Theorem prfval 16839

Description: Value of the pairing functor. (Contributed by Mario Carneiro, 12-Jan-2017.)

Hypotheses

Ref	Expression
prfval.k	|- P = ( F ⟨,⟩F G )
prfval.b	|- B = ( Base ` C )
prfval.h	|- H = ( Hom ` C )
prfval.c	|- ( ph -> F e. ( C Func D ) )
prfval.d	|- ( ph -> G e. ( C Func E ) )

Assertion

Ref	Expression
prfval	|- ( ph -> P = <. ( x e. B |-> <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. ) , ( x e. B , y e. B |-> ( h e. ( x H y ) |-> <. ( ( x ( 2nd ` F ) y ) ` h ) , ( ( x ( 2nd ` G ) y ) ` h ) >. ) ) >. )

Distinct variable groups:   x, h, y, B   x, C, y   h, F, x, y   ph, h, x, y   x, D, y   h, G, x, y   h, H, x, y

Allowed substitution hints:   C( h)   D( h)   P( x, y, h)   E( x, y, h)

Proof of Theorem prfval

Dummy variables f b g are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		prfval.k	. 2 |- P = ( F ⟨,⟩F G )
2		df-prf 16815	. . . 4 |- ⟨,⟩F = ( f e. _V , g e. _V |-> [_ dom ( 1st ` f ) / b ]_ <. ( x e. b |-> <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. ) , ( x e. b , y e. b |-> ( h e. dom ( x ( 2nd ` f ) y ) |-> <. ( ( x ( 2nd ` f ) y ) ` h ) , ( ( x ( 2nd ` g ) y ) ` h ) >. ) ) >. )
3	2	a1i 11	. . 3 |- ( ph -> ⟨,⟩F = ( f e. _V , g e. _V |-> [_ dom ( 1st ` f ) / b ]_ <. ( x e. b |-> <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. ) , ( x e. b , y e. b |-> ( h e. dom ( x ( 2nd ` f ) y ) |-> <. ( ( x ( 2nd ` f ) y ) ` h ) , ( ( x ( 2nd ` g ) y ) ` h ) >. ) ) >. ) )
4		fvex 6201	. . . . . 6 |- ( 1st ` f ) e. _V
5	4	dmex 7099	. . . . 5 |- dom ( 1st ` f ) e. _V
6	5	a1i 11	. . . 4 |- ( ( ph /\ ( f = F /\ g = G ) ) -> dom ( 1st ` f ) e. _V )
7		simprl 794	. . . . . . 7 |- ( ( ph /\ ( f = F /\ g = G ) ) -> f = F )
8	7	fveq2d 6195	. . . . . 6 |- ( ( ph /\ ( f = F /\ g = G ) ) -> ( 1st ` f ) = ( 1st ` F ) )
9	8	dmeqd 5326	. . . . 5 |- ( ( ph /\ ( f = F /\ g = G ) ) -> dom ( 1st ` f ) = dom ( 1st ` F ) )
10		prfval.b	. . . . . . . 8 |- B = ( Base ` C )
11		eqid 2622	. . . . . . . 8 |- ( Base ` D ) = ( Base ` D )
12		relfunc 16522	. . . . . . . . 9 |- Rel ( C Func D )
13		prfval.c	. . . . . . . . 9 |- ( ph -> F e. ( C Func D ) )
14		1st2ndbr 7217	. . . . . . . . 9 |- ( ( Rel ( C Func D ) /\ F e. ( C Func D ) ) -> ( 1st ` F ) ( C Func D ) ( 2nd ` F ) )
15	12, 13, 14	sylancr 695	. . . . . . . 8 |- ( ph -> ( 1st ` F ) ( C Func D ) ( 2nd ` F ) )
16	10, 11, 15	funcf1 16526	. . . . . . 7 |- ( ph -> ( 1st ` F ) : B --> ( Base ` D ) )
17		fdm 6051	. . . . . . 7 |- ( ( 1st ` F ) : B --> ( Base ` D ) -> dom ( 1st ` F ) = B )
18	16, 17	syl 17	. . . . . 6 |- ( ph -> dom ( 1st ` F ) = B )
19	18	adantr 481	. . . . 5 |- ( ( ph /\ ( f = F /\ g = G ) ) -> dom ( 1st ` F ) = B )
20	9, 19	eqtrd 2656	. . . 4 |- ( ( ph /\ ( f = F /\ g = G ) ) -> dom ( 1st ` f ) = B )
21		simpr 477	. . . . . 6 |- ( ( ( ph /\ ( f = F /\ g = G ) ) /\ b = B ) -> b = B )
22		simplrl 800	. . . . . . . . 9 |- ( ( ( ph /\ ( f = F /\ g = G ) ) /\ b = B ) -> f = F )
23	22	fveq2d 6195	. . . . . . . 8 |- ( ( ( ph /\ ( f = F /\ g = G ) ) /\ b = B ) -> ( 1st ` f ) = ( 1st ` F ) )
24	23	fveq1d 6193	. . . . . . 7 |- ( ( ( ph /\ ( f = F /\ g = G ) ) /\ b = B ) -> ( ( 1st ` f ) ` x ) = ( ( 1st ` F ) ` x ) )
25		simplrr 801	. . . . . . . . 9 |- ( ( ( ph /\ ( f = F /\ g = G ) ) /\ b = B ) -> g = G )
26	25	fveq2d 6195	. . . . . . . 8 |- ( ( ( ph /\ ( f = F /\ g = G ) ) /\ b = B ) -> ( 1st ` g ) = ( 1st ` G ) )
27	26	fveq1d 6193	. . . . . . 7 |- ( ( ( ph /\ ( f = F /\ g = G ) ) /\ b = B ) -> ( ( 1st ` g ) ` x ) = ( ( 1st ` G ) ` x ) )
28	24, 27	opeq12d 4410	. . . . . 6 |- ( ( ( ph /\ ( f = F /\ g = G ) ) /\ b = B ) -> <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. = <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. )
29	21, 28	mpteq12dv 4733	. . . . 5 |- ( ( ( ph /\ ( f = F /\ g = G ) ) /\ b = B ) -> ( x e. b |-> <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. ) = ( x e. B |-> <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. ) )
30		eqidd 2623	. . . . . . 7 |- ( ( ( ph /\ ( f = F /\ g = G ) ) /\ b = B ) -> ( h e. dom ( x ( 2nd ` f ) y ) |-> <. ( ( x ( 2nd ` f ) y ) ` h ) , ( ( x ( 2nd ` g ) y ) ` h ) >. ) = ( h e. dom ( x ( 2nd ` f ) y ) |-> <. ( ( x ( 2nd ` f ) y ) ` h ) , ( ( x ( 2nd ` g ) y ) ` h ) >. ) )
31	21, 21, 30	mpt2eq123dv 6717	. . . . . 6 |- ( ( ( ph /\ ( f = F /\ g = G ) ) /\ b = B ) -> ( x e. b , y e. b |-> ( h e. dom ( x ( 2nd ` f ) y ) |-> <. ( ( x ( 2nd ` f ) y ) ` h ) , ( ( x ( 2nd ` g ) y ) ` h ) >. ) ) = ( x e. B , y e. B |-> ( h e. dom ( x ( 2nd ` f ) y ) |-> <. ( ( x ( 2nd ` f ) y ) ` h ) , ( ( x ( 2nd ` g ) y ) ` h ) >. ) ) )
32	22	ad2antrr 762	. . . . . . . . . . . . 13 |- ( ( ( ( ( ph /\ ( f = F /\ g = G ) ) /\ b = B ) /\ x e. B ) /\ y e. B ) -> f = F )
33	32	fveq2d 6195	. . . . . . . . . . . 12 |- ( ( ( ( ( ph /\ ( f = F /\ g = G ) ) /\ b = B ) /\ x e. B ) /\ y e. B ) -> ( 2nd ` f ) = ( 2nd ` F ) )
34	33	oveqd 6667	. . . . . . . . . . 11 |- ( ( ( ( ( ph /\ ( f = F /\ g = G ) ) /\ b = B ) /\ x e. B ) /\ y e. B ) -> ( x ( 2nd ` f ) y ) = ( x ( 2nd ` F ) y ) )
35	34	dmeqd 5326	. . . . . . . . . 10 |- ( ( ( ( ( ph /\ ( f = F /\ g = G ) ) /\ b = B ) /\ x e. B ) /\ y e. B ) -> dom ( x ( 2nd ` f ) y ) = dom ( x ( 2nd ` F ) y ) )
36		prfval.h	. . . . . . . . . . . 12 |- H = ( Hom ` C )
37		eqid 2622	. . . . . . . . . . . 12 |- ( Hom ` D ) = ( Hom ` D )
38	15	ad4antr 768	. . . . . . . . . . . 12 |- ( ( ( ( ( ph /\ ( f = F /\ g = G ) ) /\ b = B ) /\ x e. B ) /\ y e. B ) -> ( 1st ` F ) ( C Func D ) ( 2nd ` F ) )
39		simplr 792	. . . . . . . . . . . 12 |- ( ( ( ( ( ph /\ ( f = F /\ g = G ) ) /\ b = B ) /\ x e. B ) /\ y e. B ) -> x e. B )
40		simpr 477	. . . . . . . . . . . 12 |- ( ( ( ( ( ph /\ ( f = F /\ g = G ) ) /\ b = B ) /\ x e. B ) /\ y e. B ) -> y e. B )
41	10, 36, 37, 38, 39, 40	funcf2 16528	. . . . . . . . . . 11 |- ( ( ( ( ( ph /\ ( f = F /\ g = G ) ) /\ b = B ) /\ x e. B ) /\ y e. B ) -> ( x ( 2nd ` F ) y ) : ( x H y ) --> ( ( ( 1st ` F ) ` x ) ( Hom ` D ) ( ( 1st ` F ) ` y ) ) )
42		fdm 6051	. . . . . . . . . . 11 |- ( ( x ( 2nd ` F ) y ) : ( x H y ) --> ( ( ( 1st ` F ) ` x ) ( Hom ` D ) ( ( 1st ` F ) ` y ) ) -> dom ( x ( 2nd ` F ) y ) = ( x H y ) )
43	41, 42	syl 17	. . . . . . . . . 10 |- ( ( ( ( ( ph /\ ( f = F /\ g = G ) ) /\ b = B ) /\ x e. B ) /\ y e. B ) -> dom ( x ( 2nd ` F ) y ) = ( x H y ) )
44	35, 43	eqtrd 2656	. . . . . . . . 9 |- ( ( ( ( ( ph /\ ( f = F /\ g = G ) ) /\ b = B ) /\ x e. B ) /\ y e. B ) -> dom ( x ( 2nd ` f ) y ) = ( x H y ) )
45	34	fveq1d 6193	. . . . . . . . . 10 |- ( ( ( ( ( ph /\ ( f = F /\ g = G ) ) /\ b = B ) /\ x e. B ) /\ y e. B ) -> ( ( x ( 2nd ` f ) y ) ` h ) = ( ( x ( 2nd ` F ) y ) ` h ) )
46	25	ad2antrr 762	. . . . . . . . . . . . 13 |- ( ( ( ( ( ph /\ ( f = F /\ g = G ) ) /\ b = B ) /\ x e. B ) /\ y e. B ) -> g = G )
47	46	fveq2d 6195	. . . . . . . . . . . 12 |- ( ( ( ( ( ph /\ ( f = F /\ g = G ) ) /\ b = B ) /\ x e. B ) /\ y e. B ) -> ( 2nd ` g ) = ( 2nd ` G ) )
48	47	oveqd 6667	. . . . . . . . . . 11 |- ( ( ( ( ( ph /\ ( f = F /\ g = G ) ) /\ b = B ) /\ x e. B ) /\ y e. B ) -> ( x ( 2nd ` g ) y ) = ( x ( 2nd ` G ) y ) )
49	48	fveq1d 6193	. . . . . . . . . 10 |- ( ( ( ( ( ph /\ ( f = F /\ g = G ) ) /\ b = B ) /\ x e. B ) /\ y e. B ) -> ( ( x ( 2nd ` g ) y ) ` h ) = ( ( x ( 2nd ` G ) y ) ` h ) )
50	45, 49	opeq12d 4410	. . . . . . . . 9 |- ( ( ( ( ( ph /\ ( f = F /\ g = G ) ) /\ b = B ) /\ x e. B ) /\ y e. B ) -> <. ( ( x ( 2nd ` f ) y ) ` h ) , ( ( x ( 2nd ` g ) y ) ` h ) >. = <. ( ( x ( 2nd ` F ) y ) ` h ) , ( ( x ( 2nd ` G ) y ) ` h ) >. )
51	44, 50	mpteq12dv 4733	. . . . . . . 8 |- ( ( ( ( ( ph /\ ( f = F /\ g = G ) ) /\ b = B ) /\ x e. B ) /\ y e. B ) -> ( h e. dom ( x ( 2nd ` f ) y ) |-> <. ( ( x ( 2nd ` f ) y ) ` h ) , ( ( x ( 2nd ` g ) y ) ` h ) >. ) = ( h e. ( x H y ) |-> <. ( ( x ( 2nd ` F ) y ) ` h ) , ( ( x ( 2nd ` G ) y ) ` h ) >. ) )
52	51	3impa 1259	. . . . . . 7 |- ( ( ( ( ph /\ ( f = F /\ g = G ) ) /\ b = B ) /\ x e. B /\ y e. B ) -> ( h e. dom ( x ( 2nd ` f ) y ) |-> <. ( ( x ( 2nd ` f ) y ) ` h ) , ( ( x ( 2nd ` g ) y ) ` h ) >. ) = ( h e. ( x H y ) |-> <. ( ( x ( 2nd ` F ) y ) ` h ) , ( ( x ( 2nd ` G ) y ) ` h ) >. ) )
53	52	mpt2eq3dva 6719	. . . . . 6 |- ( ( ( ph /\ ( f = F /\ g = G ) ) /\ b = B ) -> ( x e. B , y e. B |-> ( h e. dom ( x ( 2nd ` f ) y ) |-> <. ( ( x ( 2nd ` f ) y ) ` h ) , ( ( x ( 2nd ` g ) y ) ` h ) >. ) ) = ( x e. B , y e. B |-> ( h e. ( x H y ) |-> <. ( ( x ( 2nd ` F ) y ) ` h ) , ( ( x ( 2nd ` G ) y ) ` h ) >. ) ) )
54	31, 53	eqtrd 2656	. . . . 5 |- ( ( ( ph /\ ( f = F /\ g = G ) ) /\ b = B ) -> ( x e. b , y e. b |-> ( h e. dom ( x ( 2nd ` f ) y ) |-> <. ( ( x ( 2nd ` f ) y ) ` h ) , ( ( x ( 2nd ` g ) y ) ` h ) >. ) ) = ( x e. B , y e. B |-> ( h e. ( x H y ) |-> <. ( ( x ( 2nd ` F ) y ) ` h ) , ( ( x ( 2nd ` G ) y ) ` h ) >. ) ) )
55	29, 54	opeq12d 4410	. . . 4 |- ( ( ( ph /\ ( f = F /\ g = G ) ) /\ b = B ) -> <. ( x e. b |-> <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. ) , ( x e. b , y e. b |-> ( h e. dom ( x ( 2nd ` f ) y ) |-> <. ( ( x ( 2nd ` f ) y ) ` h ) , ( ( x ( 2nd ` g ) y ) ` h ) >. ) ) >. = <. ( x e. B |-> <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. ) , ( x e. B , y e. B |-> ( h e. ( x H y ) |-> <. ( ( x ( 2nd ` F ) y ) ` h ) , ( ( x ( 2nd ` G ) y ) ` h ) >. ) ) >. )
56	6, 20, 55	csbied2 3561	. . 3 |- ( ( ph /\ ( f = F /\ g = G ) ) -> [_ dom ( 1st ` f ) / b ]_ <. ( x e. b |-> <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. ) , ( x e. b , y e. b |-> ( h e. dom ( x ( 2nd ` f ) y ) |-> <. ( ( x ( 2nd ` f ) y ) ` h ) , ( ( x ( 2nd ` g ) y ) ` h ) >. ) ) >. = <. ( x e. B |-> <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. ) , ( x e. B , y e. B |-> ( h e. ( x H y ) |-> <. ( ( x ( 2nd ` F ) y ) ` h ) , ( ( x ( 2nd ` G ) y ) ` h ) >. ) ) >. )
57		elex 3212	. . . 4 |- ( F e. ( C Func D ) -> F e. _V )
58	13, 57	syl 17	. . 3 |- ( ph -> F e. _V )
59		prfval.d	. . . 4 |- ( ph -> G e. ( C Func E ) )
60		elex 3212	. . . 4 |- ( G e. ( C Func E ) -> G e. _V )
61	59, 60	syl 17	. . 3 |- ( ph -> G e. _V )
62		opex 4932	. . . 4 |- <. ( x e. B |-> <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. ) , ( x e. B , y e. B |-> ( h e. ( x H y ) |-> <. ( ( x ( 2nd ` F ) y ) ` h ) , ( ( x ( 2nd ` G ) y ) ` h ) >. ) ) >. e. _V
63	62	a1i 11	. . 3 |- ( ph -> <. ( x e. B |-> <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. ) , ( x e. B , y e. B |-> ( h e. ( x H y ) |-> <. ( ( x ( 2nd ` F ) y ) ` h ) , ( ( x ( 2nd ` G ) y ) ` h ) >. ) ) >. e. _V )
64	3, 56, 58, 61, 63	ovmpt2d 6788	. 2 |- ( ph -> ( F ⟨,⟩F G ) = <. ( x e. B |-> <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. ) , ( x e. B , y e. B |-> ( h e. ( x H y ) |-> <. ( ( x ( 2nd ` F ) y ) ` h ) , ( ( x ( 2nd ` G ) y ) ` h ) >. ) ) >. )
65	1, 64	syl5eq 2668	1 |- ( ph -> P = <. ( x e. B |-> <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. ) , ( x e. B , y e. B |-> ( h e. ( x H y ) |-> <. ( ( x ( 2nd ` F ) y ) ` h ) , ( ( x ( 2nd ` G ) y ) ` h ) >. ) ) >. )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   [_csb 3533   <.cop 4183   class class class wbr 4653   |-> cmpt 4729   dom cdm 5114   Rel wrel 5119   -->wf 5884   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   1stc1st 7166   2ndc2nd 7167   Basecbs 15857   Hom chom 15952   Func cfunc 16514   ⟨,⟩_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

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  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-ral 2917  df-rex 2918  df-reu 2919  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-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  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-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-map 7859  df-ixp 7909  df-func 16518  df-prf 16815

This theorem is referenced by:  prf1  16840  prf2fval  16841  prfcl  16843  prf1st  16844  prf2nd  16845  1st2ndprf  16846