Theorem 1st2ndprf 16846

Description: Break a functor into a product category into first and second projections. (Contributed by Mario Carneiro, 12-Jan-2017.)

Hypotheses

Ref	Expression
1st2ndprf.t	|- T = ( D X.c E )
1st2ndprf.f	|- ( ph -> F e. ( C Func T ) )
1st2ndprf.d	|- ( ph -> D e. Cat )
1st2ndprf.e	|- ( ph -> E e. Cat )

Assertion

Ref	Expression
1st2ndprf	|- ( ph -> F = ( ( ( D 1stF E ) o.func F ) ⟨,⟩F ( ( D 2ndF E ) o.func F ) ) )

Proof of Theorem 1st2ndprf

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

Step	Hyp	Ref	Expression
1		eqid 2622	. . . . . 6 |- ( Base ` C ) = ( Base ` C )
2		1st2ndprf.t	. . . . . . 7 |- T = ( D X.c E )
3		eqid 2622	. . . . . . 7 |- ( Base ` D ) = ( Base ` D )
4		eqid 2622	. . . . . . 7 |- ( Base ` E ) = ( Base ` E )
5	2, 3, 4	xpcbas 16818	. . . . . 6 |- ( ( Base ` D ) X. ( Base ` E ) ) = ( Base ` T )
6		relfunc 16522	. . . . . . 7 |- Rel ( C Func T )
7		1st2ndprf.f	. . . . . . 7 |- ( ph -> F e. ( C Func T ) )
8		1st2ndbr 7217	. . . . . . 7 |- ( ( Rel ( C Func T ) /\ F e. ( C Func T ) ) -> ( 1st ` F ) ( C Func T ) ( 2nd ` F ) )
9	6, 7, 8	sylancr 695	. . . . . 6 |- ( ph -> ( 1st ` F ) ( C Func T ) ( 2nd ` F ) )
10	1, 5, 9	funcf1 16526	. . . . 5 |- ( ph -> ( 1st ` F ) : ( Base ` C ) --> ( ( Base ` D ) X. ( Base ` E ) ) )
11	10	feqmptd 6249	. . . 4 |- ( ph -> ( 1st ` F ) = ( x e. ( Base ` C ) |-> ( ( 1st ` F ) ` x ) ) )
12	10	ffvelrnda 6359	. . . . . . 7 |- ( ( ph /\ x e. ( Base ` C ) ) -> ( ( 1st ` F ) ` x ) e. ( ( Base ` D ) X. ( Base ` E ) ) )
13		1st2nd2 7205	. . . . . . 7 |- ( ( ( 1st ` F ) ` x ) e. ( ( Base ` D ) X. ( Base ` E ) ) -> ( ( 1st ` F ) ` x ) = <. ( 1st ` ( ( 1st ` F ) ` x ) ) , ( 2nd ` ( ( 1st ` F ) ` x ) ) >. )
14	12, 13	syl 17	. . . . . 6 |- ( ( ph /\ x e. ( Base ` C ) ) -> ( ( 1st ` F ) ` x ) = <. ( 1st ` ( ( 1st ` F ) ` x ) ) , ( 2nd ` ( ( 1st ` F ) ` x ) ) >. )
15	7	adantr 481	. . . . . . . . 9 |- ( ( ph /\ x e. ( Base ` C ) ) -> F e. ( C Func T ) )
16		1st2ndprf.d	. . . . . . . . . . 11 |- ( ph -> D e. Cat )
17		1st2ndprf.e	. . . . . . . . . . 11 |- ( ph -> E e. Cat )
18		eqid 2622	. . . . . . . . . . 11 |- ( D 1stF E ) = ( D 1stF E )
19	2, 16, 17, 18	1stfcl 16837	. . . . . . . . . 10 |- ( ph -> ( D 1stF E ) e. ( T Func D ) )
20	19	adantr 481	. . . . . . . . 9 |- ( ( ph /\ x e. ( Base ` C ) ) -> ( D 1stF E ) e. ( T Func D ) )
21		simpr 477	. . . . . . . . 9 |- ( ( ph /\ x e. ( Base ` C ) ) -> x e. ( Base ` C ) )
22	1, 15, 20, 21	cofu1 16544	. . . . . . . 8 |- ( ( ph /\ x e. ( Base ` C ) ) -> ( ( 1st ` ( ( D 1stF E ) o.func F ) ) ` x ) = ( ( 1st ` ( D 1stF E ) ) ` ( ( 1st ` F ) ` x ) ) )
23		eqid 2622	. . . . . . . . 9 |- ( Hom ` T ) = ( Hom ` T )
24	16	adantr 481	. . . . . . . . 9 |- ( ( ph /\ x e. ( Base ` C ) ) -> D e. Cat )
25	17	adantr 481	. . . . . . . . 9 |- ( ( ph /\ x e. ( Base ` C ) ) -> E e. Cat )
26	2, 5, 23, 24, 25, 18, 12	1stf1 16832	. . . . . . . 8 |- ( ( ph /\ x e. ( Base ` C ) ) -> ( ( 1st ` ( D 1stF E ) ) ` ( ( 1st ` F ) ` x ) ) = ( 1st ` ( ( 1st ` F ) ` x ) ) )
27	22, 26	eqtrd 2656	. . . . . . 7 |- ( ( ph /\ x e. ( Base ` C ) ) -> ( ( 1st ` ( ( D 1stF E ) o.func F ) ) ` x ) = ( 1st ` ( ( 1st ` F ) ` x ) ) )
28		eqid 2622	. . . . . . . . . . 11 |- ( D 2ndF E ) = ( D 2ndF E )
29	2, 16, 17, 28	2ndfcl 16838	. . . . . . . . . 10 |- ( ph -> ( D 2ndF E ) e. ( T Func E ) )
30	29	adantr 481	. . . . . . . . 9 |- ( ( ph /\ x e. ( Base ` C ) ) -> ( D 2ndF E ) e. ( T Func E ) )
31	1, 15, 30, 21	cofu1 16544	. . . . . . . 8 |- ( ( ph /\ x e. ( Base ` C ) ) -> ( ( 1st ` ( ( D 2ndF E ) o.func F ) ) ` x ) = ( ( 1st ` ( D 2ndF E ) ) ` ( ( 1st ` F ) ` x ) ) )
32	2, 5, 23, 24, 25, 28, 12	2ndf1 16835	. . . . . . . 8 |- ( ( ph /\ x e. ( Base ` C ) ) -> ( ( 1st ` ( D 2ndF E ) ) ` ( ( 1st ` F ) ` x ) ) = ( 2nd ` ( ( 1st ` F ) ` x ) ) )
33	31, 32	eqtrd 2656	. . . . . . 7 |- ( ( ph /\ x e. ( Base ` C ) ) -> ( ( 1st ` ( ( D 2ndF E ) o.func F ) ) ` x ) = ( 2nd ` ( ( 1st ` F ) ` x ) ) )
34	27, 33	opeq12d 4410	. . . . . 6 |- ( ( ph /\ x e. ( Base ` C ) ) -> <. ( ( 1st ` ( ( D 1stF E ) o.func F ) ) ` x ) , ( ( 1st ` ( ( D 2ndF E ) o.func F ) ) ` x ) >. = <. ( 1st ` ( ( 1st ` F ) ` x ) ) , ( 2nd ` ( ( 1st ` F ) ` x ) ) >. )
35	14, 34	eqtr4d 2659	. . . . 5 |- ( ( ph /\ x e. ( Base ` C ) ) -> ( ( 1st ` F ) ` x ) = <. ( ( 1st ` ( ( D 1stF E ) o.func F ) ) ` x ) , ( ( 1st ` ( ( D 2ndF E ) o.func F ) ) ` x ) >. )
36	35	mpteq2dva 4744	. . . 4 |- ( ph -> ( x e. ( Base ` C ) |-> ( ( 1st ` F ) ` x ) ) = ( x e. ( Base ` C ) |-> <. ( ( 1st ` ( ( D 1stF E ) o.func F ) ) ` x ) , ( ( 1st ` ( ( D 2ndF E ) o.func F ) ) ` x ) >. ) )
37	11, 36	eqtrd 2656	. . 3 |- ( ph -> ( 1st ` F ) = ( x e. ( Base ` C ) |-> <. ( ( 1st ` ( ( D 1stF E ) o.func F ) ) ` x ) , ( ( 1st ` ( ( D 2ndF E ) o.func F ) ) ` x ) >. ) )
38	1, 9	funcfn2 16529	. . . . 5 |- ( ph -> ( 2nd ` F ) Fn ( ( Base ` C ) X. ( Base ` C ) ) )
39		fnov 6768	. . . . 5 |- ( ( 2nd ` F ) Fn ( ( Base ` C ) X. ( Base ` C ) ) <-> ( 2nd ` F ) = ( x e. ( Base ` C ) , y e. ( Base ` C ) |-> ( x ( 2nd ` F ) y ) ) )
40	38, 39	sylib 208	. . . 4 |- ( ph -> ( 2nd ` F ) = ( x e. ( Base ` C ) , y e. ( Base ` C ) |-> ( x ( 2nd ` F ) y ) ) )
41		eqid 2622	. . . . . . . . 9 |- ( Hom ` C ) = ( Hom ` C )
42	9	adantr 481	. . . . . . . . 9 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> ( 1st ` F ) ( C Func T ) ( 2nd ` F ) )
43		simprl 794	. . . . . . . . 9 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> x e. ( Base ` C ) )
44		simprr 796	. . . . . . . . 9 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> y e. ( Base ` C ) )
45	1, 41, 23, 42, 43, 44	funcf2 16528	. . . . . . . 8 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> ( x ( 2nd ` F ) y ) : ( x ( Hom ` C ) y ) --> ( ( ( 1st ` F ) ` x ) ( Hom ` T ) ( ( 1st ` F ) ` y ) ) )
46	45	feqmptd 6249	. . . . . . 7 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> ( x ( 2nd ` F ) y ) = ( f e. ( x ( Hom ` C ) y ) |-> ( ( x ( 2nd ` F ) y ) ` f ) ) )
47	2, 23	relxpchom 16821	. . . . . . . . . 10 |- Rel ( ( ( 1st ` F ) ` x ) ( Hom ` T ) ( ( 1st ` F ) ` y ) )
48	45	ffvelrnda 6359	. . . . . . . . . 10 |- ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ f e. ( x ( Hom ` C ) y ) ) -> ( ( x ( 2nd ` F ) y ) ` f ) e. ( ( ( 1st ` F ) ` x ) ( Hom ` T ) ( ( 1st ` F ) ` y ) ) )
49		1st2nd 7214	. . . . . . . . . 10 |- ( ( Rel ( ( ( 1st ` F ) ` x ) ( Hom ` T ) ( ( 1st ` F ) ` y ) ) /\ ( ( x ( 2nd ` F ) y ) ` f ) e. ( ( ( 1st ` F ) ` x ) ( Hom ` T ) ( ( 1st ` F ) ` y ) ) ) -> ( ( x ( 2nd ` F ) y ) ` f ) = <. ( 1st ` ( ( x ( 2nd ` F ) y ) ` f ) ) , ( 2nd ` ( ( x ( 2nd ` F ) y ) ` f ) ) >. )
50	47, 48, 49	sylancr 695	. . . . . . . . 9 |- ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ f e. ( x ( Hom ` C ) y ) ) -> ( ( x ( 2nd ` F ) y ) ` f ) = <. ( 1st ` ( ( x ( 2nd ` F ) y ) ` f ) ) , ( 2nd ` ( ( x ( 2nd ` F ) y ) ` f ) ) >. )
51	7	ad2antrr 762	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ f e. ( x ( Hom ` C ) y ) ) -> F e. ( C Func T ) )
52	19	ad2antrr 762	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ f e. ( x ( Hom ` C ) y ) ) -> ( D 1stF E ) e. ( T Func D ) )
53	43	adantr 481	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ f e. ( x ( Hom ` C ) y ) ) -> x e. ( Base ` C ) )
54	44	adantr 481	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ f e. ( x ( Hom ` C ) y ) ) -> y e. ( Base ` C ) )
55		simpr 477	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ f e. ( x ( Hom ` C ) y ) ) -> f e. ( x ( Hom ` C ) y ) )
56	1, 51, 52, 53, 54, 41, 55	cofu2 16546	. . . . . . . . . . 11 |- ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ f e. ( x ( Hom ` C ) y ) ) -> ( ( x ( 2nd ` ( ( D 1stF E ) o.func F ) ) y ) ` f ) = ( ( ( ( 1st ` F ) ` x ) ( 2nd ` ( D 1stF E ) ) ( ( 1st ` F ) ` y ) ) ` ( ( x ( 2nd ` F ) y ) ` f ) ) )
57	16	adantr 481	. . . . . . . . . . . . . 14 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> D e. Cat )
58	17	adantr 481	. . . . . . . . . . . . . 14 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> E e. Cat )
59	12	adantrr 753	. . . . . . . . . . . . . 14 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> ( ( 1st ` F ) ` x ) e. ( ( Base ` D ) X. ( Base ` E ) ) )
60	10	ffvelrnda 6359	. . . . . . . . . . . . . . 15 |- ( ( ph /\ y e. ( Base ` C ) ) -> ( ( 1st ` F ) ` y ) e. ( ( Base ` D ) X. ( Base ` E ) ) )
61	60	adantrl 752	. . . . . . . . . . . . . 14 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> ( ( 1st ` F ) ` y ) e. ( ( Base ` D ) X. ( Base ` E ) ) )
62	2, 5, 23, 57, 58, 18, 59, 61	1stf2 16833	. . . . . . . . . . . . 13 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> ( ( ( 1st ` F ) ` x ) ( 2nd ` ( D 1stF E ) ) ( ( 1st ` F ) ` y ) ) = ( 1st |` ( ( ( 1st ` F ) ` x ) ( Hom ` T ) ( ( 1st ` F ) ` y ) ) ) )
63	62	adantr 481	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ f e. ( x ( Hom ` C ) y ) ) -> ( ( ( 1st ` F ) ` x ) ( 2nd ` ( D 1stF E ) ) ( ( 1st ` F ) ` y ) ) = ( 1st |` ( ( ( 1st ` F ) ` x ) ( Hom ` T ) ( ( 1st ` F ) ` y ) ) ) )
64	63	fveq1d 6193	. . . . . . . . . . 11 |- ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ f e. ( x ( Hom ` C ) y ) ) -> ( ( ( ( 1st ` F ) ` x ) ( 2nd ` ( D 1stF E ) ) ( ( 1st ` F ) ` y ) ) ` ( ( x ( 2nd ` F ) y ) ` f ) ) = ( ( 1st |` ( ( ( 1st ` F ) ` x ) ( Hom ` T ) ( ( 1st ` F ) ` y ) ) ) ` ( ( x ( 2nd ` F ) y ) ` f ) ) )
65		fvres 6207	. . . . . . . . . . . 12 |- ( ( ( x ( 2nd ` F ) y ) ` f ) e. ( ( ( 1st ` F ) ` x ) ( Hom ` T ) ( ( 1st ` F ) ` y ) ) -> ( ( 1st |` ( ( ( 1st ` F ) ` x ) ( Hom ` T ) ( ( 1st ` F ) ` y ) ) ) ` ( ( x ( 2nd ` F ) y ) ` f ) ) = ( 1st ` ( ( x ( 2nd ` F ) y ) ` f ) ) )
66	48, 65	syl 17	. . . . . . . . . . 11 |- ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ f e. ( x ( Hom ` C ) y ) ) -> ( ( 1st |` ( ( ( 1st ` F ) ` x ) ( Hom ` T ) ( ( 1st ` F ) ` y ) ) ) ` ( ( x ( 2nd ` F ) y ) ` f ) ) = ( 1st ` ( ( x ( 2nd ` F ) y ) ` f ) ) )
67	56, 64, 66	3eqtrd 2660	. . . . . . . . . 10 |- ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ f e. ( x ( Hom ` C ) y ) ) -> ( ( x ( 2nd ` ( ( D 1stF E ) o.func F ) ) y ) ` f ) = ( 1st ` ( ( x ( 2nd ` F ) y ) ` f ) ) )
68	29	ad2antrr 762	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ f e. ( x ( Hom ` C ) y ) ) -> ( D 2ndF E ) e. ( T Func E ) )
69	1, 51, 68, 53, 54, 41, 55	cofu2 16546	. . . . . . . . . . 11 |- ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ f e. ( x ( Hom ` C ) y ) ) -> ( ( x ( 2nd ` ( ( D 2ndF E ) o.func F ) ) y ) ` f ) = ( ( ( ( 1st ` F ) ` x ) ( 2nd ` ( D 2ndF E ) ) ( ( 1st ` F ) ` y ) ) ` ( ( x ( 2nd ` F ) y ) ` f ) ) )
70	2, 5, 23, 57, 58, 28, 59, 61	2ndf2 16836	. . . . . . . . . . . . 13 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> ( ( ( 1st ` F ) ` x ) ( 2nd ` ( D 2ndF E ) ) ( ( 1st ` F ) ` y ) ) = ( 2nd |` ( ( ( 1st ` F ) ` x ) ( Hom ` T ) ( ( 1st ` F ) ` y ) ) ) )
71	70	adantr 481	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ f e. ( x ( Hom ` C ) y ) ) -> ( ( ( 1st ` F ) ` x ) ( 2nd ` ( D 2ndF E ) ) ( ( 1st ` F ) ` y ) ) = ( 2nd |` ( ( ( 1st ` F ) ` x ) ( Hom ` T ) ( ( 1st ` F ) ` y ) ) ) )
72	71	fveq1d 6193	. . . . . . . . . . 11 |- ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ f e. ( x ( Hom ` C ) y ) ) -> ( ( ( ( 1st ` F ) ` x ) ( 2nd ` ( D 2ndF E ) ) ( ( 1st ` F ) ` y ) ) ` ( ( x ( 2nd ` F ) y ) ` f ) ) = ( ( 2nd |` ( ( ( 1st ` F ) ` x ) ( Hom ` T ) ( ( 1st ` F ) ` y ) ) ) ` ( ( x ( 2nd ` F ) y ) ` f ) ) )
73		fvres 6207	. . . . . . . . . . . 12 |- ( ( ( x ( 2nd ` F ) y ) ` f ) e. ( ( ( 1st ` F ) ` x ) ( Hom ` T ) ( ( 1st ` F ) ` y ) ) -> ( ( 2nd |` ( ( ( 1st ` F ) ` x ) ( Hom ` T ) ( ( 1st ` F ) ` y ) ) ) ` ( ( x ( 2nd ` F ) y ) ` f ) ) = ( 2nd ` ( ( x ( 2nd ` F ) y ) ` f ) ) )
74	48, 73	syl 17	. . . . . . . . . . 11 |- ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ f e. ( x ( Hom ` C ) y ) ) -> ( ( 2nd |` ( ( ( 1st ` F ) ` x ) ( Hom ` T ) ( ( 1st ` F ) ` y ) ) ) ` ( ( x ( 2nd ` F ) y ) ` f ) ) = ( 2nd ` ( ( x ( 2nd ` F ) y ) ` f ) ) )
75	69, 72, 74	3eqtrd 2660	. . . . . . . . . 10 |- ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ f e. ( x ( Hom ` C ) y ) ) -> ( ( x ( 2nd ` ( ( D 2ndF E ) o.func F ) ) y ) ` f ) = ( 2nd ` ( ( x ( 2nd ` F ) y ) ` f ) ) )
76	67, 75	opeq12d 4410	. . . . . . . . 9 |- ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ f e. ( x ( Hom ` C ) y ) ) -> <. ( ( x ( 2nd ` ( ( D 1stF E ) o.func F ) ) y ) ` f ) , ( ( x ( 2nd ` ( ( D 2ndF E ) o.func F ) ) y ) ` f ) >. = <. ( 1st ` ( ( x ( 2nd ` F ) y ) ` f ) ) , ( 2nd ` ( ( x ( 2nd ` F ) y ) ` f ) ) >. )
77	50, 76	eqtr4d 2659	. . . . . . . 8 |- ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ f e. ( x ( Hom ` C ) y ) ) -> ( ( x ( 2nd ` F ) y ) ` f ) = <. ( ( x ( 2nd ` ( ( D 1stF E ) o.func F ) ) y ) ` f ) , ( ( x ( 2nd ` ( ( D 2ndF E ) o.func F ) ) y ) ` f ) >. )
78	77	mpteq2dva 4744	. . . . . . 7 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> ( f e. ( x ( Hom ` C ) y ) |-> ( ( x ( 2nd ` F ) y ) ` f ) ) = ( f e. ( x ( Hom ` C ) y ) |-> <. ( ( x ( 2nd ` ( ( D 1stF E ) o.func F ) ) y ) ` f ) , ( ( x ( 2nd ` ( ( D 2ndF E ) o.func F ) ) y ) ` f ) >. ) )
79	46, 78	eqtrd 2656	. . . . . 6 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> ( x ( 2nd ` F ) y ) = ( f e. ( x ( Hom ` C ) y ) |-> <. ( ( x ( 2nd ` ( ( D 1stF E ) o.func F ) ) y ) ` f ) , ( ( x ( 2nd ` ( ( D 2ndF E ) o.func F ) ) y ) ` f ) >. ) )
80	79	3impb 1260	. . . . 5 |- ( ( ph /\ x e. ( Base ` C ) /\ y e. ( Base ` C ) ) -> ( x ( 2nd ` F ) y ) = ( f e. ( x ( Hom ` C ) y ) |-> <. ( ( x ( 2nd ` ( ( D 1stF E ) o.func F ) ) y ) ` f ) , ( ( x ( 2nd ` ( ( D 2ndF E ) o.func F ) ) y ) ` f ) >. ) )
81	80	mpt2eq3dva 6719	. . . 4 |- ( ph -> ( x e. ( Base ` C ) , y e. ( Base ` C ) |-> ( x ( 2nd ` F ) y ) ) = ( x e. ( Base ` C ) , y e. ( Base ` C ) |-> ( f e. ( x ( Hom ` C ) y ) |-> <. ( ( x ( 2nd ` ( ( D 1stF E ) o.func F ) ) y ) ` f ) , ( ( x ( 2nd ` ( ( D 2ndF E ) o.func F ) ) y ) ` f ) >. ) ) )
82	40, 81	eqtrd 2656	. . 3 |- ( ph -> ( 2nd ` F ) = ( x e. ( Base ` C ) , y e. ( Base ` C ) |-> ( f e. ( x ( Hom ` C ) y ) |-> <. ( ( x ( 2nd ` ( ( D 1stF E ) o.func F ) ) y ) ` f ) , ( ( x ( 2nd ` ( ( D 2ndF E ) o.func F ) ) y ) ` f ) >. ) ) )
83	37, 82	opeq12d 4410	. 2 |- ( ph -> <. ( 1st ` F ) , ( 2nd ` F ) >. = <. ( x e. ( Base ` C ) |-> <. ( ( 1st ` ( ( D 1stF E ) o.func F ) ) ` x ) , ( ( 1st ` ( ( D 2ndF E ) o.func F ) ) ` x ) >. ) , ( x e. ( Base ` C ) , y e. ( Base ` C ) |-> ( f e. ( x ( Hom ` C ) y ) |-> <. ( ( x ( 2nd ` ( ( D 1stF E ) o.func F ) ) y ) ` f ) , ( ( x ( 2nd ` ( ( D 2ndF E ) o.func F ) ) y ) ` f ) >. ) ) >. )
84		1st2nd 7214	. . 3 |- ( ( Rel ( C Func T ) /\ F e. ( C Func T ) ) -> F = <. ( 1st ` F ) , ( 2nd ` F ) >. )
85	6, 7, 84	sylancr 695	. 2 |- ( ph -> F = <. ( 1st ` F ) , ( 2nd ` F ) >. )
86		eqid 2622	. . 3 |- ( ( ( D 1stF E ) o.func F ) ⟨,⟩F ( ( D 2ndF E ) o.func F ) ) = ( ( ( D 1stF E ) o.func F ) ⟨,⟩F ( ( D 2ndF E ) o.func F ) )
87	7, 19	cofucl 16548	. . 3 |- ( ph -> ( ( D 1stF E ) o.func F ) e. ( C Func D ) )
88	7, 29	cofucl 16548	. . 3 |- ( ph -> ( ( D 2ndF E ) o.func F ) e. ( C Func E ) )
89	86, 1, 41, 87, 88	prfval 16839	. 2 |- ( ph -> ( ( ( D 1stF E ) o.func F ) ⟨,⟩F ( ( D 2ndF E ) o.func F ) ) = <. ( x e. ( Base ` C ) |-> <. ( ( 1st ` ( ( D 1stF E ) o.func F ) ) ` x ) , ( ( 1st ` ( ( D 2ndF E ) o.func F ) ) ` x ) >. ) , ( x e. ( Base ` C ) , y e. ( Base ` C ) |-> ( f e. ( x ( Hom ` C ) y ) |-> <. ( ( x ( 2nd ` ( ( D 1stF E ) o.func F ) ) y ) ` f ) , ( ( x ( 2nd ` ( ( D 2ndF E ) o.func F ) ) y ) ` f ) >. ) ) >. )
90	83, 85, 89	3eqtr4d 2666	1 |- ( ph -> F = ( ( ( D 1stF E ) o.func F ) ⟨,⟩F ( ( D 2ndF E ) o.func F ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   <.cop 4183   class class class wbr 4653   |-> cmpt 4729   X. cxp 5112   |` cres 5116   Rel wrel 5119   Fn wfn 5883   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   1stc1st 7166   2ndc2nd 7167   Basecbs 15857   Hom chom 15952   Catccat 16325   Func cfunc 16514   o._func ccofu 16516   X._c cxpc 16808   1st_F c1stf 16809   2nd_F c2ndf 16810   ⟨,⟩_F cprf 16811

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-fz 12327  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-hom 15966  df-cco 15967  df-cat 16329  df-cid 16330  df-func 16518  df-cofu 16520  df-xpc 16812  df-1stf 16813  df-2ndf 16814  df-prf 16815

This theorem is referenced by: (None)