Theorem 1stfcl 15003

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 2441	. . . . 5 |- ( Base ` C ) = ( Base ` C )
3		eqid 2441	. . . . 5 |- ( Base ` D ) = ( Base ` D )
4	1, 2, 3	xpcbas 14984	. . . 4 |- ( ( Base ` C ) X. ( Base ` D ) ) = ( Base ` T )
5		eqid 2441	. . . 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 14997	. . 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 6595	. . . . . . . 8 |- 1st : _V -onto-> _V
11		fofun 5618	. . . . . . . 8 |- ( 1st : _V -onto-> _V -> Fun 1st )
12	10, 11	ax-mp 5	. . . . . . 7 |- Fun 1st
13		fvex 5698	. . . . . . . 8 |- ( Base ` C ) e. _V
14		fvex 5698	. . . . . . . 8 |- ( Base ` D ) e. _V
15	13, 14	xpex 6507	. . . . . . 7 |- ( ( Base ` C ) X. ( Base ` D ) ) e. _V
16		resfunexg 5940	. . . . . . 7 |- ( ( Fun 1st /\ ( ( Base ` C ) X. ( Base ` D ) ) e. _V ) -> ( 1st |` ( ( Base ` C ) X. ( Base ` D ) ) ) e. _V )
17	12, 15, 16	mp2an 667	. . . . . 6 |- ( 1st |` ( ( Base ` C ) X. ( Base ` D ) ) ) e. _V
18	15, 15	mpt2ex 6649	. . . . . 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 6587	. . . . 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 4063	. . 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 2476	. 2 |- ( ph -> P = <. ( 1st |` ( ( Base ` C ) X. ( Base ` D ) ) ) , ( 2nd ` P ) >. )
23		eqid 2441	. . . 4 |- ( Hom ` C ) = ( Hom ` C )
24		eqid 2441	. . . 4 |- ( Id ` T ) = ( Id ` T )
25		eqid 2441	. . . 4 |- ( Id ` C ) = ( Id ` C )
26		eqid 2441	. . . 4 |- (comp ` T ) = (comp ` T )
27		eqid 2441	. . . 4 |- (comp ` C ) = (comp ` C )
28	1, 6, 7	xpccat 14996	. . . 4 |- ( ph -> T e. Cat )
29		f1stres 6597	. . . . 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 2441	. . . . . 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 6115	. . . . . . 7 |- ( x ( Hom ` T ) y ) e. _V
33		resfunexg 5940	. . . . . . 7 |- ( ( Fun 1st /\ ( x ( Hom ` T ) y ) e. _V ) -> ( 1st |` ( x ( Hom ` T ) y ) ) e. _V )
34	12, 32, 33	mp2an 667	. . . . . 6 |- ( 1st |` ( x ( Hom ` T ) y ) ) e. _V
35	31, 34	fnmpt2i 6642	. . . . 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 5498	. . . . 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 6597	. . . . . 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 462	. . . . . . . . 9 |- ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` D ) ) /\ y e. ( ( Base ` C ) X. ( Base ` D ) ) ) ) -> C e. Cat )
40	7	adantr 462	. . . . . . . . 9 |- ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` D ) ) /\ y e. ( ( Base ` C ) X. ( Base ` D ) ) ) ) -> D e. Cat )
41		simprl 750	. . . . . . . . 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 751	. . . . . . . . 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 14999	. . . . . . . 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 2441	. . . . . . . . . 10 |- ( Hom ` D ) = ( Hom ` D )
45	1, 4, 23, 44, 5, 41, 42	xpchom 14986	. . . . . . . . 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 5106	. . . . . . . 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 2473	. . . . . . 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 5543	. . . . . 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 5701	. . . . . . . 8 |- ( x e. ( ( Base ` C ) X. ( Base ` D ) ) -> ( ( 1st |` ( ( Base ` C ) X. ( Base ` D ) ) ) ` x ) = ( 1st ` x ) )
51	50	ad2antrl 722	. . . . . . 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 5701	. . . . . . . 8 |- ( y e. ( ( Base ` C ) X. ( Base ` D ) ) -> ( ( 1st |` ( ( Base ` C ) X. ( Base ` D ) ) ) ` y ) = ( 1st ` y ) )
53	52	ad2antll 723	. . . . . . 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 6108	. . . . . 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 5551	. . . . 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 462	. . . . . . . 8 |- ( ( ph /\ x e. ( ( Base ` C ) X. ( Base ` D ) ) ) -> T e. Cat )
58		simpr 458	. . . . . . . 8 |- ( ( ph /\ x e. ( ( Base ` C ) X. ( Base ` D ) ) ) -> x e. ( ( Base ` C ) X. ( Base ` D ) ) )
59	4, 5, 24, 57, 58	catidcl 14616	. . . . . . 7 |- ( ( ph /\ x e. ( ( Base ` C ) X. ( Base ` D ) ) ) -> ( ( Id ` T ) ` x ) e. ( x ( Hom ` T ) x ) )
60		fvres 5701	. . . . . . 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 6612	. . . . . . . . . 10 |- ( x e. ( ( Base ` C ) X. ( Base ` D ) ) -> x = <. ( 1st ` x ) , ( 2nd ` x ) >. )
63	62	adantl 463	. . . . . . . . 9 |- ( ( ph /\ x e. ( ( Base ` C ) X. ( Base ` D ) ) ) -> x = <. ( 1st ` x ) , ( 2nd ` x ) >. )
64	63	fveq2d 5692	. . . . . . . 8 |- ( ( ph /\ x e. ( ( Base ` C ) X. ( Base ` D ) ) ) -> ( ( Id ` T ) ` x ) = ( ( Id ` T ) ` <. ( 1st ` x ) , ( 2nd ` x ) >. ) )
65	6	adantr 462	. . . . . . . . 9 |- ( ( ph /\ x e. ( ( Base ` C ) X. ( Base ` D ) ) ) -> C e. Cat )
66	7	adantr 462	. . . . . . . . 9 |- ( ( ph /\ x e. ( ( Base ` C ) X. ( Base ` D ) ) ) -> D e. Cat )
67		eqid 2441	. . . . . . . . 9 |- ( Id ` D ) = ( Id ` D )
68		xp1st 6605	. . . . . . . . . 10 |- ( x e. ( ( Base ` C ) X. ( Base ` D ) ) -> ( 1st ` x ) e. ( Base ` C ) )
69	68	adantl 463	. . . . . . . . 9 |- ( ( ph /\ x e. ( ( Base ` C ) X. ( Base ` D ) ) ) -> ( 1st ` x ) e. ( Base ` C ) )
70		xp2nd 6606	. . . . . . . . . 10 |- ( x e. ( ( Base ` C ) X. ( Base ` D ) ) -> ( 2nd ` x ) e. ( Base ` D ) )
71	70	adantl 463	. . . . . . . . 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 14995	. . . . . . . 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 2473	. . . . . . 7 |- ( ( ph /\ x e. ( ( Base ` C ) X. ( Base ` D ) ) ) -> ( ( Id ` T ) ` x ) = <. ( ( Id ` C ) ` ( 1st ` x ) ) , ( ( Id ` D ) ` ( 2nd ` x ) ) >. )
74		fvex 5698	. . . . . . . 8 |- ( ( Id ` C ) ` ( 1st ` x ) ) e. _V
75		fvex 5698	. . . . . . . 8 |- ( ( Id ` D ) ` ( 2nd ` x ) ) e. _V
76	74, 75	op1std 6586	. . . . . . 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 2473	. . . . 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 14999	. . . . . 6 |- ( ( ph /\ x e. ( ( Base ` C ) X. ( Base ` D ) ) ) -> ( x ( 2nd ` P ) x ) = ( 1st |` ( x ( Hom ` T ) x ) ) )
80	79	fveq1d 5690	. . . . 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 463	. . . . . 6 |- ( ( ph /\ x e. ( ( Base ` C ) X. ( Base ` D ) ) ) -> ( ( 1st |` ( ( Base ` C ) X. ( Base ` D ) ) ) ` x ) = ( 1st ` x ) )
82	81	fveq2d 5692	. . . . 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 2483	. . . 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 1004	. . . . . . . 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 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 ) ) ) -> x e. ( ( Base ` C ) X. ( Base ` D ) ) )
86		simp22 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 ) ) ) -> y e. ( ( Base ` C ) X. ( Base ` D ) ) )
87		simp23 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 ) ) ) -> z e. ( ( Base ` C ) X. ( Base ` D ) ) )
88		simp3l 1011	. . . . . . . 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 1012	. . . . . . . 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 14619	. . . . . . 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 5701	. . . . . . 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 14990	. . . . . 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 2473	. . . . 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 1004	. . . . . . 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 1004	. . . . . . 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 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 ) ) ) -> ( x ( 2nd ` P ) z ) = ( 1st |` ( x ( Hom ` T ) z ) ) )
98	97	fveq1d 5690	. . . . 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 4064	. . . . . . 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 5701	. . . . . . . 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 6108	. . . . . 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 14999	. . . . . . . 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 5690	. . . . . . 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 5701	. . . . . . . 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 2473	. . . . . 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 14999	. . . . . . . 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 5690	. . . . . . 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 5701	. . . . . . . 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 2473	. . . . . 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 6111	. . . . 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 2483	. . . 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 14771	. . 3 |- ( ph -> ( 1st |` ( ( Base ` C ) X. ( Base ` D ) ) ) ( T Func C ) ( 2nd ` P ) )
118		df-br 4290	. . 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 2515	1 |- ( ph -> P e. ( T Func C ) )

Syntax hints:   -> wi 4   /\ wa 369   /\ w3a 960   = wceq 1364   e. wcel 1761   _Vcvv 2970   <.cop 3880   class class class wbr 4289   X. cxp 4834   |` cres 4838   Fun wfun 5409   Fn wfn 5410   -->wf 5411   -onto->wfo 5413   ` cfv 5415  (class class class)co 6090   e. cmpt2 6092   1stc1st 6574   2ndc2nd 6575   Basecbs 14170   Hom chom 14245  compcco 14246   Catccat 14598   Idccid 14599   Func cfunc 14760   X._c cxpc 14974   1st_F c1stf 14975

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-fal 1370  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-1o 6916  df-oadd 6920  df-er 7097  df-map 7212  df-ixp 7260  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-nn 10319  df-2 10376  df-3 10377  df-4 10378  df-5 10379  df-6 10380  df-7 10381  df-8 10382  df-9 10383  df-10 10384  df-n0 10576  df-z 10643  df-dec 10752  df-uz 10858  df-fz 11434  df-struct 14172  df-ndx 14173  df-slot 14174  df-base 14175  df-hom 14258  df-cco 14259  df-cat 14602  df-cid 14603  df-func 14764  df-xpc 14978  df-1stf 14979

This theorem is referenced by:  prf1st  15010  1st2ndprf  15012  uncfcl  15041  uncf1  15042  uncf2  15043  diagcl  15047  diag11  15049  diag12  15050  diag2  15051  yonedalem1  15078  yonedalem21  15079  yonedalem22  15084