Theorem 1stfcl 15012

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 2443	. . . . 5 |- ( Base ` C ) = ( Base ` C )
3		eqid 2443	. . . . 5 |- ( Base ` D ) = ( Base ` D )
4	1, 2, 3	xpcbas 14993	. . . 4 |- ( ( Base ` C ) X. ( Base ` D ) ) = ( Base ` T )
5		eqid 2443	. . . 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 15006	. . 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 6601	. . . . . . . 8 |- 1st : _V -onto-> _V
11		fofun 5626	. . . . . . . 8 |- ( 1st : _V -onto-> _V -> Fun 1st )
12	10, 11	ax-mp 5	. . . . . . 7 |- Fun 1st
13		fvex 5706	. . . . . . . 8 |- ( Base ` C ) e. _V
14		fvex 5706	. . . . . . . 8 |- ( Base ` D ) e. _V
15	13, 14	xpex 6513	. . . . . . 7 |- ( ( Base ` C ) X. ( Base ` D ) ) e. _V
16		resfunexg 5948	. . . . . . 7 |- ( ( Fun 1st /\ ( ( Base ` C ) X. ( Base ` D ) ) e. _V ) -> ( 1st |` ( ( Base ` C ) X. ( Base ` D ) ) ) e. _V )
17	12, 15, 16	mp2an 672	. . . . . 6 |- ( 1st |` ( ( Base ` C ) X. ( Base ` D ) ) ) e. _V
18	15, 15	mpt2ex 6655	. . . . . 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 6593	. . . . 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 16	. . . 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 4071	. . 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 2478	. 2 |- ( ph -> P = <. ( 1st |` ( ( Base ` C ) X. ( Base ` D ) ) ) , ( 2nd ` P ) >. )
23		eqid 2443	. . . 4 |- ( Hom ` C ) = ( Hom ` C )
24		eqid 2443	. . . 4 |- ( Id ` T ) = ( Id ` T )
25		eqid 2443	. . . 4 |- ( Id ` C ) = ( Id ` C )
26		eqid 2443	. . . 4 |- (comp ` T ) = (comp ` T )
27		eqid 2443	. . . 4 |- (comp ` C ) = (comp ` C )
28	1, 6, 7	xpccat 15005	. . . 4 |- ( ph -> T e. Cat )
29		f1stres 6603	. . . . 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 2443	. . . . . 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 6121	. . . . . . 7 |- ( x ( Hom ` T ) y ) e. _V
33		resfunexg 5948	. . . . . . 7 |- ( ( Fun 1st /\ ( x ( Hom ` T ) y ) e. _V ) -> ( 1st |` ( x ( Hom ` T ) y ) ) e. _V )
34	12, 32, 33	mp2an 672	. . . . . 6 |- ( 1st |` ( x ( Hom ` T ) y ) ) e. _V
35	31, 34	fnmpt2i 6648	. . . . 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 5506	. . . . 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 6603	. . . . . 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 465	. . . . . . . . 9 |- ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` D ) ) /\ y e. ( ( Base ` C ) X. ( Base ` D ) ) ) ) -> C e. Cat )
40	7	adantr 465	. . . . . . . . 9 |- ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` D ) ) /\ y e. ( ( Base ` C ) X. ( Base ` D ) ) ) ) -> D e. Cat )
41		simprl 755	. . . . . . . . 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 756	. . . . . . . . 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 15008	. . . . . . . 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 2443	. . . . . . . . . 10 |- ( Hom ` D ) = ( Hom ` D )
45	1, 4, 23, 44, 5, 41, 42	xpchom 14995	. . . . . . . . 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 5115	. . . . . . . 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 2475	. . . . . . 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 5551	. . . . . 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 5709	. . . . . . . 8 |- ( x e. ( ( Base ` C ) X. ( Base ` D ) ) -> ( ( 1st |` ( ( Base ` C ) X. ( Base ` D ) ) ) ` x ) = ( 1st ` x ) )
51	50	ad2antrl 727	. . . . . . 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 5709	. . . . . . . 8 |- ( y e. ( ( Base ` C ) X. ( Base ` D ) ) -> ( ( 1st |` ( ( Base ` C ) X. ( Base ` D ) ) ) ` y ) = ( 1st ` y ) )
53	52	ad2antll 728	. . . . . . 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 6114	. . . . . 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 5559	. . . . 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 465	. . . . . . . 8 |- ( ( ph /\ x e. ( ( Base ` C ) X. ( Base ` D ) ) ) -> T e. Cat )
58		simpr 461	. . . . . . . 8 |- ( ( ph /\ x e. ( ( Base ` C ) X. ( Base ` D ) ) ) -> x e. ( ( Base ` C ) X. ( Base ` D ) ) )
59	4, 5, 24, 57, 58	catidcl 14625	. . . . . . 7 |- ( ( ph /\ x e. ( ( Base ` C ) X. ( Base ` D ) ) ) -> ( ( Id ` T ) ` x ) e. ( x ( Hom ` T ) x ) )
60		fvres 5709	. . . . . . 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 16	. . . . . 6 |- ( ( ph /\ x e. ( ( Base ` C ) X. ( Base ` D ) ) ) -> ( ( 1st |` ( x ( Hom ` T ) x ) ) ` ( ( Id ` T ) ` x ) ) = ( 1st ` ( ( Id ` T ) ` x ) ) )
62		1st2nd2 6618	. . . . . . . . . 10 |- ( x e. ( ( Base ` C ) X. ( Base ` D ) ) -> x = <. ( 1st ` x ) , ( 2nd ` x ) >. )
63	62	adantl 466	. . . . . . . . 9 |- ( ( ph /\ x e. ( ( Base ` C ) X. ( Base ` D ) ) ) -> x = <. ( 1st ` x ) , ( 2nd ` x ) >. )
64	63	fveq2d 5700	. . . . . . . 8 |- ( ( ph /\ x e. ( ( Base ` C ) X. ( Base ` D ) ) ) -> ( ( Id ` T ) ` x ) = ( ( Id ` T ) ` <. ( 1st ` x ) , ( 2nd ` x ) >. ) )
65	6	adantr 465	. . . . . . . . 9 |- ( ( ph /\ x e. ( ( Base ` C ) X. ( Base ` D ) ) ) -> C e. Cat )
66	7	adantr 465	. . . . . . . . 9 |- ( ( ph /\ x e. ( ( Base ` C ) X. ( Base ` D ) ) ) -> D e. Cat )
67		eqid 2443	. . . . . . . . 9 |- ( Id ` D ) = ( Id ` D )
68		xp1st 6611	. . . . . . . . . 10 |- ( x e. ( ( Base ` C ) X. ( Base ` D ) ) -> ( 1st ` x ) e. ( Base ` C ) )
69	68	adantl 466	. . . . . . . . 9 |- ( ( ph /\ x e. ( ( Base ` C ) X. ( Base ` D ) ) ) -> ( 1st ` x ) e. ( Base ` C ) )
70		xp2nd 6612	. . . . . . . . . 10 |- ( x e. ( ( Base ` C ) X. ( Base ` D ) ) -> ( 2nd ` x ) e. ( Base ` D ) )
71	70	adantl 466	. . . . . . . . 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 15004	. . . . . . . 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 2475	. . . . . . 7 |- ( ( ph /\ x e. ( ( Base ` C ) X. ( Base ` D ) ) ) -> ( ( Id ` T ) ` x ) = <. ( ( Id ` C ) ` ( 1st ` x ) ) , ( ( Id ` D ) ` ( 2nd ` x ) ) >. )
74		fvex 5706	. . . . . . . 8 |- ( ( Id ` C ) ` ( 1st ` x ) ) e. _V
75		fvex 5706	. . . . . . . 8 |- ( ( Id ` D ) ` ( 2nd ` x ) ) e. _V
76	74, 75	op1std 6592	. . . . . . 7 |- ( ( ( Id ` T ) ` x ) = <. ( ( Id ` C ) ` ( 1st ` x ) ) , ( ( Id ` D ) ` ( 2nd ` x ) ) >. -> ( 1st ` ( ( Id ` T ) ` x ) ) = ( ( Id ` C ) ` ( 1st ` x ) ) )
77	73, 76	syl 16	. . . . . 6 |- ( ( ph /\ x e. ( ( Base ` C ) X. ( Base ` D ) ) ) -> ( 1st ` ( ( Id ` T ) ` x ) ) = ( ( Id ` C ) ` ( 1st ` x ) ) )
78	61, 77	eqtrd 2475	. . . . 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 15008	. . . . . 6 |- ( ( ph /\ x e. ( ( Base ` C ) X. ( Base ` D ) ) ) -> ( x ( 2nd ` P ) x ) = ( 1st |` ( x ( Hom ` T ) x ) ) )
80	79	fveq1d 5698	. . . . 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 466	. . . . . 6 |- ( ( ph /\ x e. ( ( Base ` C ) X. ( Base ` D ) ) ) -> ( ( 1st |` ( ( Base ` C ) X. ( Base ` D ) ) ) ` x ) = ( 1st ` x ) )
82	81	fveq2d 5700	. . . . 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 2485	. . . 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 1009	. . . . . . . 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 1021	. . . . . . . 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 1022	. . . . . . . 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 1023	. . . . . . . 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 1016	. . . . . . . 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 1017	. . . . . . . 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 14628	. . . . . . 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 5709	. . . . . . 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 16	. . . . . 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 14999	. . . . . 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 2475	. . . . 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 1009	. . . . . . 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 1009	. . . . . . 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 15008	. . . . . 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 5698	. . . . 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 16	. . . . . . . 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 16	. . . . . . . 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 4072	. . . . . . 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 5709	. . . . . . . 8 |- ( z e. ( ( Base ` C ) X. ( Base ` D ) ) -> ( ( 1st |` ( ( Base ` C ) X. ( Base ` D ) ) ) ` z ) = ( 1st ` z ) )
103	87, 102	syl 16	. . . . . . 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 6114	. . . . . 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 15008	. . . . . . . 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 5698	. . . . . . 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 5709	. . . . . . . 8 |- ( g e. ( y ( Hom ` T ) z ) -> ( ( 1st |` ( y ( Hom ` T ) z ) ) ` g ) = ( 1st ` g ) )
108	89, 107	syl 16	. . . . . . 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 2475	. . . . . 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 15008	. . . . . . . 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 5698	. . . . . . 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 5709	. . . . . . . 8 |- ( f e. ( x ( Hom ` T ) y ) -> ( ( 1st |` ( x ( Hom ` T ) y ) ) ` f ) = ( 1st ` f ) )
113	88, 112	syl 16	. . . . . . 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 2475	. . . . . 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 6117	. . . . 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 2485	. . . 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 14780	. . 3 |- ( ph -> ( 1st |` ( ( Base ` C ) X. ( Base ` D ) ) ) ( T Func C ) ( 2nd ` P ) )
118		df-br 4298	. . 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 2517	1 |- ( ph -> P e. ( T Func C ) )

Syntax hints:   -> wi 4   /\ wa 369   /\ w3a 965   = wceq 1369   e. wcel 1756   _Vcvv 2977   <.cop 3888   class class class wbr 4297   X. cxp 4843   |` cres 4847   Fun wfun 5417   Fn wfn 5418   -->wf 5419   -onto->wfo 5421   ` cfv 5423  (class class class)co 6096   e. cmpt2 6098   1stc1st 6580   2ndc2nd 6581   Basecbs 14179   Hom chom 14254  compcco 14255   Catccat 14607   Idccid 14608   Func cfunc 14769   X._c cxpc 14983   1st_F c1stf 14984

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-1o 6925  df-oadd 6929  df-er 7106  df-map 7221  df-ixp 7269  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-nn 10328  df-2 10385  df-3 10386  df-4 10387  df-5 10388  df-6 10389  df-7 10390  df-8 10391  df-9 10392  df-10 10393  df-n0 10585  df-z 10652  df-dec 10761  df-uz 10867  df-fz 11443  df-struct 14181  df-ndx 14182  df-slot 14183  df-base 14184  df-hom 14267  df-cco 14268  df-cat 14611  df-cid 14612  df-func 14773  df-xpc 14987  df-1stf 14988

This theorem is referenced by:  prf1st  15019  1st2ndprf  15021  uncfcl  15050  uncf1  15051  uncf2  15052  diagcl  15056  diag11  15058  diag12  15059  diag2  15060  yonedalem1  15087  yonedalem21  15088  yonedalem22  15093