Theorem catcval 14956

Description: Value of the category of categories (in a universe). (Contributed by Mario Carneiro, 3-Jan-2017.)

Hypotheses

Ref	Expression
catcval.c	|- C = (CatCat ` U )
catcval.u	|- ( ph -> U e. V )
catcval.b	|- ( ph -> B = ( U i^i Cat ) )
catcval.h	|- ( ph -> H = ( x e. B , y e. B |-> ( x Func y ) ) )
catcval.o	|- ( ph -> .x. = ( v e. ( B X. B ) , z e. B |-> ( g e. ( ( 2nd ` v ) Func z ) , f e. ( Func ` v ) |-> ( g o.func f ) ) ) )

Assertion

Ref	Expression
catcval	|- ( ph -> C = { <. ( Base ` ndx ) , B >. , <. ( Hom ` ndx ) , H >. , <. (comp ` ndx ) , .x. >. } )

Distinct variable groups:   x, v, y, z, B   ph, v, x, y, z   v, U, x, y, z   f, g, v, x, y, z

Allowed substitution hints:   ph( f, g)   B( f, g)   C( x, y, z, v, f, g)   .x. ( x, y, z, v, f, g)   U( f, g)   H( x, y, z, v, f, g)   V( x, y, z, v, f, g)

Proof of Theorem catcval

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

Step	Hyp	Ref	Expression
1		catcval.c	. 2 |- C = (CatCat ` U )
2		df-catc 14955	. . . 4 |- CatCat = ( u e. _V |-> [_ ( u i^i Cat ) / b ]_ { <. ( Base ` ndx ) , b >. , <. ( Hom ` ndx ) , ( x e. b , y e. b |-> ( x Func y ) ) >. , <. (comp ` ndx ) , ( v e. ( b X. b ) , z e. b |-> ( g e. ( ( 2nd ` v ) Func z ) , f e. ( Func ` v ) |-> ( g o.func f ) ) ) >. } )
3	2	a1i 11	. . 3 |- ( ph -> CatCat = ( u e. _V |-> [_ ( u i^i Cat ) / b ]_ { <. ( Base ` ndx ) , b >. , <. ( Hom ` ndx ) , ( x e. b , y e. b |-> ( x Func y ) ) >. , <. (comp ` ndx ) , ( v e. ( b X. b ) , z e. b |-> ( g e. ( ( 2nd ` v ) Func z ) , f e. ( Func ` v ) |-> ( g o.func f ) ) ) >. } ) )
4		vex 2970	. . . . . 6 |- u e. _V
5	4	inex1 4428	. . . . 5 |- ( u i^i Cat ) e. _V
6	5	a1i 11	. . . 4 |- ( ( ph /\ u = U ) -> ( u i^i Cat ) e. _V )
7		simpr 461	. . . . . 6 |- ( ( ph /\ u = U ) -> u = U )
8	7	ineq1d 3546	. . . . 5 |- ( ( ph /\ u = U ) -> ( u i^i Cat ) = ( U i^i Cat ) )
9		catcval.b	. . . . . 6 |- ( ph -> B = ( U i^i Cat ) )
10	9	adantr 465	. . . . 5 |- ( ( ph /\ u = U ) -> B = ( U i^i Cat ) )
11	8, 10	eqtr4d 2473	. . . 4 |- ( ( ph /\ u = U ) -> ( u i^i Cat ) = B )
12		simpr 461	. . . . . 6 |- ( ( ( ph /\ u = U ) /\ b = B ) -> b = B )
13	12	opeq2d 4061	. . . . 5 |- ( ( ( ph /\ u = U ) /\ b = B ) -> <. ( Base ` ndx ) , b >. = <. ( Base ` ndx ) , B >. )
14		eqidd 2439	. . . . . . . 8 |- ( ( ( ph /\ u = U ) /\ b = B ) -> ( x Func y ) = ( x Func y ) )
15	12, 12, 14	mpt2eq123dv 6143	. . . . . . 7 |- ( ( ( ph /\ u = U ) /\ b = B ) -> ( x e. b , y e. b |-> ( x Func y ) ) = ( x e. B , y e. B |-> ( x Func y ) ) )
16		catcval.h	. . . . . . . 8 |- ( ph -> H = ( x e. B , y e. B |-> ( x Func y ) ) )
17	16	ad2antrr 725	. . . . . . 7 |- ( ( ( ph /\ u = U ) /\ b = B ) -> H = ( x e. B , y e. B |-> ( x Func y ) ) )
18	15, 17	eqtr4d 2473	. . . . . 6 |- ( ( ( ph /\ u = U ) /\ b = B ) -> ( x e. b , y e. b |-> ( x Func y ) ) = H )
19	18	opeq2d 4061	. . . . 5 |- ( ( ( ph /\ u = U ) /\ b = B ) -> <. ( Hom ` ndx ) , ( x e. b , y e. b |-> ( x Func y ) ) >. = <. ( Hom ` ndx ) , H >. )
20	12, 12	xpeq12d 4860	. . . . . . . 8 |- ( ( ( ph /\ u = U ) /\ b = B ) -> ( b X. b ) = ( B X. B ) )
21		eqidd 2439	. . . . . . . 8 |- ( ( ( ph /\ u = U ) /\ b = B ) -> ( g e. ( ( 2nd ` v ) Func z ) , f e. ( Func ` v ) |-> ( g o.func f ) ) = ( g e. ( ( 2nd ` v ) Func z ) , f e. ( Func ` v ) |-> ( g o.func f ) ) )
22	20, 12, 21	mpt2eq123dv 6143	. . . . . . 7 |- ( ( ( ph /\ u = U ) /\ b = B ) -> ( v e. ( b X. b ) , z e. b |-> ( g e. ( ( 2nd ` v ) Func z ) , f e. ( Func ` v ) |-> ( g o.func f ) ) ) = ( v e. ( B X. B ) , z e. B |-> ( g e. ( ( 2nd ` v ) Func z ) , f e. ( Func ` v ) |-> ( g o.func f ) ) ) )
23		catcval.o	. . . . . . . 8 |- ( ph -> .x. = ( v e. ( B X. B ) , z e. B |-> ( g e. ( ( 2nd ` v ) Func z ) , f e. ( Func ` v ) |-> ( g o.func f ) ) ) )
24	23	ad2antrr 725	. . . . . . 7 |- ( ( ( ph /\ u = U ) /\ b = B ) -> .x. = ( v e. ( B X. B ) , z e. B |-> ( g e. ( ( 2nd ` v ) Func z ) , f e. ( Func ` v ) |-> ( g o.func f ) ) ) )
25	22, 24	eqtr4d 2473	. . . . . 6 |- ( ( ( ph /\ u = U ) /\ b = B ) -> ( v e. ( b X. b ) , z e. b |-> ( g e. ( ( 2nd ` v ) Func z ) , f e. ( Func ` v ) |-> ( g o.func f ) ) ) = .x. )
26	25	opeq2d 4061	. . . . 5 |- ( ( ( ph /\ u = U ) /\ b = B ) -> <. (comp ` ndx ) , ( v e. ( b X. b ) , z e. b |-> ( g e. ( ( 2nd ` v ) Func z ) , f e. ( Func ` v ) |-> ( g o.func f ) ) ) >. = <. (comp ` ndx ) , .x. >. )
27	13, 19, 26	tpeq123d 3964	. . . 4 |- ( ( ( ph /\ u = U ) /\ b = B ) -> { <. ( Base ` ndx ) , b >. , <. ( Hom ` ndx ) , ( x e. b , y e. b |-> ( x Func y ) ) >. , <. (comp ` ndx ) , ( v e. ( b X. b ) , z e. b |-> ( g e. ( ( 2nd ` v ) Func z ) , f e. ( Func ` v ) |-> ( g o.func f ) ) ) >. } = { <. ( Base ` ndx ) , B >. , <. ( Hom ` ndx ) , H >. , <. (comp ` ndx ) , .x. >. } )
28	6, 11, 27	csbied2 3310	. . 3 |- ( ( ph /\ u = U ) -> [_ ( u i^i Cat ) / b ]_ { <. ( Base ` ndx ) , b >. , <. ( Hom ` ndx ) , ( x e. b , y e. b |-> ( x Func y ) ) >. , <. (comp ` ndx ) , ( v e. ( b X. b ) , z e. b |-> ( g e. ( ( 2nd ` v ) Func z ) , f e. ( Func ` v ) |-> ( g o.func f ) ) ) >. } = { <. ( Base ` ndx ) , B >. , <. ( Hom ` ndx ) , H >. , <. (comp ` ndx ) , .x. >. } )
29		catcval.u	. . . 4 |- ( ph -> U e. V )
30		elex 2976	. . . 4 |- ( U e. V -> U e. _V )
31	29, 30	syl 16	. . 3 |- ( ph -> U e. _V )
32		tpex 6374	. . . 4 |- { <. ( Base ` ndx ) , B >. , <. ( Hom ` ndx ) , H >. , <. (comp ` ndx ) , .x. >. } e. _V
33	32	a1i 11	. . 3 |- ( ph -> { <. ( Base ` ndx ) , B >. , <. ( Hom ` ndx ) , H >. , <. (comp ` ndx ) , .x. >. } e. _V )
34	3, 28, 31, 33	fvmptd 5774	. 2 |- ( ph -> (CatCat ` U ) = { <. ( Base ` ndx ) , B >. , <. ( Hom ` ndx ) , H >. , <. (comp ` ndx ) , .x. >. } )
35	1, 34	syl5eq 2482	1 |- ( ph -> C = { <. ( Base ` ndx ) , B >. , <. ( Hom ` ndx ) , H >. , <. (comp ` ndx ) , .x. >. } )

Syntax hints:   -> wi 4   /\ wa 369   = wceq 1369   e. wcel 1756   _Vcvv 2967   [_csb 3283   i^i cin 3322   {ctp 3876   <.cop 3878   e. cmpt 4345   X. cxp 4833   ` cfv 5413  (class class class)co 6086   e. cmpt2 6088   2ndc2nd 6571   ndxcnx 14163   Basecbs 14166   Hom chom 14241  compcco 14242   Catccat 14594   Func cfunc 14756   o._func ccofu 14758  CatCatccatc 14954

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pr 4526  ax-un 6367

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-iota 5376  df-fun 5415  df-fv 5421  df-oprab 6090  df-mpt2 6091  df-catc 14955

This theorem is referenced by:  catcbas  14957  catchomfval  14958  catccofval  14960