Theorem catcval 16746

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 16745	. . . 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 3203	. . . . . 6 |- u e. _V
5	4	inex1 4799	. . . . 5 |- ( u i^i Cat ) e. _V
6	5	a1i 11	. . . 4 |- ( ( ph /\ u = U ) -> ( u i^i Cat ) e. _V )
7		simpr 477	. . . . . 6 |- ( ( ph /\ u = U ) -> u = U )
8	7	ineq1d 3813	. . . . 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 481	. . . . 5 |- ( ( ph /\ u = U ) -> B = ( U i^i Cat ) )
11	8, 10	eqtr4d 2659	. . . 4 |- ( ( ph /\ u = U ) -> ( u i^i Cat ) = B )
12		simpr 477	. . . . . 6 |- ( ( ( ph /\ u = U ) /\ b = B ) -> b = B )
13	12	opeq2d 4409	. . . . 5 |- ( ( ( ph /\ u = U ) /\ b = B ) -> <. ( Base ` ndx ) , b >. = <. ( Base ` ndx ) , B >. )
14		eqidd 2623	. . . . . . . 8 |- ( ( ( ph /\ u = U ) /\ b = B ) -> ( x Func y ) = ( x Func y ) )
15	12, 12, 14	mpt2eq123dv 6717	. . . . . . 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 762	. . . . . . 7 |- ( ( ( ph /\ u = U ) /\ b = B ) -> H = ( x e. B , y e. B |-> ( x Func y ) ) )
18	15, 17	eqtr4d 2659	. . . . . 6 |- ( ( ( ph /\ u = U ) /\ b = B ) -> ( x e. b , y e. b |-> ( x Func y ) ) = H )
19	18	opeq2d 4409	. . . . 5 |- ( ( ( ph /\ u = U ) /\ b = B ) -> <. ( Hom ` ndx ) , ( x e. b , y e. b |-> ( x Func y ) ) >. = <. ( Hom ` ndx ) , H >. )
20	12	sqxpeqd 5141	. . . . . . . 8 |- ( ( ( ph /\ u = U ) /\ b = B ) -> ( b X. b ) = ( B X. B ) )
21		eqidd 2623	. . . . . . . 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 6717	. . . . . . 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 762	. . . . . . 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 2659	. . . . . 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 4409	. . . . 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 4283	. . . 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 3561	. . 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 3212	. . . 4 |- ( U e. V -> U e. _V )
31	29, 30	syl 17	. . 3 |- ( ph -> U e. _V )
32		tpex 6957	. . . 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 6288	. 2 |- ( ph -> (CatCat ` U ) = { <. ( Base ` ndx ) , B >. , <. ( Hom ` ndx ) , H >. , <. (comp ` ndx ) , .x. >. } )
35	1, 34	syl5eq 2668	1 |- ( ph -> C = { <. ( Base ` ndx ) , B >. , <. ( Hom ` ndx ) , H >. , <. (comp ` ndx ) , .x. >. } )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   [_csb 3533   i^i cin 3573   {ctp 4181   <.cop 4183   |-> cmpt 4729   X. cxp 5112   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   2ndc2nd 7167   ndxcnx 15854   Basecbs 15857   Hom chom 15952  compcco 15953   Catccat 16325   Func cfunc 16514   o._func ccofu 16516  CatCatccatc 16744

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-oprab 6654  df-mpt2 6655  df-catc 16745

This theorem is referenced by:  catcbas  16747  catchomfval  16748  catccofval  16750