Theorem catcval 15082

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 15081	. . . 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 3079	. . . . . 6 |- u e. _V
5	4	inex1 4540	. . . . 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 3658	. . . . 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 2498	. . . 4 |- ( ( ph /\ u = U ) -> ( u i^i Cat ) = B )
12		simpr 461	. . . . . 6 |- ( ( ( ph /\ u = U ) /\ b = B ) -> b = B )
13	12	opeq2d 4173	. . . . 5 |- ( ( ( ph /\ u = U ) /\ b = B ) -> <. ( Base ` ndx ) , b >. = <. ( Base ` ndx ) , B >. )
14		eqidd 2455	. . . . . . . 8 |- ( ( ( ph /\ u = U ) /\ b = B ) -> ( x Func y ) = ( x Func y ) )
15	12, 12, 14	mpt2eq123dv 6256	. . . . . . 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 2498	. . . . . 6 |- ( ( ( ph /\ u = U ) /\ b = B ) -> ( x e. b , y e. b |-> ( x Func y ) ) = H )
19	18	opeq2d 4173	. . . . 5 |- ( ( ( ph /\ u = U ) /\ b = B ) -> <. ( Hom ` ndx ) , ( x e. b , y e. b |-> ( x Func y ) ) >. = <. ( Hom ` ndx ) , H >. )
20	12, 12	xpeq12d 4972	. . . . . . . 8 |- ( ( ( ph /\ u = U ) /\ b = B ) -> ( b X. b ) = ( B X. B ) )
21		eqidd 2455	. . . . . . . 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 6256	. . . . . . 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 2498	. . . . . 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 4173	. . . . 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 4076	. . . 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 3422	. . 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 3085	. . . 4 |- ( U e. V -> U e. _V )
31	29, 30	syl 16	. . 3 |- ( ph -> U e. _V )
32		tpex 6488	. . . 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 5887	. 2 |- ( ph -> (CatCat ` U ) = { <. ( Base ` ndx ) , B >. , <. ( Hom ` ndx ) , H >. , <. (comp ` ndx ) , .x. >. } )
35	1, 34	syl5eq 2507	1 |- ( ph -> C = { <. ( Base ` ndx ) , B >. , <. ( Hom ` ndx ) , H >. , <. (comp ` ndx ) , .x. >. } )

Syntax hints:   -> wi 4   /\ wa 369   = wceq 1370   e. wcel 1758   _Vcvv 3076   [_csb 3394   i^i cin 3434   {ctp 3988   <.cop 3990   |-> cmpt 4457   X. cxp 4945   ` cfv 5525  (class class class)co 6199   |-> cmpt2 6201   2ndc2nd 6685   ndxcnx 14288   Basecbs 14291   Hom chom 14367  compcco 14368   Catccat 14720   Func cfunc 14882   o._func ccofu 14884  CatCatccatc 15080

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4520  ax-nul 4528  ax-pr 4638  ax-un 6481

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-ral 2803  df-rex 2804  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-nul 3745  df-if 3899  df-sn 3985  df-pr 3987  df-tp 3989  df-op 3991  df-uni 4199  df-br 4400  df-opab 4458  df-mpt 4459  df-id 4743  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-iota 5488  df-fun 5527  df-fv 5533  df-oprab 6203  df-mpt2 6204  df-catc 15081

This theorem is referenced by:  catcbas  15083  catchomfval  15084  catccofval  15086