Theorem fucval 16618

Description: Value of the functor category. (Contributed by Mario Carneiro, 6-Jan-2017.)

Hypotheses

Ref	Expression
fucval.q	|- Q = ( C FuncCat D )
fucval.b	|- B = ( C Func D )
fucval.n	|- N = ( C Nat D )
fucval.a	|- A = ( Base ` C )
fucval.o	|- .x. = (comp ` D )
fucval.c	|- ( ph -> C e. Cat )
fucval.d	|- ( ph -> D e. Cat )
fucval.x	|- ( ph -> .xb = ( v e. ( B X. B ) , h e. B |-> [_ ( 1st ` v ) / f ]_ [_ ( 2nd ` v ) / g ]_ ( b e. ( g N h ) , a e. ( f N g ) |-> ( x e. A |-> ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. .x. ( ( 1st ` h ) ` x ) ) ( a ` x ) ) ) ) ) )

Assertion

Ref	Expression
fucval	|- ( ph -> Q = { <. ( Base ` ndx ) , B >. , <. ( Hom ` ndx ) , N >. , <. (comp ` ndx ) , .xb >. } )

Distinct variable groups:   v, h, B   a, b, f, g, h, v, x, ph   C, a, b, f, g, h, v, x   D, a, b, f, g, h, v, x

Allowed substitution hints:   A( x, v, f, g, h, a, b)   B( x, f, g, a, b)   Q( x, v, f, g, h, a, b)   .xb ( x, v, f, g, h, a, b)   .x. ( x, v, f, g, h, a, b)   N( x, v, f, g, h, a, b)

Proof of Theorem fucval

Dummy variables t u are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fucval.q	. 2 |- Q = ( C FuncCat D )
2		df-fuc 16604	. . . 4 |- FuncCat = ( t e. Cat , u e. Cat |-> { <. ( Base ` ndx ) , ( t Func u ) >. , <. ( Hom ` ndx ) , ( t Nat u ) >. , <. (comp ` ndx ) , ( v e. ( ( t Func u ) X. ( t Func u ) ) , h e. ( t Func u ) |-> [_ ( 1st ` v ) / f ]_ [_ ( 2nd ` v ) / g ]_ ( b e. ( g ( t Nat u ) h ) , a e. ( f ( t Nat u ) g ) |-> ( x e. ( Base ` t ) |-> ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. (comp ` u ) ( ( 1st ` h ) ` x ) ) ( a ` x ) ) ) ) ) >. } )
3	2	a1i 11	. . 3 |- ( ph -> FuncCat = ( t e. Cat , u e. Cat |-> { <. ( Base ` ndx ) , ( t Func u ) >. , <. ( Hom ` ndx ) , ( t Nat u ) >. , <. (comp ` ndx ) , ( v e. ( ( t Func u ) X. ( t Func u ) ) , h e. ( t Func u ) |-> [_ ( 1st ` v ) / f ]_ [_ ( 2nd ` v ) / g ]_ ( b e. ( g ( t Nat u ) h ) , a e. ( f ( t Nat u ) g ) |-> ( x e. ( Base ` t ) |-> ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. (comp ` u ) ( ( 1st ` h ) ` x ) ) ( a ` x ) ) ) ) ) >. } ) )
4		simprl 794	. . . . . . 7 |- ( ( ph /\ ( t = C /\ u = D ) ) -> t = C )
5		simprr 796	. . . . . . 7 |- ( ( ph /\ ( t = C /\ u = D ) ) -> u = D )
6	4, 5	oveq12d 6668	. . . . . 6 |- ( ( ph /\ ( t = C /\ u = D ) ) -> ( t Func u ) = ( C Func D ) )
7		fucval.b	. . . . . 6 |- B = ( C Func D )
8	6, 7	syl6eqr 2674	. . . . 5 |- ( ( ph /\ ( t = C /\ u = D ) ) -> ( t Func u ) = B )
9	8	opeq2d 4409	. . . 4 |- ( ( ph /\ ( t = C /\ u = D ) ) -> <. ( Base ` ndx ) , ( t Func u ) >. = <. ( Base ` ndx ) , B >. )
10	4, 5	oveq12d 6668	. . . . . 6 |- ( ( ph /\ ( t = C /\ u = D ) ) -> ( t Nat u ) = ( C Nat D ) )
11		fucval.n	. . . . . 6 |- N = ( C Nat D )
12	10, 11	syl6eqr 2674	. . . . 5 |- ( ( ph /\ ( t = C /\ u = D ) ) -> ( t Nat u ) = N )
13	12	opeq2d 4409	. . . 4 |- ( ( ph /\ ( t = C /\ u = D ) ) -> <. ( Hom ` ndx ) , ( t Nat u ) >. = <. ( Hom ` ndx ) , N >. )
14	8	sqxpeqd 5141	. . . . . . 7 |- ( ( ph /\ ( t = C /\ u = D ) ) -> ( ( t Func u ) X. ( t Func u ) ) = ( B X. B ) )
15	12	oveqd 6667	. . . . . . . . . 10 |- ( ( ph /\ ( t = C /\ u = D ) ) -> ( g ( t Nat u ) h ) = ( g N h ) )
16	12	oveqd 6667	. . . . . . . . . 10 |- ( ( ph /\ ( t = C /\ u = D ) ) -> ( f ( t Nat u ) g ) = ( f N g ) )
17	4	fveq2d 6195	. . . . . . . . . . . 12 |- ( ( ph /\ ( t = C /\ u = D ) ) -> ( Base ` t ) = ( Base ` C ) )
18		fucval.a	. . . . . . . . . . . 12 |- A = ( Base ` C )
19	17, 18	syl6eqr 2674	. . . . . . . . . . 11 |- ( ( ph /\ ( t = C /\ u = D ) ) -> ( Base ` t ) = A )
20	5	fveq2d 6195	. . . . . . . . . . . . . 14 |- ( ( ph /\ ( t = C /\ u = D ) ) -> (comp ` u ) = (comp ` D ) )
21		fucval.o	. . . . . . . . . . . . . 14 |- .x. = (comp ` D )
22	20, 21	syl6eqr 2674	. . . . . . . . . . . . 13 |- ( ( ph /\ ( t = C /\ u = D ) ) -> (comp ` u ) = .x. )
23	22	oveqd 6667	. . . . . . . . . . . 12 |- ( ( ph /\ ( t = C /\ u = D ) ) -> ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. (comp ` u ) ( ( 1st ` h ) ` x ) ) = ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. .x. ( ( 1st ` h ) ` x ) ) )
24	23	oveqd 6667	. . . . . . . . . . 11 |- ( ( ph /\ ( t = C /\ u = D ) ) -> ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. (comp ` u ) ( ( 1st ` h ) ` x ) ) ( a ` x ) ) = ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. .x. ( ( 1st ` h ) ` x ) ) ( a ` x ) ) )
25	19, 24	mpteq12dv 4733	. . . . . . . . . 10 |- ( ( ph /\ ( t = C /\ u = D ) ) -> ( x e. ( Base ` t ) |-> ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. (comp ` u ) ( ( 1st ` h ) ` x ) ) ( a ` x ) ) ) = ( x e. A |-> ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. .x. ( ( 1st ` h ) ` x ) ) ( a ` x ) ) ) )
26	15, 16, 25	mpt2eq123dv 6717	. . . . . . . . 9 |- ( ( ph /\ ( t = C /\ u = D ) ) -> ( b e. ( g ( t Nat u ) h ) , a e. ( f ( t Nat u ) g ) |-> ( x e. ( Base ` t ) |-> ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. (comp ` u ) ( ( 1st ` h ) ` x ) ) ( a ` x ) ) ) ) = ( b e. ( g N h ) , a e. ( f N g ) |-> ( x e. A |-> ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. .x. ( ( 1st ` h ) ` x ) ) ( a ` x ) ) ) ) )
27	26	csbeq2dv 3992	. . . . . . . 8 |- ( ( ph /\ ( t = C /\ u = D ) ) -> [_ ( 2nd ` v ) / g ]_ ( b e. ( g ( t Nat u ) h ) , a e. ( f ( t Nat u ) g ) |-> ( x e. ( Base ` t ) |-> ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. (comp ` u ) ( ( 1st ` h ) ` x ) ) ( a ` x ) ) ) ) = [_ ( 2nd ` v ) / g ]_ ( b e. ( g N h ) , a e. ( f N g ) |-> ( x e. A |-> ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. .x. ( ( 1st ` h ) ` x ) ) ( a ` x ) ) ) ) )
28	27	csbeq2dv 3992	. . . . . . 7 |- ( ( ph /\ ( t = C /\ u = D ) ) -> [_ ( 1st ` v ) / f ]_ [_ ( 2nd ` v ) / g ]_ ( b e. ( g ( t Nat u ) h ) , a e. ( f ( t Nat u ) g ) |-> ( x e. ( Base ` t ) |-> ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. (comp ` u ) ( ( 1st ` h ) ` x ) ) ( a ` x ) ) ) ) = [_ ( 1st ` v ) / f ]_ [_ ( 2nd ` v ) / g ]_ ( b e. ( g N h ) , a e. ( f N g ) |-> ( x e. A |-> ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. .x. ( ( 1st ` h ) ` x ) ) ( a ` x ) ) ) ) )
29	14, 8, 28	mpt2eq123dv 6717	. . . . . 6 |- ( ( ph /\ ( t = C /\ u = D ) ) -> ( v e. ( ( t Func u ) X. ( t Func u ) ) , h e. ( t Func u ) |-> [_ ( 1st ` v ) / f ]_ [_ ( 2nd ` v ) / g ]_ ( b e. ( g ( t Nat u ) h ) , a e. ( f ( t Nat u ) g ) |-> ( x e. ( Base ` t ) |-> ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. (comp ` u ) ( ( 1st ` h ) ` x ) ) ( a ` x ) ) ) ) ) = ( v e. ( B X. B ) , h e. B |-> [_ ( 1st ` v ) / f ]_ [_ ( 2nd ` v ) / g ]_ ( b e. ( g N h ) , a e. ( f N g ) |-> ( x e. A |-> ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. .x. ( ( 1st ` h ) ` x ) ) ( a ` x ) ) ) ) ) )
30		fucval.x	. . . . . . 7 |- ( ph -> .xb = ( v e. ( B X. B ) , h e. B |-> [_ ( 1st ` v ) / f ]_ [_ ( 2nd ` v ) / g ]_ ( b e. ( g N h ) , a e. ( f N g ) |-> ( x e. A |-> ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. .x. ( ( 1st ` h ) ` x ) ) ( a ` x ) ) ) ) ) )
31	30	adantr 481	. . . . . 6 |- ( ( ph /\ ( t = C /\ u = D ) ) -> .xb = ( v e. ( B X. B ) , h e. B |-> [_ ( 1st ` v ) / f ]_ [_ ( 2nd ` v ) / g ]_ ( b e. ( g N h ) , a e. ( f N g ) |-> ( x e. A |-> ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. .x. ( ( 1st ` h ) ` x ) ) ( a ` x ) ) ) ) ) )
32	29, 31	eqtr4d 2659	. . . . 5 |- ( ( ph /\ ( t = C /\ u = D ) ) -> ( v e. ( ( t Func u ) X. ( t Func u ) ) , h e. ( t Func u ) |-> [_ ( 1st ` v ) / f ]_ [_ ( 2nd ` v ) / g ]_ ( b e. ( g ( t Nat u ) h ) , a e. ( f ( t Nat u ) g ) |-> ( x e. ( Base ` t ) |-> ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. (comp ` u ) ( ( 1st ` h ) ` x ) ) ( a ` x ) ) ) ) ) = .xb )
33	32	opeq2d 4409	. . . 4 |- ( ( ph /\ ( t = C /\ u = D ) ) -> <. (comp ` ndx ) , ( v e. ( ( t Func u ) X. ( t Func u ) ) , h e. ( t Func u ) |-> [_ ( 1st ` v ) / f ]_ [_ ( 2nd ` v ) / g ]_ ( b e. ( g ( t Nat u ) h ) , a e. ( f ( t Nat u ) g ) |-> ( x e. ( Base ` t ) |-> ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. (comp ` u ) ( ( 1st ` h ) ` x ) ) ( a ` x ) ) ) ) ) >. = <. (comp ` ndx ) , .xb >. )
34	9, 13, 33	tpeq123d 4283	. . 3 |- ( ( ph /\ ( t = C /\ u = D ) ) -> { <. ( Base ` ndx ) , ( t Func u ) >. , <. ( Hom ` ndx ) , ( t Nat u ) >. , <. (comp ` ndx ) , ( v e. ( ( t Func u ) X. ( t Func u ) ) , h e. ( t Func u ) |-> [_ ( 1st ` v ) / f ]_ [_ ( 2nd ` v ) / g ]_ ( b e. ( g ( t Nat u ) h ) , a e. ( f ( t Nat u ) g ) |-> ( x e. ( Base ` t ) |-> ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. (comp ` u ) ( ( 1st ` h ) ` x ) ) ( a ` x ) ) ) ) ) >. } = { <. ( Base ` ndx ) , B >. , <. ( Hom ` ndx ) , N >. , <. (comp ` ndx ) , .xb >. } )
35		fucval.c	. . 3 |- ( ph -> C e. Cat )
36		fucval.d	. . 3 |- ( ph -> D e. Cat )
37		tpex 6957	. . . 4 |- { <. ( Base ` ndx ) , B >. , <. ( Hom ` ndx ) , N >. , <. (comp ` ndx ) , .xb >. } e. _V
38	37	a1i 11	. . 3 |- ( ph -> { <. ( Base ` ndx ) , B >. , <. ( Hom ` ndx ) , N >. , <. (comp ` ndx ) , .xb >. } e. _V )
39	3, 34, 35, 36, 38	ovmpt2d 6788	. 2 |- ( ph -> ( C FuncCat D ) = { <. ( Base ` ndx ) , B >. , <. ( Hom ` ndx ) , N >. , <. (comp ` ndx ) , .xb >. } )
40	1, 39	syl5eq 2668	1 |- ( ph -> Q = { <. ( Base ` ndx ) , B >. , <. ( Hom ` ndx ) , N >. , <. (comp ` ndx ) , .xb >. } )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   [_csb 3533   {ctp 4181   <.cop 4183   |-> cmpt 4729   X. cxp 5112   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   1stc1st 7166   2ndc2nd 7167   ndxcnx 15854   Basecbs 15857   Hom chom 15952  compcco 15953   Catccat 16325   Func cfunc 16514   Nat cnat 16601   FuncCat cfuc 16602

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-sep 4781  ax-nul 4789  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-ral 2917  df-rex 2918  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-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  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-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-fuc 16604

This theorem is referenced by:  fuccofval  16619  fucbas  16620  fuchom  16621  fucpropd  16637  catcfuccl  16759