ` ( ( 1st ` F ) ` x ) ) , ( ( Id ` E ) ` ( ( 1st ` G ) ` x ) ) >. )
98	4	adantr 481	. . . . . 6 |- ( ( ph /\ x e. ( Base ` C ) ) -> F e. ( C Func D ) )
99	5	adantr 481	. . . . . 6 |- ( ( ph /\ x e. ( Base ` C ) ) -> G e. ( C Func E ) )
100	27	adantr 481	. . . . . . 7 |- ( ( ph /\ x e. ( Base ` C ) ) -> C e. Cat )
101	2, 3, 21, 100, 92	catidcl 16343	. . . . . 6 |- ( ( ph /\ x e. ( Base ` C ) ) -> ( ( Id ` C ) ` x ) e. ( x ( Hom ` C ) x ) )
102	1, 2, 3, 98, 99, 92, 92, 101	prf2 16842	. . . . 5 |- ( ( ph /\ x e. ( Base ` C ) ) -> ( ( x ( 2nd ` P ) x ) ` ( ( Id ` C ) ` x ) ) = <. ( ( x ( 2nd ` F ) x ) ` ( ( Id ` C ) ` x ) ) , ( ( x ( 2nd ` G ) x ) ` ( ( Id ` C ) ` x ) ) >. )
103	1, 2, 3, 98, 99, 92	prf1 16840	. . . . . . 7 |- ( ( ph /\ x e. ( Base ` C ) ) -> ( ( 1st ` P ) ` x ) = <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. )
104	103	fveq2d 6195	. . . . . 6 |- ( ( ph /\ x e. ( Base ` C ) ) -> ( ( Id ` T ) ` ( ( 1st ` P ) ` x ) ) = ( ( Id ` T ) ` <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. ) )
105	28	adantr 481	. . . . . . 7 |- ( ( ph /\ x e. ( Base ` C ) ) -> D e. Cat )
106	31	adantr 481	. . . . . . 7 |- ( ( ph /\ x e. ( Base ` C ) ) -> E e. Cat )
107	16, 105, 106, 17, 18, 90, 94, 22, 37, 42	xpcid 16829	. . . . . 6 |- ( ( ph /\ x e. ( Base ` C ) ) -> ( ( Id ` T ) ` <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. ) = <. ( ( Id ` D ) ` ( ( 1st ` F ) ` x ) ) , ( ( Id ` E ) ` ( ( 1st ` G ) ` x ) ) >. )
108	104, 107	eqtrd 2656	. . . . 5 |- ( ( ph /\ x e. ( Base ` C ) ) -> ( ( Id ` T ) ` ( ( 1st ` P ) ` x ) ) = <. ( ( Id ` D ) ` ( ( 1st ` F ) ` x ) ) , ( ( Id ` E ) ` ( ( 1st ` G ) ` x ) ) >. )
109	97, 102, 108	3eqtr4d 2666	. . . 4 |- ( ( ph /\ x e. ( Base ` C ) ) -> ( ( x ( 2nd ` P ) x ) ` ( ( Id ` C ) ` x ) ) = ( ( Id ` T ) ` ( ( 1st ` P ) ` x ) ) )
110		eqid 2622	. . . . . . 7 |- (comp ` D ) = (comp ` D )
111	35	3ad2ant1 1082	. . . . . . 7 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ z e. ( Base ` C ) ) /\ ( f e. ( x ( Hom ` C ) y ) /\ g e. ( y ( Hom ` C ) z ) ) ) -> ( 1st ` F ) ( C Func D ) ( 2nd ` F ) )
112		simp21 1094	. . . . . . 7 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ z e. ( Base ` C ) ) /\ ( f e. ( x ( Hom ` C ) y ) /\ g e. ( y ( Hom ` C ) z ) ) ) -> x e. ( Base ` C ) )
113		simp22 1095	. . . . . . 7 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ z e. ( Base ` C ) ) /\ ( f e. ( x ( Hom ` C ) y ) /\ g e. ( y ( Hom ` C ) z ) ) ) -> y e. ( Base ` C ) )
114		simp23 1096	. . . . . . 7 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ z e. ( Base ` C ) ) /\ ( f e. ( x ( Hom ` C ) y ) /\ g e. ( y ( Hom ` C ) z ) ) ) -> z e. ( Base ` C ) )
115		simp3l 1089	. . . . . . 7 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ z e. ( Base ` C ) ) /\ ( f e. ( x ( Hom ` C ) y ) /\ g e. ( y ( Hom ` C ) z ) ) ) -> f e. ( x ( Hom ` C ) y ) )
116		simp3r 1090	. . . . . . 7 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ z e. ( Base ` C ) ) /\ ( f e. ( x ( Hom ` C ) y ) /\ g e. ( y ( Hom ` C ) z ) ) ) -> g e. ( y ( Hom ` C ) z ) )
117	2, 3, 23, 110, 111, 112, 113, 114, 115, 116	funcco 16531	. . . . . 6 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ z e. ( Base ` C ) ) /\ ( f e. ( x ( Hom ` C ) y ) /\ g e. ( y ( Hom ` C ) z ) ) ) -> ( ( x ( 2nd ` F ) z ) ` ( g ( <. x , y >. (comp ` C ) z ) f ) ) = ( ( ( y ( 2nd ` F ) z ) ` g ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` F ) ` y ) >. (comp ` D ) ( ( 1st ` F ) ` z ) ) ( ( x ( 2nd ` F ) y ) ` f ) ) )
118		eqid 2622	. . . . . . 7 |- (comp ` E ) = (comp ` E )
119	5	3ad2ant1 1082	. . . . . . . 8 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ z e. ( Base ` C ) ) /\ ( f e. ( x ( Hom ` C ) y ) /\ g e. ( y ( Hom ` C ) z ) ) ) -> G e. ( C Func E ) )
120	38, 119, 39	sylancr 695	. . . . . . 7 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ z e. ( Base ` C ) ) /\ ( f e. ( x ( Hom ` C ) y ) /\ g e. ( y ( Hom ` C ) z ) ) ) -> ( 1st ` G ) ( C Func E ) ( 2nd ` G ) )
121	2, 3, 23, 118, 120, 112, 113, 114, 115, 116	funcco 16531	. . . . . 6 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ z e. ( Base ` C ) ) /\ ( f e. ( x ( Hom ` C ) y ) /\ g e. ( y ( Hom ` C ) z ) ) ) -> ( ( x ( 2nd ` G ) z ) ` ( g ( <. x , y >. (comp ` C ) z ) f ) ) = ( ( ( y ( 2nd ` G ) z ) ` g ) ( <. ( ( 1st ` G ) ` x ) , ( ( 1st ` G ) ` y ) >. (comp ` E ) ( ( 1st ` G ) ` z ) ) ( ( x ( 2nd ` G ) y ) ` f ) ) )
122	117, 121	opeq12d 4410	. . . . 5 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ z e. ( Base ` C ) ) /\ ( f e. ( x ( Hom ` C ) y ) /\ g e. ( y ( Hom ` C ) z ) ) ) -> <. ( ( x ( 2nd ` F ) z ) ` ( g ( <. x , y >. (comp ` C ) z ) f ) ) , ( ( x ( 2nd ` G ) z ) ` ( g ( <. x , y >. (comp ` C ) z ) f ) ) >. = <. ( ( ( y ( 2nd ` F ) z ) ` g ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` F ) ` y ) >. (comp ` D ) ( ( 1st ` F ) ` z ) ) ( ( x ( 2nd ` F ) y ) ` f ) ) , ( ( ( y ( 2nd ` G ) z ) ` g ) ( <. ( ( 1st ` G ) ` x ) , ( ( 1st ` G ) ` y ) >. (comp ` E ) ( ( 1st ` G ) ` z ) ) ( ( x ( 2nd ` G ) y ) ` f ) ) >. )
123	4	3ad2ant1 1082	. . . . . 6 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ z e. ( Base ` C ) ) /\ ( f e. ( x ( Hom ` C ) y ) /\ g e. ( y ( Hom ` C ) z ) ) ) -> F e. ( C Func D ) )
124	27	3ad2ant1 1082	. . . . . . 7 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ z e. ( Base ` C ) ) /\ ( f e. ( x ( Hom ` C ) y ) /\ g e. ( y ( Hom ` C ) z ) ) ) -> C e. Cat )
125	2, 3, 23, 124, 112, 113, 114, 115, 116	catcocl 16346	. . . . . 6 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ z e. ( Base ` C ) ) /\ ( f e. ( x ( Hom ` C ) y ) /\ g e. ( y ( Hom ` C ) z ) ) ) -> ( g ( <. x , y >. (comp ` C ) z ) f ) e. ( x ( Hom ` C ) z ) )
126	1, 2, 3, 123, 119, 112, 114, 125	prf2 16842	. . . . 5 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ z e. ( Base ` C ) ) /\ ( f e. ( x ( Hom ` C ) y ) /\ g e. ( y ( Hom ` C ) z ) ) ) -> ( ( x ( 2nd ` P ) z ) ` ( g ( <. x , y >. (comp ` C ) z ) f ) ) = <. ( ( x ( 2nd ` F ) z ) ` ( g ( <. x , y >. (comp ` C ) z ) f ) ) , ( ( x ( 2nd ` G ) z ) ` ( g ( <. x , y >. (comp ` C ) z ) f ) ) >. )
127	1, 2, 3, 123, 119, 112	prf1 16840	. . . . . . . . 9 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ z e. ( Base ` C ) ) /\ ( f e. ( x ( Hom ` C ) y ) /\ g e. ( y ( Hom ` C ) z ) ) ) -> ( ( 1st ` P ) ` x ) = <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. )
128	1, 2, 3, 123, 119, 113	prf1 16840	. . . . . . . . 9 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ z e. ( Base ` C ) ) /\ ( f e. ( x ( Hom ` C ) y ) /\ g e. ( y ( Hom ` C ) z ) ) ) -> ( ( 1st ` P ) ` y ) = <. ( ( 1st ` F ) ` y ) , ( ( 1st ` G ) ` y ) >. )
129	127, 128	opeq12d 4410	. . . . . . . 8 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ z e. ( Base ` C ) ) /\ ( f e. ( x ( Hom ` C ) y ) /\ g e. ( y ( Hom ` C ) z ) ) ) -> <. ( ( 1st ` P ) ` x ) , ( ( 1st ` P ) ` y ) >. = <. <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. , <. ( ( 1st ` F ) ` y ) , ( ( 1st ` G ) ` y ) >. >. )
130	1, 2, 3, 123, 119, 114	prf1 16840	. . . . . . . 8 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ z e. ( Base ` C ) ) /\ ( f e. ( x ( Hom ` C ) y ) /\ g e. ( y ( Hom ` C ) z ) ) ) -> ( ( 1st ` P ) ` z ) = <. ( ( 1st ` F ) ` z ) , ( ( 1st ` G ) ` z ) >. )
131	129, 130	oveq12d 6668	. . . . . . 7 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ z e. ( Base ` C ) ) /\ ( f e. ( x ( Hom ` C ) y ) /\ g e. ( y ( Hom ` C ) z ) ) ) -> ( <. ( ( 1st ` P ) ` x ) , ( ( 1st ` P ) ` y ) >. (comp ` T ) ( ( 1st ` P ) ` z ) ) = ( <. <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. , <. ( ( 1st ` F ) ` y ) , ( ( 1st ` G ) ` y ) >. >. (comp ` T ) <. ( ( 1st ` F ) ` z ) , ( ( 1st ` G ) ` z ) >. ) )
132	1, 2, 3, 123, 119, 113, 114, 116	prf2 16842	. . . . . . 7 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ z e. ( Base ` C ) ) /\ ( f e. ( x ( Hom ` C ) y ) /\ g e. ( y ( Hom ` C ) z ) ) ) -> ( ( y ( 2nd ` P ) z ) ` g ) = <. ( ( y ( 2nd ` F ) z ) ` g ) , ( ( y ( 2nd ` G ) z ) ` g ) >. )
133	1, 2, 3, 123, 119, 112, 113, 115	prf2 16842	. . . . . . 7 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ z e. ( Base ` C ) ) /\ ( f e. ( x ( Hom ` C ) y ) /\ g e. ( y ( Hom ` C ) z ) ) ) -> ( ( x ( 2nd ` P ) y ) ` f ) = <. ( ( x ( 2nd ` F ) y ) ` f ) , ( ( x ( 2nd ` G ) y ) ` f ) >. )
134	131, 132, 133	oveq123d 6671	. . . . . 6 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ z e. ( Base ` C ) ) /\ ( f e. ( x ( Hom ` C ) y ) /\ g e. ( y ( Hom ` C ) z ) ) ) -> ( ( ( y ( 2nd ` P ) z ) ` g ) ( <. ( ( 1st ` P ) ` x ) , ( ( 1st ` P ) ` y ) >. (comp ` T ) ( ( 1st ` P ) ` z ) ) ( ( x ( 2nd ` P ) y ) ` f ) ) = ( <. ( ( y ( 2nd ` F ) z ) ` g ) , ( ( y ( 2nd ` G ) z ) ` g ) >. ( <. <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. , <. ( ( 1st ` F ) ` y ) , ( ( 1st ` G ) ` y ) >. >. (comp ` T ) <. ( ( 1st ` F ) ` z ) , ( ( 1st ` G ) ` z ) >. ) <. ( ( x ( 2nd ` F ) y ) ` f ) , ( ( x ( 2nd ` G ) y ) ` f ) >. ) )
135	36	3ad2ant1 1082	. . . . . . . 8 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ z e. ( Base ` C ) ) /\ ( f e. ( x ( Hom ` C ) y ) /\ g e. ( y ( Hom ` C ) z ) ) ) -> ( 1st ` F ) : ( Base ` C ) --> ( Base ` D ) )
136	135, 112	ffvelrnd 6360	. . . . . . 7 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ z e. ( Base ` C ) ) /\ ( f e. ( x ( Hom ` C ) y ) /\ g e. ( y ( Hom ` C ) z ) ) ) -> ( ( 1st ` F ) ` x ) e. ( Base ` D ) )
137	41	3ad2ant1 1082	. . . . . . . 8 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ z e. ( Base ` C ) ) /\ ( f e. ( x ( Hom ` C ) y ) /\ g e. ( y ( Hom ` C ) z ) ) ) -> ( 1st ` G ) : ( Base ` C ) --> ( Base ` E ) )
138	137, 112	ffvelrnd 6360	. . . . . . 7 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ z e. ( Base ` C ) ) /\ ( f e. ( x ( Hom ` C ) y ) /\ g e. ( y ( Hom ` C ) z ) ) ) -> ( ( 1st ` G ) ` x ) e. ( Base ` E ) )
139	135, 113	ffvelrnd 6360	. . . . . . 7 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ z e. ( Base ` C ) ) /\ ( f e. ( x ( Hom ` C ) y ) /\ g e. ( y ( Hom ` C ) z ) ) ) -> ( ( 1st ` F ) ` y ) e. ( Base ` D ) )
140	137, 113	ffvelrnd 6360	. . . . . . 7 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ z e. ( Base ` C ) ) /\ ( f e. ( x ( Hom ` C ) y ) /\ g e. ( y ( Hom ` C ) z ) ) ) -> ( ( 1st ` G ) ` y ) e. ( Base ` E ) )
141	135, 114	ffvelrnd 6360	. . . . . . 7 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ z e. ( Base ` C ) ) /\ ( f e. ( x ( Hom ` C ) y ) /\ g e. ( y ( Hom ` C ) z ) ) ) -> ( ( 1st ` F ) ` z ) e. ( Base ` D ) )
142	137, 114	ffvelrnd 6360	. . . . . . 7 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ z e. ( Base ` C ) ) /\ ( f e. ( x ( Hom ` C ) y ) /\ g e. ( y ( Hom ` C ) z ) ) ) -> ( ( 1st ` G ) ` z ) e. ( Base ` E ) )
143	2, 3, 55, 111, 112, 113	funcf2 16528	. . . . . . . 8 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ z e. ( Base ` C ) ) /\ ( f e. ( x ( Hom ` C ) y ) /\ g e. ( y ( Hom ` C ) z ) ) ) -> ( x ( 2nd ` F ) y ) : ( x ( Hom ` C ) y ) --> ( ( ( 1st ` F ) ` x ) ( Hom ` D ) ( ( 1st ` F ) ` y ) ) )
144	143, 115	ffvelrnd 6360	. . . . . . 7 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ z e. ( Base ` C ) ) /\ ( f e. ( x ( Hom ` C ) y ) /\ g e. ( y ( Hom ` C ) z ) ) ) -> ( ( x ( 2nd ` F ) y ) ` f ) e. ( ( ( 1st ` F ) ` x ) ( Hom ` D ) ( ( 1st ` F ) ` y ) ) )
145	2, 3, 61, 120, 112, 113	funcf2 16528	. . . . . . . 8 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ z e. ( Base ` C ) ) /\ ( f e. ( x ( Hom ` C ) y ) /\ g e. ( y ( Hom ` C ) z ) ) ) -> ( x ( 2nd ` G ) y ) : ( x ( Hom ` C ) y ) --> ( ( ( 1st ` G ) ` x ) ( Hom ` E ) ( ( 1st ` G ) ` y ) ) )
146	145, 115	ffvelrnd 6360	. . . . . . 7 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ z e. ( Base ` C ) ) /\ ( f e. ( x ( Hom ` C ) y ) /\ g e. ( y ( Hom ` C ) z ) ) ) -> ( ( x ( 2nd ` G ) y ) ` f ) e. ( ( ( 1st ` G ) ` x ) ( Hom ` E ) ( ( 1st ` G ) ` y ) ) )
147	2, 3, 55, 111, 113, 114	funcf2 16528	. . . . . . . 8 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ z e. ( Base ` C ) ) /\ ( f e. ( x ( Hom ` C ) y ) /\ g e. ( y ( Hom ` C ) z ) ) ) -> ( y ( 2nd ` F ) z ) : ( y ( Hom ` C ) z ) --> ( ( ( 1st ` F ) ` y ) ( Hom ` D ) ( ( 1st ` F ) ` z ) ) )
148	147, 116	ffvelrnd 6360	. . . . . . 7 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ z e. ( Base ` C ) ) /\ ( f e. ( x ( Hom ` C ) y ) /\ g e. ( y ( Hom ` C ) z ) ) ) -> ( ( y ( 2nd ` F ) z ) ` g ) e. ( ( ( 1st ` F ) ` y ) ( Hom ` D ) ( ( 1st ` F ) ` z ) ) )
149	2, 3, 61, 120, 113, 114	funcf2 16528	. . . . . . . 8 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ z e. ( Base ` C ) ) /\ ( f e. ( x ( Hom ` C ) y ) /\ g e. ( y ( Hom ` C ) z ) ) ) -> ( y ( 2nd ` G ) z ) : ( y ( Hom ` C ) z ) --> ( ( ( 1st ` G ) ` y ) ( Hom ` E ) ( ( 1st ` G ) ` z ) ) )
150	149, 116	ffvelrnd 6360	. . . . . . 7 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ z e. ( Base ` C ) ) /\ ( f e. ( x ( Hom ` C ) y ) /\ g e. ( y ( Hom ` C ) z ) ) ) -> ( ( y ( 2nd ` G ) z ) ` g ) e. ( ( ( 1st ` G ) ` y ) ( Hom ` E ) ( ( 1st ` G ) ` z ) ) )
151	16, 17, 18, 55, 61, 136, 138, 139, 140, 110, 118, 24, 141, 142, 144, 146, 148, 150	xpcco2 16827	. . . . . 6 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ z e. ( Base ` C ) ) /\ ( f e. ( x ( Hom ` C ) y ) /\ g e. ( y ( Hom ` C ) z ) ) ) -> ( <. ( ( y ( 2nd ` F ) z ) ` g ) , ( ( y ( 2nd ` G ) z ) ` g ) >. ( <. <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. , <. ( ( 1st ` F ) ` y ) , ( ( 1st ` G ) ` y ) >. >. (comp ` T ) <. ( ( 1st ` F ) ` z ) , ( ( 1st ` G ) ` z ) >. ) <. ( ( x ( 2nd ` F ) y ) ` f ) , ( ( x ( 2nd ` G ) y ) ` f ) >. ) = <. ( ( ( y ( 2nd ` F ) z ) ` g ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` F ) ` y ) >. (comp ` D ) ( ( 1st ` F ) ` z ) ) ( ( x ( 2nd ` F ) y ) ` f ) ) , ( ( ( y ( 2nd ` G ) z ) ` g ) ( <. ( ( 1st ` G ) ` x ) , ( ( 1st ` G ) ` y ) >. (comp ` E ) ( ( 1st ` G ) ` z ) ) ( ( x ( 2nd ` G ) y ) ` f ) ) >. )
152	134, 151	eqtrd 2656	. . . . 5 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ z e. ( Base ` C ) ) /\ ( f e. ( x ( Hom ` C ) y ) /\ g e. ( y ( Hom ` C ) z ) ) ) -> ( ( ( y ( 2nd ` P ) z ) ` g ) ( <. ( ( 1st ` P ) ` x ) , ( ( 1st ` P ) ` y ) >. (comp ` T ) ( ( 1st ` P ) ` z ) ) ( ( x ( 2nd ` P ) y ) ` f ) ) = <. ( ( ( y ( 2nd ` F ) z ) ` g ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` F ) ` y ) >. (comp ` D ) ( ( 1st ` F ) ` z ) ) ( ( x ( 2nd ` F ) y ) ` f ) ) , ( ( ( y ( 2nd ` G ) z ) ` g ) ( <. ( ( 1st ` G ) ` x ) , ( ( 1st ` G ) ` y ) >. (comp ` E ) ( ( 1st ` G ) ` z ) ) ( ( x ( 2nd ` G ) y ) ` f ) ) >. )
153	122, 126, 152	3eqtr4d 2666	. . . 4 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ z e. ( Base ` C ) ) /\ ( f e. ( x ( Hom ` C ) y ) /\ g e. ( y ( Hom ` C ) z ) ) ) -> ( ( x ( 2nd ` P ) z ) ` ( g ( <. x , y >. (comp ` C ) z ) f ) ) = ( ( ( y ( 2nd ` P ) z ) ` g ) ( <. ( ( 1st ` P ) ` x ) , ( ( 1st ` P ) ` y ) >. (comp ` T ) ( ( 1st ` P ) ` z ) ) ( ( x ( 2nd ` P ) y ) ` f ) ) )
154	2, 19, 3, 20, 21, 22, 23, 24, 27, 32, 48, 54, 89, 109, 153	isfuncd 16525	. . 3 |- ( ph -> ( 1st ` P ) ( C Func T ) ( 2nd ` P ) )
155		df-br 4654	. . 3 |- ( ( 1st ` P ) ( C Func T ) ( 2nd ` P ) <-> <. ( 1st ` P ) , ( 2nd ` P ) >. e. ( C Func T ) )
156	154, 155	sylib 208	. 2 |- ( ph -> <. ( 1st ` P ) , ( 2nd ` P ) >. e. ( C Func T ) )
157	15, 156	eqeltrd 2701	1 |- ( ph -> P e. ( C Func T ) )

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   ⟨,⟩_F cprf 16811

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-prf 16815

This theorem is referenced by:  prf1st  16844  prf2nd  16845  uncfcl  16875  uncf1  16876  uncf2  16877  yonedalem1  16912  yonedalem21  16913  yonedalem22  16918