Theorem prf1st 16844

Description: Cancellation of pairing with first projection. (Contributed by Mario Carneiro, 12-Jan-2017.)

Hypotheses

Ref	Expression
prf1st.p	|- P = ( F ⟨,⟩F G )
prf1st.c	|- ( ph -> F e. ( C Func D ) )
prf1st.d	|- ( ph -> G e. ( C Func E ) )

Assertion

Ref	Expression
prf1st	|- ( ph -> ( ( D 1stF E ) o.func P ) = F )

Proof of Theorem prf1st

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

Step	Hyp	Ref	Expression
1		eqid 2622	. . . . . . 7 |- ( D X.c E ) = ( D X.c E )
2		eqid 2622	. . . . . . . 8 |- ( Base ` D ) = ( Base ` D )
3		eqid 2622	. . . . . . . 8 |- ( Base ` E ) = ( Base ` E )
4	1, 2, 3	xpcbas 16818	. . . . . . 7 |- ( ( Base ` D ) X. ( Base ` E ) ) = ( Base ` ( D X.c E ) )
5		eqid 2622	. . . . . . 7 |- ( Hom ` ( D X.c E ) ) = ( Hom ` ( D X.c E ) )
6		prf1st.c	. . . . . . . . . 10 |- ( ph -> F e. ( C Func D ) )
7		funcrcl 16523	. . . . . . . . . 10 |- ( F e. ( C Func D ) -> ( C e. Cat /\ D e. Cat ) )
8	6, 7	syl 17	. . . . . . . . 9 |- ( ph -> ( C e. Cat /\ D e. Cat ) )
9	8	simprd 479	. . . . . . . 8 |- ( ph -> D e. Cat )
10	9	adantr 481	. . . . . . 7 |- ( ( ph /\ x e. ( Base ` C ) ) -> D e. Cat )
11		prf1st.d	. . . . . . . . . 10 |- ( ph -> G e. ( C Func E ) )
12		funcrcl 16523	. . . . . . . . . 10 |- ( G e. ( C Func E ) -> ( C e. Cat /\ E e. Cat ) )
13	11, 12	syl 17	. . . . . . . . 9 |- ( ph -> ( C e. Cat /\ E e. Cat ) )
14	13	simprd 479	. . . . . . . 8 |- ( ph -> E e. Cat )
15	14	adantr 481	. . . . . . 7 |- ( ( ph /\ x e. ( Base ` C ) ) -> E e. Cat )
16		eqid 2622	. . . . . . 7 |- ( D 1stF E ) = ( D 1stF E )
17		eqid 2622	. . . . . . . . . 10 |- ( Base ` C ) = ( Base ` C )
18		relfunc 16522	. . . . . . . . . . 11 |- Rel ( C Func D )
19		1st2ndbr 7217	. . . . . . . . . . 11 |- ( ( Rel ( C Func D ) /\ F e. ( C Func D ) ) -> ( 1st ` F ) ( C Func D ) ( 2nd ` F ) )
20	18, 6, 19	sylancr 695	. . . . . . . . . 10 |- ( ph -> ( 1st ` F ) ( C Func D ) ( 2nd ` F ) )
21	17, 2, 20	funcf1 16526	. . . . . . . . 9 |- ( ph -> ( 1st ` F ) : ( Base ` C ) --> ( Base ` D ) )
22	21	ffvelrnda 6359	. . . . . . . 8 |- ( ( ph /\ x e. ( Base ` C ) ) -> ( ( 1st ` F ) ` x ) e. ( Base ` D ) )
23		relfunc 16522	. . . . . . . . . . 11 |- Rel ( C Func E )
24		1st2ndbr 7217	. . . . . . . . . . 11 |- ( ( Rel ( C Func E ) /\ G e. ( C Func E ) ) -> ( 1st ` G ) ( C Func E ) ( 2nd ` G ) )
25	23, 11, 24	sylancr 695	. . . . . . . . . 10 |- ( ph -> ( 1st ` G ) ( C Func E ) ( 2nd ` G ) )
26	17, 3, 25	funcf1 16526	. . . . . . . . 9 |- ( ph -> ( 1st ` G ) : ( Base ` C ) --> ( Base ` E ) )
27	26	ffvelrnda 6359	. . . . . . . 8 |- ( ( ph /\ x e. ( Base ` C ) ) -> ( ( 1st ` G ) ` x ) e. ( Base ` E ) )
28		opelxpi 5148	. . . . . . . 8 |- ( ( ( ( 1st ` F ) ` x ) e. ( Base ` D ) /\ ( ( 1st ` G ) ` x ) e. ( Base ` E ) ) -> <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. e. ( ( Base ` D ) X. ( Base ` E ) ) )
29	22, 27, 28	syl2anc 693	. . . . . . 7 |- ( ( ph /\ x e. ( Base ` C ) ) -> <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. e. ( ( Base ` D ) X. ( Base ` E ) ) )
30	1, 4, 5, 10, 15, 16, 29	1stf1 16832	. . . . . 6 |- ( ( ph /\ x e. ( Base ` C ) ) -> ( ( 1st ` ( D 1stF E ) ) ` <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. ) = ( 1st ` <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. ) )
31		fvex 6201	. . . . . . 7 |- ( ( 1st ` F ) ` x ) e. _V
32		fvex 6201	. . . . . . 7 |- ( ( 1st ` G ) ` x ) e. _V
33	31, 32	op1st 7176	. . . . . 6 |- ( 1st ` <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. ) = ( ( 1st ` F ) ` x )
34	30, 33	syl6eq 2672	. . . . 5 |- ( ( ph /\ x e. ( Base ` C ) ) -> ( ( 1st ` ( D 1stF E ) ) ` <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. ) = ( ( 1st ` F ) ` x ) )
35	34	mpteq2dva 4744	. . . 4 |- ( ph -> ( x e. ( Base ` C ) |-> ( ( 1st ` ( D 1stF E ) ) ` <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. ) ) = ( x e. ( Base ` C ) |-> ( ( 1st ` F ) ` x ) ) )
36		prf1st.p	. . . . . . 7 |- P = ( F ⟨,⟩F G )
37		eqid 2622	. . . . . . 7 |- ( Hom ` C ) = ( Hom ` C )
38	36, 17, 37, 6, 11	prfval 16839	. . . . . 6 |- ( ph -> P = <. ( x e. ( Base ` C ) |-> <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. ) , ( x e. ( Base ` C ) , y e. ( Base ` C ) |-> ( h e. ( x ( Hom ` C ) y ) |-> <. ( ( x ( 2nd ` F ) y ) ` h ) , ( ( x ( 2nd ` G ) y ) ` h ) >. ) ) >. )
39		fvex 6201	. . . . . . . 8 |- ( Base ` C ) e. _V
40	39	mptex 6486	. . . . . . 7 |- ( x e. ( Base ` C ) |-> <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. ) e. _V
41	39, 39	mpt2ex 7247	. . . . . . 7 |- ( x e. ( Base ` C ) , y e. ( Base ` C ) |-> ( h e. ( x ( Hom ` C ) y ) |-> <. ( ( x ( 2nd ` F ) y ) ` h ) , ( ( x ( 2nd ` G ) y ) ` h ) >. ) ) e. _V
42	40, 41	op1std 7178	. . . . . 6 |- ( P = <. ( x e. ( Base ` C ) |-> <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. ) , ( x e. ( Base ` C ) , y e. ( Base ` C ) |-> ( h e. ( x ( Hom ` C ) y ) |-> <. ( ( x ( 2nd ` F ) y ) ` h ) , ( ( x ( 2nd ` G ) y ) ` h ) >. ) ) >. -> ( 1st ` P ) = ( x e. ( Base ` C ) |-> <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. ) )
43	38, 42	syl 17	. . . . 5 |- ( ph -> ( 1st ` P ) = ( x e. ( Base ` C ) |-> <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. ) )
44		relfunc 16522	. . . . . . . 8 |- Rel ( ( D X.c E ) Func D )
45	1, 9, 14, 16	1stfcl 16837	. . . . . . . 8 |- ( ph -> ( D 1stF E ) e. ( ( D X.c E ) Func D ) )
46		1st2ndbr 7217	. . . . . . . 8 |- ( ( Rel ( ( D X.c E ) Func D ) /\ ( D 1stF E ) e. ( ( D X.c E ) Func D ) ) -> ( 1st ` ( D 1stF E ) ) ( ( D X.c E ) Func D ) ( 2nd ` ( D 1stF E ) ) )
47	44, 45, 46	sylancr 695	. . . . . . 7 |- ( ph -> ( 1st ` ( D 1stF E ) ) ( ( D X.c E ) Func D ) ( 2nd ` ( D 1stF E ) ) )
48	4, 2, 47	funcf1 16526	. . . . . 6 |- ( ph -> ( 1st ` ( D 1stF E ) ) : ( ( Base ` D ) X. ( Base ` E ) ) --> ( Base ` D ) )
49	48	feqmptd 6249	. . . . 5 |- ( ph -> ( 1st ` ( D 1stF E ) ) = ( u e. ( ( Base ` D ) X. ( Base ` E ) ) |-> ( ( 1st ` ( D 1stF E ) ) ` u ) ) )
50		fveq2 6191	. . . . 5 |- ( u = <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. -> ( ( 1st ` ( D 1stF E ) ) ` u ) = ( ( 1st ` ( D 1stF E ) ) ` <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. ) )
51	29, 43, 49, 50	fmptco 6396	. . . 4 |- ( ph -> ( ( 1st ` ( D 1stF E ) ) o. ( 1st ` P ) ) = ( x e. ( Base ` C ) |-> ( ( 1st ` ( D 1stF E ) ) ` <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. ) ) )
52	21	feqmptd 6249	. . . 4 |- ( ph -> ( 1st ` F ) = ( x e. ( Base ` C ) |-> ( ( 1st ` F ) ` x ) ) )
53	35, 51, 52	3eqtr4d 2666	. . 3 |- ( ph -> ( ( 1st ` ( D 1stF E ) ) o. ( 1st ` P ) ) = ( 1st ` F ) )
54	9	ad2antrr 762	. . . . . . . . . . 11 |- ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ f e. ( x ( Hom ` C ) y ) ) -> D e. Cat )
55	14	ad2antrr 762	. . . . . . . . . . 11 |- ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ f e. ( x ( Hom ` C ) y ) ) -> E e. Cat )
56		relfunc 16522	. . . . . . . . . . . . . . . 16 |- Rel ( C Func ( D X.c E ) )
57	36, 1, 6, 11	prfcl 16843	. . . . . . . . . . . . . . . 16 |- ( ph -> P e. ( C Func ( D X.c E ) ) )
58		1st2ndbr 7217	. . . . . . . . . . . . . . . 16 |- ( ( Rel ( C Func ( D X.c E ) ) /\ P e. ( C Func ( D X.c E ) ) ) -> ( 1st ` P ) ( C Func ( D X.c E ) ) ( 2nd ` P ) )
59	56, 57, 58	sylancr 695	. . . . . . . . . . . . . . 15 |- ( ph -> ( 1st ` P ) ( C Func ( D X.c E ) ) ( 2nd ` P ) )
60	17, 4, 59	funcf1 16526	. . . . . . . . . . . . . 14 |- ( ph -> ( 1st ` P ) : ( Base ` C ) --> ( ( Base ` D ) X. ( Base ` E ) ) )
61	60	ffvelrnda 6359	. . . . . . . . . . . . 13 |- ( ( ph /\ x e. ( Base ` C ) ) -> ( ( 1st ` P ) ` x ) e. ( ( Base ` D ) X. ( Base ` E ) ) )
62	61	adantrr 753	. . . . . . . . . . . 12 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> ( ( 1st ` P ) ` x ) e. ( ( Base ` D ) X. ( Base ` E ) ) )
63	62	adantr 481	. . . . . . . . . . 11 |- ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ f e. ( x ( Hom ` C ) y ) ) -> ( ( 1st ` P ) ` x ) e. ( ( Base ` D ) X. ( Base ` E ) ) )
64	60	ffvelrnda 6359	. . . . . . . . . . . . 13 |- ( ( ph /\ y e. ( Base ` C ) ) -> ( ( 1st ` P ) ` y ) e. ( ( Base ` D ) X. ( Base ` E ) ) )
65	64	adantrl 752	. . . . . . . . . . . 12 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> ( ( 1st ` P ) ` y ) e. ( ( Base ` D ) X. ( Base ` E ) ) )
66	65	adantr 481	. . . . . . . . . . 11 |- ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ f e. ( x ( Hom ` C ) y ) ) -> ( ( 1st ` P ) ` y ) e. ( ( Base ` D ) X. ( Base ` E ) ) )
67	1, 4, 5, 54, 55, 16, 63, 66	1stf2 16833	. . . . . . . . . 10 |- ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ f e. ( x ( Hom ` C ) y ) ) -> ( ( ( 1st ` P ) ` x ) ( 2nd ` ( D 1stF E ) ) ( ( 1st ` P ) ` y ) ) = ( 1st |` ( ( ( 1st ` P ) ` x ) ( Hom ` ( D X.c E ) ) ( ( 1st ` P ) ` y ) ) ) )
68	67	fveq1d 6193	. . . . . . . . 9 |- ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ f e. ( x ( Hom ` C ) y ) ) -> ( ( ( ( 1st ` P ) ` x ) ( 2nd ` ( D 1stF E ) ) ( ( 1st ` P ) ` y ) ) ` ( ( x ( 2nd ` P ) y ) ` f ) ) = ( ( 1st |` ( ( ( 1st ` P ) ` x ) ( Hom ` ( D X.c E ) ) ( ( 1st ` P ) ` y ) ) ) ` ( ( x ( 2nd ` P ) y ) ` f ) ) )
69	59	adantr 481	. . . . . . . . . . . 12 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> ( 1st ` P ) ( C Func ( D X.c E ) ) ( 2nd ` P ) )
70		simprl 794	. . . . . . . . . . . 12 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> x e. ( Base ` C ) )
71		simprr 796	. . . . . . . . . . . 12 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> y e. ( Base ` C ) )
72	17, 37, 5, 69, 70, 71	funcf2 16528	. . . . . . . . . . 11 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> ( x ( 2nd ` P ) y ) : ( x ( Hom ` C ) y ) --> ( ( ( 1st ` P ) ` x ) ( Hom ` ( D X.c E ) ) ( ( 1st ` P ) ` y ) ) )
73	72	ffvelrnda 6359	. . . . . . . . . 10 |- ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ f e. ( x ( Hom ` C ) y ) ) -> ( ( x ( 2nd ` P ) y ) ` f ) e. ( ( ( 1st ` P ) ` x ) ( Hom ` ( D X.c E ) ) ( ( 1st ` P ) ` y ) ) )
74		fvres 6207	. . . . . . . . . 10 |- ( ( ( x ( 2nd ` P ) y ) ` f ) e. ( ( ( 1st ` P ) ` x ) ( Hom ` ( D X.c E ) ) ( ( 1st ` P ) ` y ) ) -> ( ( 1st |` ( ( ( 1st ` P ) ` x ) ( Hom ` ( D X.c E ) ) ( ( 1st ` P ) ` y ) ) ) ` ( ( x ( 2nd ` P ) y ) ` f ) ) = ( 1st ` ( ( x ( 2nd ` P ) y ) ` f ) ) )
75	73, 74	syl 17	. . . . . . . . 9 |- ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ f e. ( x ( Hom ` C ) y ) ) -> ( ( 1st |` ( ( ( 1st ` P ) ` x ) ( Hom ` ( D X.c E ) ) ( ( 1st ` P ) ` y ) ) ) ` ( ( x ( 2nd ` P ) y ) ` f ) ) = ( 1st ` ( ( x ( 2nd ` P ) y ) ` f ) ) )
76	6	ad2antrr 762	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ f e. ( x ( Hom ` C ) y ) ) -> F e. ( C Func D ) )
77	11	ad2antrr 762	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ f e. ( x ( Hom ` C ) y ) ) -> G e. ( C Func E ) )
78	70	adantr 481	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ f e. ( x ( Hom ` C ) y ) ) -> x e. ( Base ` C ) )
79	71	adantr 481	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ f e. ( x ( Hom ` C ) y ) ) -> y e. ( Base ` C ) )
80		simpr 477	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ f e. ( x ( Hom ` C ) y ) ) -> f e. ( x ( Hom ` C ) y ) )
81	36, 17, 37, 76, 77, 78, 79, 80	prf2 16842	. . . . . . . . . . 11 |- ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ f e. ( x ( Hom ` C ) y ) ) -> ( ( x ( 2nd ` P ) y ) ` f ) = <. ( ( x ( 2nd ` F ) y ) ` f ) , ( ( x ( 2nd ` G ) y ) ` f ) >. )
82	81	fveq2d 6195	. . . . . . . . . 10 |- ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ f e. ( x ( Hom ` C ) y ) ) -> ( 1st ` ( ( x ( 2nd ` P ) y ) ` f ) ) = ( 1st ` <. ( ( x ( 2nd ` F ) y ) ` f ) , ( ( x ( 2nd ` G ) y ) ` f ) >. ) )
83		fvex 6201	. . . . . . . . . . 11 |- ( ( x ( 2nd ` F ) y ) ` f ) e. _V
84		fvex 6201	. . . . . . . . . . 11 |- ( ( x ( 2nd ` G ) y ) ` f ) e. _V
85	83, 84	op1st 7176	. . . . . . . . . 10 |- ( 1st ` <. ( ( x ( 2nd ` F ) y ) ` f ) , ( ( x ( 2nd ` G ) y ) ` f ) >. ) = ( ( x ( 2nd ` F ) y ) ` f )
86	82, 85	syl6eq 2672	. . . . . . . . 9 |- ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ f e. ( x ( Hom ` C ) y ) ) -> ( 1st ` ( ( x ( 2nd ` P ) y ) ` f ) ) = ( ( x ( 2nd ` F ) y ) ` f ) )
87	68, 75, 86	3eqtrd 2660	. . . . . . . 8 |- ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ f e. ( x ( Hom ` C ) y ) ) -> ( ( ( ( 1st ` P ) ` x ) ( 2nd ` ( D 1stF E ) ) ( ( 1st ` P ) ` y ) ) ` ( ( x ( 2nd ` P ) y ) ` f ) ) = ( ( x ( 2nd ` F ) y ) ` f ) )
88	87	mpteq2dva 4744	. . . . . . 7 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> ( f e. ( x ( Hom ` C ) y ) |-> ( ( ( ( 1st ` P ) ` x ) ( 2nd ` ( D 1stF E ) ) ( ( 1st ` P ) ` y ) ) ` ( ( x ( 2nd ` P ) y ) ` f ) ) ) = ( f e. ( x ( Hom ` C ) y ) |-> ( ( x ( 2nd ` F ) y ) ` f ) ) )
89		eqid 2622	. . . . . . . . 9 |- ( Hom ` D ) = ( Hom ` D )
90	47	adantr 481	. . . . . . . . 9 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> ( 1st ` ( D 1stF E ) ) ( ( D X.c E ) Func D ) ( 2nd ` ( D 1stF E ) ) )
91	4, 5, 89, 90, 62, 65	funcf2 16528	. . . . . . . 8 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> ( ( ( 1st ` P ) ` x ) ( 2nd ` ( D 1stF E ) ) ( ( 1st ` P ) ` y ) ) : ( ( ( 1st ` P ) ` x ) ( Hom ` ( D X.c E ) ) ( ( 1st ` P ) ` y ) ) --> ( ( ( 1st ` ( D 1stF E ) ) ` ( ( 1st ` P ) ` x ) ) ( Hom ` D ) ( ( 1st ` ( D 1stF E ) ) ` ( ( 1st ` P ) ` y ) ) ) )
92		fcompt 6400	. . . . . . . 8 |- ( ( ( ( ( 1st ` P ) ` x ) ( 2nd ` ( D 1stF E ) ) ( ( 1st ` P ) ` y ) ) : ( ( ( 1st ` P ) ` x ) ( Hom ` ( D X.c E ) ) ( ( 1st ` P ) ` y ) ) --> ( ( ( 1st ` ( D 1stF E ) ) ` ( ( 1st ` P ) ` x ) ) ( Hom ` D ) ( ( 1st ` ( D 1stF E ) ) ` ( ( 1st ` P ) ` y ) ) ) /\ ( x ( 2nd ` P ) y ) : ( x ( Hom ` C ) y ) --> ( ( ( 1st ` P ) ` x ) ( Hom ` ( D X.c E ) ) ( ( 1st ` P ) ` y ) ) ) -> ( ( ( ( 1st ` P ) ` x ) ( 2nd ` ( D 1stF E ) ) ( ( 1st ` P ) ` y ) ) o. ( x ( 2nd ` P ) y ) ) = ( f e. ( x ( Hom ` C ) y ) |-> ( ( ( ( 1st ` P ) ` x ) ( 2nd ` ( D 1stF E ) ) ( ( 1st ` P ) ` y ) ) ` ( ( x ( 2nd ` P ) y ) ` f ) ) ) )
93	91, 72, 92	syl2anc 693	. . . . . . 7 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> ( ( ( ( 1st ` P ) ` x ) ( 2nd ` ( D 1stF E ) ) ( ( 1st ` P ) ` y ) ) o. ( x ( 2nd ` P ) y ) ) = ( f e. ( x ( Hom ` C ) y ) |-> ( ( ( ( 1st ` P ) ` x ) ( 2nd ` ( D 1stF E ) ) ( ( 1st ` P ) ` y ) ) ` ( ( x ( 2nd ` P ) y ) ` f ) ) ) )
94	20	adantr 481	. . . . . . . . 9 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> ( 1st ` F ) ( C Func D ) ( 2nd ` F ) )
95	17, 37, 89, 94, 70, 71	funcf2 16528	. . . . . . . 8 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> ( x ( 2nd ` F ) y ) : ( x ( Hom ` C ) y ) --> ( ( ( 1st ` F ) ` x ) ( Hom ` D ) ( ( 1st ` F ) ` y ) ) )
96	95	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 ) ) )
97	88, 93, 96	3eqtr4d 2666	. . . . . 6 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> ( ( ( ( 1st ` P ) ` x ) ( 2nd ` ( D 1stF E ) ) ( ( 1st ` P ) ` y ) ) o. ( x ( 2nd ` P ) y ) ) = ( x ( 2nd ` F ) y ) )
98	97	3impb 1260	. . . . 5 |- ( ( ph /\ x e. ( Base ` C ) /\ y e. ( Base ` C ) ) -> ( ( ( ( 1st ` P ) ` x ) ( 2nd ` ( D 1stF E ) ) ( ( 1st ` P ) ` y ) ) o. ( x ( 2nd ` P ) y ) ) = ( x ( 2nd ` F ) y ) )
99	98	mpt2eq3dva 6719	. . . 4 |- ( ph -> ( x e. ( Base ` C ) , y e. ( Base ` C ) |-> ( ( ( ( 1st ` P ) ` x ) ( 2nd ` ( D 1stF E ) ) ( ( 1st ` P ) ` y ) ) o. ( x ( 2nd ` P ) y ) ) ) = ( x e. ( Base ` C ) , y e. ( Base ` C ) |-> ( x ( 2nd ` F ) y ) ) )
100	17, 20	funcfn2 16529	. . . . 5 |- ( ph -> ( 2nd ` F ) Fn ( ( Base ` C ) X. ( Base ` C ) ) )
101		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 ) ) )
102	100, 101	sylib 208	. . . 4 |- ( ph -> ( 2nd ` F ) = ( x e. ( Base ` C ) , y e. ( Base ` C ) |-> ( x ( 2nd ` F ) y ) ) )
103	99, 102	eqtr4d 2659	. . 3 |- ( ph -> ( x e. ( Base ` C ) , y e. ( Base ` C ) |-> ( ( ( ( 1st ` P ) ` x ) ( 2nd ` ( D 1stF E ) ) ( ( 1st ` P ) ` y ) ) o. ( x ( 2nd ` P ) y ) ) ) = ( 2nd ` F ) )
104	53, 103	opeq12d 4410	. 2 |- ( ph -> <. ( ( 1st ` ( D 1stF E ) ) o. ( 1st ` P ) ) , ( x e. ( Base ` C ) , y e. ( Base ` C ) |-> ( ( ( ( 1st ` P ) ` x ) ( 2nd ` ( D 1stF E ) ) ( ( 1st ` P ) ` y ) ) o. ( x ( 2nd ` P ) y ) ) ) >. = <. ( 1st ` F ) , ( 2nd ` F ) >. )
105	17, 57, 45	cofuval 16542	. 2 |- ( ph -> ( ( D 1stF E ) o.func P ) = <. ( ( 1st ` ( D 1stF E ) ) o. ( 1st ` P ) ) , ( x e. ( Base ` C ) , y e. ( Base ` C ) |-> ( ( ( ( 1st ` P ) ` x ) ( 2nd ` ( D 1stF E ) ) ( ( 1st ` P ) ` y ) ) o. ( x ( 2nd ` P ) y ) ) ) >. )
106		1st2nd 7214	. . 3 |- ( ( Rel ( C Func D ) /\ F e. ( C Func D ) ) -> F = <. ( 1st ` F ) , ( 2nd ` F ) >. )
107	18, 6, 106	sylancr 695	. 2 |- ( ph -> F = <. ( 1st ` F ) , ( 2nd ` F ) >. )
108	104, 105, 107	3eqtr4d 2666	1 |- ( ph -> ( ( D 1stF E ) o.func P ) = 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   o. ccom 5118   Rel wrel 5119   Fn wfn 5883   -->wf 5884   ` 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   ⟨,⟩_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-prf 16815

This theorem is referenced by: (None)