Theorem 1stfcl 15682

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 2402	. . . . 5 |- ( Base ` C ) = ( Base ` C )
3		eqid 2402	. . . . 5 |- ( Base ` D ) = ( Base ` D )
4	1, 2, 3	xpcbas 15663	. . . 4 |- ( ( Base ` C ) X. ( Base ` D ) ) = ( Base ` T )
5		eqid 2402	. . . 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 15676	. . 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 6758	. . . . . . . 8 |- 1st : _V -onto-> _V
11		fofun 5735	. . . . . . . 8 |- ( 1st : _V -onto-> _V -> Fun 1st )
12	10, 11	ax-mp 5	. . . . . . 7 |- Fun 1st
13		fvex 5815	. . . . . . . 8 |- ( Base ` C ) e. _V
14		fvex 5815	. . . . . . . 8 |- ( Base ` D ) e. _V
15	13, 14	xpex 6542	. . . . . . 7 |- ( ( Base ` C ) X. ( Base ` D ) ) e. _V
16		resfunexg 6074	. . . . . . 7 |- ( ( Fun 1st /\ ( ( Base ` C ) X. ( Base ` D ) ) e. _V ) -> ( 1st |` ( ( Base ` C ) X. ( Base ` D ) ) ) e. _V )
17	12, 15, 16	mp2an 670	. . . . . 6 |- ( 1st |` ( ( Base ` C ) X. ( Base ` D ) ) ) e. _V
18	15, 15	mpt2ex 6815	. . . . . 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 6749	. . . . 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 4165	. . 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 2446	. 2 |- ( ph -> P = <. ( 1st |` ( ( Base ` C ) X. ( Base ` D ) ) ) , ( 2nd ` P ) >. )
23		eqid 2402	. . . 4 |- ( Hom ` C ) = ( Hom ` C )
24		eqid 2402	. . . 4 |- ( Id ` T ) = ( Id ` T )
25		eqid 2402	. . . 4 |- ( Id ` C ) = ( Id ` C )
26		eqid 2402	. . . 4 |- (comp ` T ) = (comp ` T )
27		eqid 2402	. . . 4 |- (comp ` C ) = (comp ` C )
28	1, 6, 7	xpccat 15675	. . . 4 |- ( ph -> T e. Cat )
29		f1stres 6760	. . . . 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 2402	. . . . . 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 6262	. . . . . . 7 |- ( x ( Hom ` T ) y ) e. _V
33		resfunexg 6074	. . . . . . 7 |- ( ( Fun 1st /\ ( x ( Hom ` T ) y ) e. _V ) -> ( 1st |` ( x ( Hom ` T ) y ) ) e. _V )
34	12, 32, 33	mp2an 670	. . . . . 6 |- ( 1st |` ( x ( Hom ` T ) y ) ) e. _V
35	31, 34	fnmpt2i 6807	. . . . 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 5608	. . . . 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 233	. . . 4 |- ( ph -> ( 2nd ` P ) Fn ( ( ( Base ` C ) X. ( Base ` D ) ) X. ( ( Base ` C ) X. ( Base ` D ) ) ) )
38		f1stres 6760	. . . . . 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 463	. . . . . . . . 9 |- ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` D ) ) /\ y e. ( ( Base ` C ) X. ( Base ` D ) ) ) ) -> C e. Cat )
40	7	adantr 463	. . . . . . . . 9 |- ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` D ) ) /\ y e. ( ( Base ` C ) X. ( Base ` D ) ) ) ) -> D e. Cat )
41		simprl 756	. . . . . . . . 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 758	. . . . . . . . 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 15678	. . . . . . . 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 2402	. . . . . . . . . 10 |- ( Hom ` D ) = ( Hom ` D )
45	1, 4, 23, 44, 5, 41, 42	xpchom 15665	. . . . . . . . 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 5215	. . . . . . . 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 2443	. . . . . . 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 5656	. . . . . 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 233	. . . . 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 5819	. . . . . . . 8 |- ( x e. ( ( Base ` C ) X. ( Base ` D ) ) -> ( ( 1st |` ( ( Base ` C ) X. ( Base ` D ) ) ) ` x ) = ( 1st ` x ) )
51	50	ad2antrl 726	. . . . . . 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 5819	. . . . . . . 8 |- ( y e. ( ( Base ` C ) X. ( Base ` D ) ) -> ( ( 1st |` ( ( Base ` C ) X. ( Base ` D ) ) ) ` y ) = ( 1st ` y ) )
53	52	ad2antll 727	. . . . . . 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 6252	. . . . . 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 5665	. . . . 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 232	. . . 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 463	. . . . . . . 8 |- ( ( ph /\ x e. ( ( Base ` C ) X. ( Base ` D ) ) ) -> T e. Cat )
58		simpr 459	. . . . . . . 8 |- ( ( ph /\ x e. ( ( Base ` C ) X. ( Base ` D ) ) ) -> x e. ( ( Base ` C ) X. ( Base ` D ) ) )
59	4, 5, 24, 57, 58	catidcl 15188	. . . . . . 7 |- ( ( ph /\ x e. ( ( Base ` C ) X. ( Base ` D ) ) ) -> ( ( Id ` T ) ` x ) e. ( x ( Hom ` T ) x ) )
60		fvres 5819	. . . . . . 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 6775	. . . . . . . . . 10 |- ( x e. ( ( Base ` C ) X. ( Base ` D ) ) -> x = <. ( 1st ` x ) , ( 2nd ` x ) >. )
63	62	adantl 464	. . . . . . . . 9 |- ( ( ph /\ x e. ( ( Base ` C ) X. ( Base ` D ) ) ) -> x = <. ( 1st ` x ) , ( 2nd ` x ) >. )
64	63	fveq2d 5809	. . . . . . . 8 |- ( ( ph /\ x e. ( ( Base ` C ) X. ( Base ` D ) ) ) -> ( ( Id ` T ) ` x ) = ( ( Id ` T ) ` <. ( 1st ` x ) , ( 2nd ` x ) >. ) )
65	6	adantr 463	. . . . . . . . 9 |- ( ( ph /\ x e. ( ( Base ` C ) X. ( Base ` D ) ) ) -> C e. Cat )
66	7	adantr 463	. . . . . . . . 9 |- ( ( ph /\ x e. ( ( Base ` C ) X. ( Base ` D ) ) ) -> D e. Cat )
67		eqid 2402	. . . . . . . . 9 |- ( Id ` D ) = ( Id ` D )
68		xp1st 6768	. . . . . . . . . 10 |- ( x e. ( ( Base ` C ) X. ( Base ` D ) ) -> ( 1st ` x ) e. ( Base ` C ) )
69	68	adantl 464	. . . . . . . . 9 |- ( ( ph /\ x e. ( ( Base ` C ) X. ( Base ` D ) ) ) -> ( 1st ` x ) e. ( Base ` C ) )
70		xp2nd 6769	. . . . . . . . . 10 |- ( x e. ( ( Base ` C ) X. ( Base ` D ) ) -> ( 2nd ` x ) e. ( Base ` D ) )
71	70	adantl 464	. . . . . . . . 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 15674	. . . . . . . 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 2443	. . . . . . 7 |- ( ( ph /\ x e. ( ( Base ` C ) X. ( Base ` D ) ) ) -> ( ( Id ` T ) ` x ) = <. ( ( Id ` C ) ` ( 1st ` x ) ) , ( ( Id ` D ) ` ( 2nd ` x ) ) >. )
74		fvex 5815	. . . . . . . 8 |- ( ( Id ` C ) ` ( 1st ` x ) ) e. _V
75		fvex 5815	. . . . . . . 8 |- ( ( Id ` D ) ` ( 2nd ` x ) ) e. _V
76	74, 75	op1std 6748	. . . . . . 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 2443	. . . . 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 15678	. . . . . 6 |- ( ( ph /\ x e. ( ( Base ` C ) X. ( Base ` D ) ) ) -> ( x ( 2nd ` P ) x ) = ( 1st |` ( x ( Hom ` T ) x ) ) )
80	79	fveq1d 5807	. . . . 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 464	. . . . . 6 |- ( ( ph /\ x e. ( ( Base ` C ) X. ( Base ` D ) ) ) -> ( ( 1st |` ( ( Base ` C ) X. ( Base ` D ) ) ) ` x ) = ( 1st ` x ) )
82	81	fveq2d 5809	. . . . 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 2453	. . . 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 1018	. . . . . . . 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 1030	. . . . . . . 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 1031	. . . . . . . 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 1032	. . . . . . . 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 1025	. . . . . . . 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 1026	. . . . . . . 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 15191	. . . . . . 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 5819	. . . . . . 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 15669	. . . . . 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 2443	. . . . 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 1018	. . . . . . 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 1018	. . . . . . 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 15678	. . . . . 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 5807	. . . . 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 4166	. . . . . . 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 5819	. . . . . . . 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 6252	. . . . . 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 15678	. . . . . . . 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 5807	. . . . . . 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 5819	. . . . . . . 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 2443	. . . . . 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 15678	. . . . . . . 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 5807	. . . . . . 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 5819	. . . . . . . 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 2443	. . . . . 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 6255	. . . . 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 2453	. . . 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 15370	. . 3 |- ( ph -> ( 1st |` ( ( Base ` C ) X. ( Base ` D ) ) ) ( T Func C ) ( 2nd ` P ) )
118		df-br 4395	. . 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 196	. 2 |- ( ph -> <. ( 1st |` ( ( Base ` C ) X. ( Base ` D ) ) ) , ( 2nd ` P ) >. e. ( T Func C ) )
120	22, 119	eqeltrd 2490	1 |- ( ph -> P e. ( T Func C ) )

Syntax hints:   -> wi 4   /\ wa 367   /\ w3a 974   = wceq 1405   e. wcel 1842   _Vcvv 3058   <.cop 3977   class class class wbr 4394   X. cxp 4940   |` cres 4944   Fun wfun 5519   Fn wfn 5520   -->wf 5521   -onto->wfo 5523   ` cfv 5525  (class class class)co 6234   |-> cmpt2 6236   1stc1st 6736   2ndc2nd 6737   Basecbs 14733   Hom chom 14812  compcco 14813   Catccat 15170   Idccid 15171   Func cfunc 15359   X._c cxpc 15653   1st_F c1stf 15654

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6530  ax-cnex 9498  ax-resscn 9499  ax-1cn 9500  ax-icn 9501  ax-addcl 9502  ax-addrcl 9503  ax-mulcl 9504  ax-mulrcl 9505  ax-mulcom 9506  ax-addass 9507  ax-mulass 9508  ax-distr 9509  ax-i2m1 9510  ax-1ne0 9511  ax-1rid 9512  ax-rnegex 9513  ax-rrecex 9514  ax-cnre 9515  ax-pre-lttri 9516  ax-pre-lttrn 9517  ax-pre-ltadd 9518  ax-pre-mulgt0 9519

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-fal 1411  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-int 4227  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-we 4783  df-ord 4824  df-on 4825  df-lim 4826  df-suc 4827  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  df-res 4954  df-ima 4955  df-iota 5489  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-riota 6196  df-ov 6237  df-oprab 6238  df-mpt2 6239  df-om 6639  df-1st 6738  df-2nd 6739  df-recs 6999  df-rdg 7033  df-1o 7087  df-oadd 7091  df-er 7268  df-map 7379  df-ixp 7428  df-en 7475  df-dom 7476  df-sdom 7477  df-fin 7478  df-pnf 9580  df-mnf 9581  df-xr 9582  df-ltxr 9583  df-le 9584  df-sub 9763  df-neg 9764  df-nn 10497  df-2 10555  df-3 10556  df-4 10557  df-5 10558  df-6 10559  df-7 10560  df-8 10561  df-9 10562  df-10 10563  df-n0 10757  df-z 10826  df-dec 10940  df-uz 11046  df-fz 11644  df-struct 14735  df-ndx 14736  df-slot 14737  df-base 14738  df-hom 14825  df-cco 14826  df-cat 15174  df-cid 15175  df-func 15363  df-xpc 15657  df-1stf 15658

This theorem is referenced by:  prf1st  15689  1st2ndprf  15691  uncfcl  15720  uncf1  15721  uncf2  15722  diagcl  15726  diag11  15728  diag12  15729  diag2  15730  yonedalem1  15757  yonedalem21  15758  yonedalem22  15763