Theorem 1stfcl 16082

Description: The first projection functor is a functor onto the left argument. (Contributed by Mario Carneiro, 11-Jan-2017.)

Hypotheses

Ref	Expression
1stfcl.t	|- T = ( C X.c D )
1stfcl.c	|- ( ph -> C e. Cat )
1stfcl.d	|- ( ph -> D e. Cat )
1stfcl.p	|- P = ( C 1stF D )

Assertion

Ref	Expression
1stfcl	|- ( ph -> P e. ( T Func C ) )

Proof of Theorem 1stfcl

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

Step	Hyp	Ref	Expression
1		1stfcl.t	. . . 4 |- T = ( C X.c D )
2		eqid 2451	. . . . 5 |- ( Base ` C ) = ( Base ` C )
3		eqid 2451	. . . . 5 |- ( Base ` D ) = ( Base ` D )
4	1, 2, 3	xpcbas 16063	. . . 4 |- ( ( Base ` C ) X. ( Base ` D ) ) = ( Base ` T )
5		eqid 2451	. . . 4 |- ( Hom ` T ) = ( Hom ` T )
6		1stfcl.c	. . . 4 |- ( ph -> C e. Cat )
7		1stfcl.d	. . . 4 |- ( ph -> D e. Cat )
8		1stfcl.p	. . . 4 |- P = ( C 1stF D )
9	1, 4, 5, 6, 7, 8	1stfval 16076	. . 3 |- ( ph -> P = <. ( 1st |` ( ( Base ` C ) X. ( Base ` D ) ) ) , ( x e. ( ( Base ` C ) X. ( Base ` D ) ) , y e. ( ( Base ` C ) X. ( Base ` D ) ) |-> ( 1st |` ( x ( Hom ` T ) y ) ) ) >. )
10		fo1st 6813	. . . . . . . 8 |- 1st : _V -onto-> _V
11		fofun 5794	. . . . . . . 8 |- ( 1st : _V -onto-> _V -> Fun 1st )
12	10, 11	ax-mp 5	. . . . . . 7 |- Fun 1st
13		fvex 5875	. . . . . . . 8 |- ( Base ` C ) e. _V
14		fvex 5875	. . . . . . . 8 |- ( Base ` D ) e. _V
15	13, 14	xpex 6595	. . . . . . 7 |- ( ( Base ` C ) X. ( Base ` D ) ) e. _V
16		resfunexg 6130	. . . . . . 7 |- ( ( Fun 1st /\ ( ( Base ` C ) X. ( Base ` D ) ) e. _V ) -> ( 1st |` ( ( Base ` C ) X. ( Base ` D ) ) ) e. _V )
17	12, 15, 16	mp2an 678	. . . . . 6 |- ( 1st |` ( ( Base ` C ) X. ( Base ` D ) ) ) e. _V
18	15, 15	mpt2ex 6870	. . . . . 6 |- ( x e. ( ( Base ` C ) X. ( Base ` D ) ) , y e. ( ( Base ` C ) X. ( Base ` D ) ) |-> ( 1st |` ( x ( Hom ` T ) y ) ) ) e. _V
19	17, 18	op2ndd 6804	. . . . 5 |- ( P = <. ( 1st |` ( ( Base ` C ) X. ( Base ` D ) ) ) , ( x e. ( ( Base ` C ) X. ( Base ` D ) ) , y e. ( ( Base ` C ) X. ( Base ` D ) ) |-> ( 1st |` ( x ( Hom ` T ) y ) ) ) >. -> ( 2nd ` P ) = ( x e. ( ( Base ` C ) X. ( Base ` D ) ) , y e. ( ( Base ` C ) X. ( Base ` D ) ) |-> ( 1st |` ( x ( Hom ` T ) y ) ) ) )
20	9, 19	syl 17	. . . 4 |- ( ph -> ( 2nd ` P ) = ( x e. ( ( Base ` C ) X. ( Base ` D ) ) , y e. ( ( Base ` C ) X. ( Base ` D ) ) |-> ( 1st |` ( x ( Hom ` T ) y ) ) ) )
21	20	opeq2d 4173	. . 3 |- ( ph -> <. ( 1st |` ( ( Base ` C ) X. ( Base ` D ) ) ) , ( 2nd ` P ) >. = <. ( 1st |` ( ( Base ` C ) X. ( Base ` D ) ) ) , ( x e. ( ( Base ` C ) X. ( Base ` D ) ) , y e. ( ( Base ` C ) X. ( Base ` D ) ) |-> ( 1st |` ( x ( Hom ` T ) y ) ) ) >. )
22	9, 21	eqtr4d 2488	. 2 |- ( ph -> P = <. ( 1st |` ( ( Base ` C ) X. ( Base ` D ) ) ) , ( 2nd ` P ) >. )
23		eqid 2451	. . . 4 |- ( Hom ` C ) = ( Hom ` C )
24		eqid 2451	. . . 4 |- ( Id ` T ) = ( Id ` T )
25		eqid 2451	. . . 4 |- ( Id ` C ) = ( Id ` C )
26		eqid 2451	. . . 4 |- (comp ` T ) = (comp ` T )
27		eqid 2451	. . . 4 |- (comp ` C ) = (comp ` C )
28	1, 6, 7	xpccat 16075	. . . 4 |- ( ph -> T e. Cat )
29		f1stres 6815	. . . . 5 |- ( 1st |` ( ( Base ` C ) X. ( Base ` D ) ) ) : ( ( Base ` C ) X. ( Base ` D ) ) --> ( Base ` C )
30	29	a1i 11	. . . 4 |- ( ph -> ( 1st |` ( ( Base ` C ) X. ( Base ` D ) ) ) : ( ( Base ` C ) X. ( Base ` D ) ) --> ( Base ` C ) )
31		eqid 2451	. . . . . 6 |- ( x e. ( ( Base ` C ) X. ( Base ` D ) ) , y e. ( ( Base ` C ) X. ( Base ` D ) ) |-> ( 1st |` ( x ( Hom ` T ) y ) ) ) = ( x e. ( ( Base ` C ) X. ( Base ` D ) ) , y e. ( ( Base ` C ) X. ( Base ` D ) ) |-> ( 1st |` ( x ( Hom ` T ) y ) ) )
32		ovex 6318	. . . . . . 7 |- ( x ( Hom ` T ) y ) e. _V
33		resfunexg 6130	. . . . . . 7 |- ( ( Fun 1st /\ ( x ( Hom ` T ) y ) e. _V ) -> ( 1st |` ( x ( Hom ` T ) y ) ) e. _V )
34	12, 32, 33	mp2an 678	. . . . . 6 |- ( 1st |` ( x ( Hom ` T ) y ) ) e. _V
35	31, 34	fnmpt2i 6862	. . . . 5 |- ( x e. ( ( Base ` C ) X. ( Base ` D ) ) , y e. ( ( Base ` C ) X. ( Base ` D ) ) |-> ( 1st |` ( x ( Hom ` T ) y ) ) ) Fn ( ( ( Base ` C ) X. ( Base ` D ) ) X. ( ( Base ` C ) X. ( Base ` D ) ) )
36	20	fneq1d 5666	. . . . 5 |- ( ph -> ( ( 2nd ` P ) Fn ( ( ( Base ` C ) X. ( Base ` D ) ) X. ( ( Base ` C ) X. ( Base ` D ) ) ) <-> ( x e. ( ( Base ` C ) X. ( Base ` D ) ) , y e. ( ( Base ` C ) X. ( Base ` D ) ) |-> ( 1st |` ( x ( Hom ` T ) y ) ) ) Fn ( ( ( Base ` C ) X. ( Base ` D ) ) X. ( ( Base ` C ) X. ( Base ` D ) ) ) ) )
37	35, 36	mpbiri 237	. . . 4 |- ( ph -> ( 2nd ` P ) Fn ( ( ( Base ` C ) X. ( Base ` D ) ) X. ( ( Base ` C ) X. ( Base ` D ) ) ) )
38		f1stres 6815	. . . . . 6 |- ( 1st |` ( ( ( 1st ` x ) ( Hom ` C ) ( 1st ` y ) ) X. ( ( 2nd ` x ) ( Hom ` D ) ( 2nd ` y ) ) ) ) : ( ( ( 1st ` x ) ( Hom ` C ) ( 1st ` y ) ) X. ( ( 2nd ` x ) ( Hom ` D ) ( 2nd ` y ) ) ) --> ( ( 1st ` x ) ( Hom ` C ) ( 1st ` y ) )
39	6	adantr 467	. . . . . . . . 9 |- ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` D ) ) /\ y e. ( ( Base ` C ) X. ( Base ` D ) ) ) ) -> C e. Cat )
40	7	adantr 467	. . . . . . . . 9 |- ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` D ) ) /\ y e. ( ( Base ` C ) X. ( Base ` D ) ) ) ) -> D e. Cat )
41		simprl 764	. . . . . . . . 9 |- ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` D ) ) /\ y e. ( ( Base ` C ) X. ( Base ` D ) ) ) ) -> x e. ( ( Base ` C ) X. ( Base ` D ) ) )
42		simprr 766	. . . . . . . . 9 |- ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` D ) ) /\ y e. ( ( Base ` C ) X. ( Base ` D ) ) ) ) -> y e. ( ( Base ` C ) X. ( Base ` D ) ) )
43	1, 4, 5, 39, 40, 8, 41, 42	1stf2 16078	. . . . . . . 8 |- ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` D ) ) /\ y e. ( ( Base ` C ) X. ( Base ` D ) ) ) ) -> ( x ( 2nd ` P ) y ) = ( 1st |` ( x ( Hom ` T ) y ) ) )
44		eqid 2451	. . . . . . . . . 10 |- ( Hom ` D ) = ( Hom ` D )
45	1, 4, 23, 44, 5, 41, 42	xpchom 16065	. . . . . . . . 9 |- ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` D ) ) /\ y e. ( ( Base ` C ) X. ( Base ` D ) ) ) ) -> ( x ( Hom ` T ) y ) = ( ( ( 1st ` x ) ( Hom ` C ) ( 1st ` y ) ) X. ( ( 2nd ` x ) ( Hom ` D ) ( 2nd ` y ) ) ) )
46	45	reseq2d 5105	. . . . . . . 8 |- ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` D ) ) /\ y e. ( ( Base ` C ) X. ( Base ` D ) ) ) ) -> ( 1st |` ( x ( Hom ` T ) y ) ) = ( 1st |` ( ( ( 1st ` x ) ( Hom ` C ) ( 1st ` y ) ) X. ( ( 2nd ` x ) ( Hom ` D ) ( 2nd ` y ) ) ) ) )
47	43, 46	eqtrd 2485	. . . . . . 7 |- ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` D ) ) /\ y e. ( ( Base ` C ) X. ( Base ` D ) ) ) ) -> ( x ( 2nd ` P ) y ) = ( 1st |` ( ( ( 1st ` x ) ( Hom ` C ) ( 1st ` y ) ) X. ( ( 2nd ` x ) ( Hom ` D ) ( 2nd ` y ) ) ) ) )
48	47	feq1d 5714	. . . . . 6 |- ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` D ) ) /\ y e. ( ( Base ` C ) X. ( Base ` D ) ) ) ) -> ( ( x ( 2nd ` P ) y ) : ( ( ( 1st ` x ) ( Hom ` C ) ( 1st ` y ) ) X. ( ( 2nd ` x ) ( Hom ` D ) ( 2nd ` y ) ) ) --> ( ( 1st ` x ) ( Hom ` C ) ( 1st ` y ) ) <-> ( 1st |` ( ( ( 1st ` x ) ( Hom ` C ) ( 1st ` y ) ) X. ( ( 2nd ` x ) ( Hom ` D ) ( 2nd ` y ) ) ) ) : ( ( ( 1st ` x ) ( Hom ` C ) ( 1st ` y ) ) X. ( ( 2nd ` x ) ( Hom ` D ) ( 2nd ` y ) ) ) --> ( ( 1st ` x ) ( Hom ` C ) ( 1st ` y ) ) ) )
49	38, 48	mpbiri 237	. . . . 5 |- ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` D ) ) /\ y e. ( ( Base ` C ) X. ( Base ` D ) ) ) ) -> ( x ( 2nd ` P ) y ) : ( ( ( 1st ` x ) ( Hom ` C ) ( 1st ` y ) ) X. ( ( 2nd ` x ) ( Hom ` D ) ( 2nd ` y ) ) ) --> ( ( 1st ` x ) ( Hom ` C ) ( 1st ` y ) ) )
50		fvres 5879	. . . . . . . 8 |- ( x e. ( ( Base ` C ) X. ( Base ` D ) ) -> ( ( 1st |` ( ( Base ` C ) X. ( Base ` D ) ) ) ` x ) = ( 1st ` x ) )
51	50	ad2antrl 734	. . . . . . 7 |- ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` D ) ) /\ y e. ( ( Base ` C ) X. ( Base ` D ) ) ) ) -> ( ( 1st |` ( ( Base ` C ) X. ( Base ` D ) ) ) ` x ) = ( 1st ` x ) )
52		fvres 5879	. . . . . . . 8 |- ( y e. ( ( Base ` C ) X. ( Base ` D ) ) -> ( ( 1st |` ( ( Base ` C ) X. ( Base ` D ) ) ) ` y ) = ( 1st ` y ) )
53	52	ad2antll 735	. . . . . . 7 |- ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` D ) ) /\ y e. ( ( Base ` C ) X. ( Base ` D ) ) ) ) -> ( ( 1st |` ( ( Base ` C ) X. ( Base ` D ) ) ) ` y ) = ( 1st ` y ) )
54	51, 53	oveq12d 6308	. . . . . 6 |- ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` D ) ) /\ y e. ( ( Base ` C ) X. ( Base ` D ) ) ) ) -> ( ( ( 1st |` ( ( Base ` C ) X. ( Base ` D ) ) ) ` x ) ( Hom ` C ) ( ( 1st |` ( ( Base ` C ) X. ( Base ` D ) ) ) ` y ) ) = ( ( 1st ` x ) ( Hom ` C ) ( 1st ` y ) ) )
55	45, 54	feq23d 5723	. . . . 5 |- ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` D ) ) /\ y e. ( ( Base ` C ) X. ( Base ` D ) ) ) ) -> ( ( x ( 2nd ` P ) y ) : ( x ( Hom ` T ) y ) --> ( ( ( 1st |` ( ( Base ` C ) X. ( Base ` D ) ) ) ` x ) ( Hom ` C ) ( ( 1st |` ( ( Base ` C ) X. ( Base ` D ) ) ) ` y ) ) <-> ( x ( 2nd ` P ) y ) : ( ( ( 1st ` x ) ( Hom ` C ) ( 1st ` y ) ) X. ( ( 2nd ` x ) ( Hom ` D ) ( 2nd ` y ) ) ) --> ( ( 1st ` x ) ( Hom ` C ) ( 1st ` y ) ) ) )
56	49, 55	mpbird 236	. . . 4 |- ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` D ) ) /\ y e. ( ( Base ` C ) X. ( Base ` D ) ) ) ) -> ( x ( 2nd ` P ) y ) : ( x ( Hom ` T ) y ) --> ( ( ( 1st |` ( ( Base ` C ) X. ( Base ` D ) ) ) ` x ) ( Hom ` C ) ( ( 1st |` ( ( Base ` C ) X. ( Base ` D ) ) ) ` y ) ) )
57	28	adantr 467	. . . . . . . 8 |- ( ( ph /\ x e. ( ( Base ` C ) X. ( Base ` D ) ) ) -> T e. Cat )
58		simpr 463	. . . . . . . 8 |- ( ( ph /\ x e. ( ( Base ` C ) X. ( Base ` D ) ) ) -> x e. ( ( Base ` C ) X. ( Base ` D ) ) )
59	4, 5, 24, 57, 58	catidcl 15588	. . . . . . 7 |- ( ( ph /\ x e. ( ( Base ` C ) X. ( Base ` D ) ) ) -> ( ( Id ` T ) ` x ) e. ( x ( Hom ` T ) x ) )
60		fvres 5879	. . . . . . 7 |- ( ( ( Id ` T ) ` x ) e. ( x ( Hom ` T ) x ) -> ( ( 1st |` ( x ( Hom ` T ) x ) ) ` ( ( Id ` T ) ` x ) ) = ( 1st ` ( ( Id ` T ) ` x ) ) )
61	59, 60	syl 17	. . . . . 6 |- ( ( ph /\ x e. ( ( Base ` C ) X. ( Base ` D ) ) ) -> ( ( 1st |` ( x ( Hom ` T ) x ) ) ` ( ( Id ` T ) ` x ) ) = ( 1st ` ( ( Id ` T ) ` x ) ) )
62		1st2nd2 6830	. . . . . . . . . 10 |- ( x e. ( ( Base ` C ) X. ( Base ` D ) ) -> x = <. ( 1st ` x ) , ( 2nd ` x ) >. )
63	62	adantl 468	. . . . . . . . 9 |- ( ( ph /\ x e. ( ( Base ` C ) X. ( Base ` D ) ) ) -> x = <. ( 1st ` x ) , ( 2nd ` x ) >. )
64	63	fveq2d 5869	. . . . . . . 8 |- ( ( ph /\ x e. ( ( Base ` C ) X. ( Base ` D ) ) ) -> ( ( Id ` T ) ` x ) = ( ( Id ` T ) ` <. ( 1st ` x ) , ( 2nd ` x ) >. ) )
65	6	adantr 467	. . . . . . . . 9 |- ( ( ph /\ x e. ( ( Base ` C ) X. ( Base ` D ) ) ) -> C e. Cat )
66	7	adantr 467	. . . . . . . . 9 |- ( ( ph /\ x e. ( ( Base ` C ) X. ( Base ` D ) ) ) -> D e. Cat )
67		eqid 2451	. . . . . . . . 9 |- ( Id ` D ) = ( Id ` D )
68		xp1st 6823	. . . . . . . . . 10 |- ( x e. ( ( Base ` C ) X. ( Base ` D ) ) -> ( 1st ` x ) e. ( Base ` C ) )
69	68	adantl 468	. . . . . . . . 9 |- ( ( ph /\ x e. ( ( Base ` C ) X. ( Base ` D ) ) ) -> ( 1st ` x ) e. ( Base ` C ) )
70		xp2nd 6824	. . . . . . . . . 10 |- ( x e. ( ( Base ` C ) X. ( Base ` D ) ) -> ( 2nd ` x ) e. ( Base ` D ) )
71	70	adantl 468	. . . . . . . . 9 |- ( ( ph /\ x e. ( ( Base ` C ) X. ( Base ` D ) ) ) -> ( 2nd ` x ) e. ( Base ` D ) )
72	1, 65, 66, 2, 3, 25, 67, 24, 69, 71	xpcid 16074	. . . . . . . 8 |- ( ( ph /\ x e. ( ( Base ` C ) X. ( Base ` D ) ) ) -> ( ( Id ` T ) ` <. ( 1st ` x ) , ( 2nd ` x ) >. ) = <. ( ( Id ` C ) ` ( 1st ` x ) ) , ( ( Id ` D ) ` ( 2nd ` x ) ) >. )
73	64, 72	eqtrd 2485	. . . . . . 7 |- ( ( ph /\ x e. ( ( Base ` C ) X. ( Base ` D ) ) ) -> ( ( Id ` T ) ` x ) = <. ( ( Id ` C ) ` ( 1st ` x ) ) , ( ( Id ` D ) ` ( 2nd ` x ) ) >. )
74		fvex 5875	. . . . . . . 8 |- ( ( Id ` C ) ` ( 1st ` x ) ) e. _V
75		fvex 5875	. . . . . . . 8 |- ( ( Id ` D ) ` ( 2nd ` x ) ) e. _V
76	74, 75	op1std 6803	. . . . . . 7 |- ( ( ( Id ` T ) ` x ) = <. ( ( Id ` C ) ` ( 1st ` x ) ) , ( ( Id ` D ) ` ( 2nd ` x ) ) >. -> ( 1st ` ( ( Id ` T ) ` x ) ) = ( ( Id ` C ) ` ( 1st ` x ) ) )
77	73, 76	syl 17	. . . . . 6 |- ( ( ph /\ x e. ( ( Base ` C ) X. ( Base ` D ) ) ) -> ( 1st ` ( ( Id ` T ) ` x ) ) = ( ( Id ` C ) ` ( 1st ` x ) ) )
78	61, 77	eqtrd 2485	. . . . 5 |- ( ( ph /\ x e. ( ( Base ` C ) X. ( Base ` D ) ) ) -> ( ( 1st |` ( x ( Hom ` T ) x ) ) ` ( ( Id ` T ) ` x ) ) = ( ( Id ` C ) ` ( 1st ` x ) ) )
79	1, 4, 5, 65, 66, 8, 58, 58	1stf2 16078	. . . . . 6 |- ( ( ph /\ x e. ( ( Base ` C ) X. ( Base ` D ) ) ) -> ( x ( 2nd ` P ) x ) = ( 1st |` ( x ( Hom ` T ) x ) ) )
80	79	fveq1d 5867	. . . . 5 |- ( ( ph /\ x e. ( ( Base ` C ) X. ( Base ` D ) ) ) -> ( ( x ( 2nd ` P ) x ) ` ( ( Id ` T ) ` x ) ) = ( ( 1st |` ( x ( Hom ` T ) x ) ) ` ( ( Id ` T ) ` x ) ) )
81	50	adantl 468	. . . . . 6 |- ( ( ph /\ x e. ( ( Base ` C ) X. ( Base ` D ) ) ) -> ( ( 1st |` ( ( Base ` C ) X. ( Base ` D ) ) ) ` x ) = ( 1st ` x ) )
82	81	fveq2d 5869	. . . . 5 |- ( ( ph /\ x e. ( ( Base ` C ) X. ( Base ` D ) ) ) -> ( ( Id ` C ) ` ( ( 1st |` ( ( Base ` C ) X. ( Base ` D ) ) ) ` x ) ) = ( ( Id ` C ) ` ( 1st ` x ) ) )
83	78, 80, 82	3eqtr4d 2495	. . . 4 |- ( ( ph /\ x e. ( ( Base ` C ) X. ( Base ` D ) ) ) -> ( ( x ( 2nd ` P ) x ) ` ( ( Id ` T ) ` x ) ) = ( ( Id ` C ) ` ( ( 1st |` ( ( Base ` C ) X. ( Base ` D ) ) ) ` x ) ) )
84	28	3ad2ant1 1029	. . . . . . . 8 |- ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` D ) ) /\ y e. ( ( Base ` C ) X. ( Base ` D ) ) /\ z e. ( ( Base ` C ) X. ( Base ` D ) ) ) /\ ( f e. ( x ( Hom ` T ) y ) /\ g e. ( y ( Hom ` T ) z ) ) ) -> T e. Cat )
85		simp21 1041	. . . . . . . 8 |- ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` D ) ) /\ y e. ( ( Base ` C ) X. ( Base ` D ) ) /\ z e. ( ( Base ` C ) X. ( Base ` D ) ) ) /\ ( f e. ( x ( Hom ` T ) y ) /\ g e. ( y ( Hom ` T ) z ) ) ) -> x e. ( ( Base ` C ) X. ( Base ` D ) ) )
86		simp22 1042	. . . . . . . 8 |- ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` D ) ) /\ y e. ( ( Base ` C ) X. ( Base ` D ) ) /\ z e. ( ( Base ` C ) X. ( Base ` D ) ) ) /\ ( f e. ( x ( Hom ` T ) y ) /\ g e. ( y ( Hom ` T ) z ) ) ) -> y e. ( ( Base ` C ) X. ( Base ` D ) ) )
87		simp23 1043	. . . . . . . 8 |- ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` D ) ) /\ y e. ( ( Base ` C ) X. ( Base ` D ) ) /\ z e. ( ( Base ` C ) X. ( Base ` D ) ) ) /\ ( f e. ( x ( Hom ` T ) y ) /\ g e. ( y ( Hom ` T ) z ) ) ) -> z e. ( ( Base ` C ) X. ( Base ` D ) ) )
88		simp3l 1036	. . . . . . . 8 |- ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` D ) ) /\ y e. ( ( Base ` C ) X. ( Base ` D ) ) /\ z e. ( ( Base ` C ) X. ( Base ` D ) ) ) /\ ( f e. ( x ( Hom ` T ) y ) /\ g e. ( y ( Hom ` T ) z ) ) ) -> f e. ( x ( Hom ` T ) y ) )
89		simp3r 1037	. . . . . . . 8 |- ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` D ) ) /\ y e. ( ( Base ` C ) X. ( Base ` D ) ) /\ z e. ( ( Base ` C ) X. ( Base ` D ) ) ) /\ ( f e. ( x ( Hom ` T ) y ) /\ g e. ( y ( Hom ` T ) z ) ) ) -> g e. ( y ( Hom ` T ) z ) )
90	4, 5, 26, 84, 85, 86, 87, 88, 89	catcocl 15591	. . . . . . 7 |- ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` D ) ) /\ y e. ( ( Base ` C ) X. ( Base ` D ) ) /\ z e. ( ( Base ` C ) X. ( Base ` D ) ) ) /\ ( f e. ( x ( Hom ` T ) y ) /\ g e. ( y ( Hom ` T ) z ) ) ) -> ( g ( <. x , y >. (comp ` T ) z ) f ) e. ( x ( Hom ` T ) z ) )
91		fvres 5879	. . . . . . 7 |- ( ( g ( <. x , y >. (comp ` T ) z ) f ) e. ( x ( Hom ` T ) z ) -> ( ( 1st |` ( x ( Hom ` T ) z ) ) ` ( g ( <. x , y >. (comp ` T ) z ) f ) ) = ( 1st ` ( g ( <. x , y >. (comp ` T ) z ) f ) ) )
92	90, 91	syl 17	. . . . . 6 |- ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` D ) ) /\ y e. ( ( Base ` C ) X. ( Base ` D ) ) /\ z e. ( ( Base ` C ) X. ( Base ` D ) ) ) /\ ( f e. ( x ( Hom ` T ) y ) /\ g e. ( y ( Hom ` T ) z ) ) ) -> ( ( 1st |` ( x ( Hom ` T ) z ) ) ` ( g ( <. x , y >. (comp ` T ) z ) f ) ) = ( 1st ` ( g ( <. x , y >. (comp ` T ) z ) f ) ) )
93	1, 4, 5, 26, 85, 86, 87, 88, 89, 27	xpcco1st 16069	. . . . . 6 |- ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` D ) ) /\ y e. ( ( Base ` C ) X. ( Base ` D ) ) /\ z e. ( ( Base ` C ) X. ( Base ` D ) ) ) /\ ( f e. ( x ( Hom ` T ) y ) /\ g e. ( y ( Hom ` T ) z ) ) ) -> ( 1st ` ( g ( <. x , y >. (comp ` T ) z ) f ) ) = ( ( 1st ` g ) ( <. ( 1st ` x ) , ( 1st ` y ) >. (comp ` C ) ( 1st ` z ) ) ( 1st ` f ) ) )
94	92, 93	eqtrd 2485	. . . . 5 |- ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` D ) ) /\ y e. ( ( Base ` C ) X. ( Base ` D ) ) /\ z e. ( ( Base ` C ) X. ( Base ` D ) ) ) /\ ( f e. ( x ( Hom ` T ) y ) /\ g e. ( y ( Hom ` T ) z ) ) ) -> ( ( 1st |` ( x ( Hom ` T ) z ) ) ` ( g ( <. x , y >. (comp ` T ) z ) f ) ) = ( ( 1st ` g ) ( <. ( 1st ` x ) , ( 1st ` y ) >. (comp ` C ) ( 1st ` z ) ) ( 1st ` f ) ) )
95	6	3ad2ant1 1029	. . . . . . 7 |- ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` D ) ) /\ y e. ( ( Base ` C ) X. ( Base ` D ) ) /\ z e. ( ( Base ` C ) X. ( Base ` D ) ) ) /\ ( f e. ( x ( Hom ` T ) y ) /\ g e. ( y ( Hom ` T ) z ) ) ) -> C e. Cat )
96	7	3ad2ant1 1029	. . . . . . 7 |- ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` D ) ) /\ y e. ( ( Base ` C ) X. ( Base ` D ) ) /\ z e. ( ( Base ` C ) X. ( Base ` D ) ) ) /\ ( f e. ( x ( Hom ` T ) y ) /\ g e. ( y ( Hom ` T ) z ) ) ) -> D e. Cat )
97	1, 4, 5, 95, 96, 8, 85, 87	1stf2 16078	. . . . . 6 |- ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` D ) ) /\ y e. ( ( Base ` C ) X. ( Base ` D ) ) /\ z e. ( ( Base ` C ) X. ( Base ` D ) ) ) /\ ( f e. ( x ( Hom ` T ) y ) /\ g e. ( y ( Hom ` T ) z ) ) ) -> ( x ( 2nd ` P ) z ) = ( 1st |` ( x ( Hom ` T ) z ) ) )
98	97	fveq1d 5867	. . . . 5 |- ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` D ) ) /\ y e. ( ( Base ` C ) X. ( Base ` D ) ) /\ z e. ( ( Base ` C ) X. ( Base ` D ) ) ) /\ ( f e. ( x ( Hom ` T ) y ) /\ g e. ( y ( Hom ` T ) z ) ) ) -> ( ( x ( 2nd ` P ) z ) ` ( g ( <. x , y >. (comp ` T ) z ) f ) ) = ( ( 1st |` ( x ( Hom ` T ) z ) ) ` ( g ( <. x , y >. (comp ` T ) z ) f ) ) )
99	85, 50	syl 17	. . . . . . . 8 |- ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` D ) ) /\ y e. ( ( Base ` C ) X. ( Base ` D ) ) /\ z e. ( ( Base ` C ) X. ( Base ` D ) ) ) /\ ( f e. ( x ( Hom ` T ) y ) /\ g e. ( y ( Hom ` T ) z ) ) ) -> ( ( 1st |` ( ( Base ` C ) X. ( Base ` D ) ) ) ` x ) = ( 1st ` x ) )
100	86, 52	syl 17	. . . . . . . 8 |- ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` D ) ) /\ y e. ( ( Base ` C ) X. ( Base ` D ) ) /\ z e. ( ( Base ` C ) X. ( Base ` D ) ) ) /\ ( f e. ( x ( Hom ` T ) y ) /\ g e. ( y ( Hom ` T ) z ) ) ) -> ( ( 1st |` ( ( Base ` C ) X. ( Base ` D ) ) ) ` y ) = ( 1st ` y ) )
101	99, 100	opeq12d 4174	. . . . . . 7 |- ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` D ) ) /\ y e. ( ( Base ` C ) X. ( Base ` D ) ) /\ z e. ( ( Base ` C ) X. ( Base ` D ) ) ) /\ ( f e. ( x ( Hom ` T ) y ) /\ g e. ( y ( Hom ` T ) z ) ) ) -> <. ( ( 1st |` ( ( Base ` C ) X. ( Base ` D ) ) ) ` x ) , ( ( 1st |` ( ( Base ` C ) X. ( Base ` D ) ) ) ` y ) >. = <. ( 1st ` x ) , ( 1st ` y ) >. )
102		fvres 5879	. . . . . . . 8 |- ( z e. ( ( Base ` C ) X. ( Base ` D ) ) -> ( ( 1st |` ( ( Base ` C ) X. ( Base ` D ) ) ) ` z ) = ( 1st ` z ) )
103	87, 102	syl 17	. . . . . . 7 |- ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` D ) ) /\ y e. ( ( Base ` C ) X. ( Base ` D ) ) /\ z e. ( ( Base ` C ) X. ( Base ` D ) ) ) /\ ( f e. ( x ( Hom ` T ) y ) /\ g e. ( y ( Hom ` T ) z ) ) ) -> ( ( 1st |` ( ( Base ` C ) X. ( Base ` D ) ) ) ` z ) = ( 1st ` z ) )
104	101, 103	oveq12d 6308	. . . . . 6 |- ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` D ) ) /\ y e. ( ( Base ` C ) X. ( Base ` D ) ) /\ z e. ( ( Base ` C ) X. ( Base ` D ) ) ) /\ ( f e. ( x ( Hom ` T ) y ) /\ g e. ( y ( Hom ` T ) z ) ) ) -> ( <. ( ( 1st |` ( ( Base ` C ) X. ( Base ` D ) ) ) ` x ) , ( ( 1st |` ( ( Base ` C ) X. ( Base ` D ) ) ) ` y ) >. (comp ` C ) ( ( 1st |` ( ( Base ` C ) X. ( Base ` D ) ) ) ` z ) ) = ( <. ( 1st ` x ) , ( 1st ` y ) >. (comp ` C ) ( 1st ` z ) ) )
105	1, 4, 5, 95, 96, 8, 86, 87	1stf2 16078	. . . . . . . 8 |- ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` D ) ) /\ y e. ( ( Base ` C ) X. ( Base ` D ) ) /\ z e. ( ( Base ` C ) X. ( Base ` D ) ) ) /\ ( f e. ( x ( Hom ` T ) y ) /\ g e. ( y ( Hom ` T ) z ) ) ) -> ( y ( 2nd ` P ) z ) = ( 1st |` ( y ( Hom ` T ) z ) ) )
106	105	fveq1d 5867	. . . . . . 7 |- ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` D ) ) /\ y e. ( ( Base ` C ) X. ( Base ` D ) ) /\ z e. ( ( Base ` C ) X. ( Base ` D ) ) ) /\ ( f e. ( x ( Hom ` T ) y ) /\ g e. ( y ( Hom ` T ) z ) ) ) -> ( ( y ( 2nd ` P ) z ) ` g ) = ( ( 1st |` ( y ( Hom ` T ) z ) ) ` g ) )
107		fvres 5879	. . . . . . . 8 |- ( g e. ( y ( Hom ` T ) z ) -> ( ( 1st |` ( y ( Hom ` T ) z ) ) ` g ) = ( 1st ` g ) )
108	89, 107	syl 17	. . . . . . 7 |- ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` D ) ) /\ y e. ( ( Base ` C ) X. ( Base ` D ) ) /\ z e. ( ( Base ` C ) X. ( Base ` D ) ) ) /\ ( f e. ( x ( Hom ` T ) y ) /\ g e. ( y ( Hom ` T ) z ) ) ) -> ( ( 1st |` ( y ( Hom ` T ) z ) ) ` g ) = ( 1st ` g ) )
109	106, 108	eqtrd 2485	. . . . . 6 |- ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` D ) ) /\ y e. ( ( Base ` C ) X. ( Base ` D ) ) /\ z e. ( ( Base ` C ) X. ( Base ` D ) ) ) /\ ( f e. ( x ( Hom ` T ) y ) /\ g e. ( y ( Hom ` T ) z ) ) ) -> ( ( y ( 2nd ` P ) z ) ` g ) = ( 1st ` g ) )
110	1, 4, 5, 95, 96, 8, 85, 86	1stf2 16078	. . . . . . . 8 |- ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` D ) ) /\ y e. ( ( Base ` C ) X. ( Base ` D ) ) /\ z e. ( ( Base ` C ) X. ( Base ` D ) ) ) /\ ( f e. ( x ( Hom ` T ) y ) /\ g e. ( y ( Hom ` T ) z ) ) ) -> ( x ( 2nd ` P ) y ) = ( 1st |` ( x ( Hom ` T ) y ) ) )
111	110	fveq1d 5867	. . . . . . 7 |- ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` D ) ) /\ y e. ( ( Base ` C ) X. ( Base ` D ) ) /\ z e. ( ( Base ` C ) X. ( Base ` D ) ) ) /\ ( f e. ( x ( Hom ` T ) y ) /\ g e. ( y ( Hom ` T ) z ) ) ) -> ( ( x ( 2nd ` P ) y ) ` f ) = ( ( 1st |` ( x ( Hom ` T ) y ) ) ` f ) )
112		fvres 5879	. . . . . . . 8 |- ( f e. ( x ( Hom ` T ) y ) -> ( ( 1st |` ( x ( Hom ` T ) y ) ) ` f ) = ( 1st ` f ) )
113	88, 112	syl 17	. . . . . . 7 |- ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` D ) ) /\ y e. ( ( Base ` C ) X. ( Base ` D ) ) /\ z e. ( ( Base ` C ) X. ( Base ` D ) ) ) /\ ( f e. ( x ( Hom ` T ) y ) /\ g e. ( y ( Hom ` T ) z ) ) ) -> ( ( 1st |` ( x ( Hom ` T ) y ) ) ` f ) = ( 1st ` f ) )
114	111, 113	eqtrd 2485	. . . . . 6 |- ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` D ) ) /\ y e. ( ( Base ` C ) X. ( Base ` D ) ) /\ z e. ( ( Base ` C ) X. ( Base ` D ) ) ) /\ ( f e. ( x ( Hom ` T ) y ) /\ g e. ( y ( Hom ` T ) z ) ) ) -> ( ( x ( 2nd ` P ) y ) ` f ) = ( 1st ` f ) )
115	104, 109, 114	oveq123d 6311	. . . . 5 |- ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` D ) ) /\ y e. ( ( Base ` C ) X. ( Base ` D ) ) /\ z e. ( ( Base ` C ) X. ( Base ` D ) ) ) /\ ( f e. ( x ( Hom ` T ) y ) /\ g e. ( y ( Hom ` T ) z ) ) ) -> ( ( ( y ( 2nd ` P ) z ) ` g ) ( <. ( ( 1st |` ( ( Base ` C ) X. ( Base ` D ) ) ) ` x ) , ( ( 1st |` ( ( Base ` C ) X. ( Base ` D ) ) ) ` y ) >. (comp ` C ) ( ( 1st |` ( ( Base ` C ) X. ( Base ` D ) ) ) ` z ) ) ( ( x ( 2nd ` P ) y ) ` f ) ) = ( ( 1st ` g ) ( <. ( 1st ` x ) , ( 1st ` y ) >. (comp ` C ) ( 1st ` z ) ) ( 1st ` f ) ) )
116	94, 98, 115	3eqtr4d 2495	. . . 4 |- ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` D ) ) /\ y e. ( ( Base ` C ) X. ( Base ` D ) ) /\ z e. ( ( Base ` C ) X. ( Base ` D ) ) ) /\ ( f e. ( x ( Hom ` T ) y ) /\ g e. ( y ( Hom ` T ) z ) ) ) -> ( ( x ( 2nd ` P ) z ) ` ( g ( <. x , y >. (comp ` T ) z ) f ) ) = ( ( ( y ( 2nd ` P ) z ) ` g ) ( <. ( ( 1st |` ( ( Base ` C ) X. ( Base ` D ) ) ) ` x ) , ( ( 1st |` ( ( Base ` C ) X. ( Base ` D ) ) ) ` y ) >. (comp ` C ) ( ( 1st |` ( ( Base ` C ) X. ( Base ` D ) ) ) ` z ) ) ( ( x ( 2nd ` P ) y ) ` f ) ) )
117	4, 2, 5, 23, 24, 25, 26, 27, 28, 6, 30, 37, 56, 83, 116	isfuncd 15770	. . 3 |- ( ph -> ( 1st |` ( ( Base ` C ) X. ( Base ` D ) ) ) ( T Func C ) ( 2nd ` P ) )
118		df-br 4403	. . 3 |- ( ( 1st |` ( ( Base ` C ) X. ( Base ` D ) ) ) ( T Func C ) ( 2nd ` P ) <-> <. ( 1st |` ( ( Base ` C ) X. ( Base ` D ) ) ) , ( 2nd ` P ) >. e. ( T Func C ) )
119	117, 118	sylib 200	. 2 |- ( ph -> <. ( 1st |` ( ( Base ` C ) X. ( Base ` D ) ) ) , ( 2nd ` P ) >. e. ( T Func C ) )
120	22, 119	eqeltrd 2529	1 |- ( ph -> P e. ( T Func C ) )

Syntax hints:   -> wi 4   /\ wa 371   /\ w3a 985   = wceq 1444   e. wcel 1887   _Vcvv 3045   <.cop 3974   class class class wbr 4402   X. cxp 4832   |` cres 4836   Fun wfun 5576   Fn wfn 5577   -->wf 5578   -onto->wfo 5580   ` cfv 5582  (class class class)co 6290   |-> cmpt2 6292   1stc1st 6791   2ndc2nd 6792   Basecbs 15121   Hom chom 15201  compcco 15202   Catccat 15570   Idccid 15571   Func cfunc 15759   X._c cxpc 16053   1st_F c1stf 16054

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-fal 1450  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-oadd 7186  df-er 7363  df-map 7474  df-ixp 7523  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-fz 11785  df-struct 15123  df-ndx 15124  df-slot 15125  df-base 15126  df-hom 15214  df-cco 15215  df-cat 15574  df-cid 15575  df-func 15763  df-xpc 16057  df-1stf 16058

This theorem is referenced by:  prf1st  16089  1st2ndprf  16091  uncfcl  16120  uncf1  16121  uncf2  16122  diagcl  16126  diag11  16128  diag12  16129  diag2  16130  yonedalem1  16157  yonedalem21  16158  yonedalem22  16163