Theorem estrcval 16020

Description: Value of the category of extensible structures (in a universe). (Contributed by AV, 7-Mar-2020.)

Hypotheses

Ref	Expression
estrcval.c	|- C = (ExtStrCat ` U )
estrcval.u	|- ( ph -> U e. V )
estrcval.h	|- ( ph -> H = ( x e. U , y e. U |-> ( ( Base ` y ) ^m ( Base ` x ) ) ) )
estrcval.o	|- ( ph -> .x. = ( v e. ( U X. U ) , z e. U |-> ( g e. ( ( Base ` z ) ^m ( Base ` ( 2nd ` v ) ) ) , f e. ( ( Base ` ( 2nd ` v ) ) ^m ( Base ` ( 1st ` v ) ) ) |-> ( g o. f ) ) ) )

Assertion

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

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

Allowed substitution hints:   ph( 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 estrcval

Dummy variable u is distinct from all other variables.

Step	Hyp	Ref	Expression
1		estrcval.c	. 2 |- C = (ExtStrCat ` U )
2		df-estrc 16019	. . . 4 |- ExtStrCat = ( u e. _V |-> { <. ( Base ` ndx ) , u >. , <. ( Hom ` ndx ) , ( x e. u , y e. u |-> ( ( Base ` y ) ^m ( Base ` x ) ) ) >. , <. (comp ` ndx ) , ( v e. ( u X. u ) , z e. u |-> ( g e. ( ( Base ` z ) ^m ( Base ` ( 2nd ` v ) ) ) , f e. ( ( Base ` ( 2nd ` v ) ) ^m ( Base ` ( 1st ` v ) ) ) |-> ( g o. f ) ) ) >. } )
3	2	a1i 11	. . 3 |- ( ph -> ExtStrCat = ( u e. _V |-> { <. ( Base ` ndx ) , u >. , <. ( Hom ` ndx ) , ( x e. u , y e. u |-> ( ( Base ` y ) ^m ( Base ` x ) ) ) >. , <. (comp ` ndx ) , ( v e. ( u X. u ) , z e. u |-> ( g e. ( ( Base ` z ) ^m ( Base ` ( 2nd ` v ) ) ) , f e. ( ( Base ` ( 2nd ` v ) ) ^m ( Base ` ( 1st ` v ) ) ) |-> ( g o. f ) ) ) >. } ) )
4		simpr 467	. . . . 5 |- ( ( ph /\ u = U ) -> u = U )
5	4	opeq2d 4143	. . . 4 |- ( ( ph /\ u = U ) -> <. ( Base ` ndx ) , u >. = <. ( Base ` ndx ) , U >. )
6		eqidd 2453	. . . . . . 7 |- ( ( ph /\ u = U ) -> ( ( Base ` y ) ^m ( Base ` x ) ) = ( ( Base ` y ) ^m ( Base ` x ) ) )
7	4, 4, 6	mpt2eq123dv 6341	. . . . . 6 |- ( ( ph /\ u = U ) -> ( x e. u , y e. u |-> ( ( Base ` y ) ^m ( Base ` x ) ) ) = ( x e. U , y e. U |-> ( ( Base ` y ) ^m ( Base ` x ) ) ) )
8		estrcval.h	. . . . . . 7 |- ( ph -> H = ( x e. U , y e. U |-> ( ( Base ` y ) ^m ( Base ` x ) ) ) )
9	8	adantr 471	. . . . . 6 |- ( ( ph /\ u = U ) -> H = ( x e. U , y e. U |-> ( ( Base ` y ) ^m ( Base ` x ) ) ) )
10	7, 9	eqtr4d 2489	. . . . 5 |- ( ( ph /\ u = U ) -> ( x e. u , y e. u |-> ( ( Base ` y ) ^m ( Base ` x ) ) ) = H )
11	10	opeq2d 4143	. . . 4 |- ( ( ph /\ u = U ) -> <. ( Hom ` ndx ) , ( x e. u , y e. u |-> ( ( Base ` y ) ^m ( Base ` x ) ) ) >. = <. ( Hom ` ndx ) , H >. )
12	4	sqxpeqd 4838	. . . . . . 7 |- ( ( ph /\ u = U ) -> ( u X. u ) = ( U X. U ) )
13		eqidd 2453	. . . . . . 7 |- ( ( ph /\ u = U ) -> ( g e. ( ( Base ` z ) ^m ( Base ` ( 2nd ` v ) ) ) , f e. ( ( Base ` ( 2nd ` v ) ) ^m ( Base ` ( 1st ` v ) ) ) |-> ( g o. f ) ) = ( g e. ( ( Base ` z ) ^m ( Base ` ( 2nd ` v ) ) ) , f e. ( ( Base ` ( 2nd ` v ) ) ^m ( Base ` ( 1st ` v ) ) ) |-> ( g o. f ) ) )
14	12, 4, 13	mpt2eq123dv 6341	. . . . . 6 |- ( ( ph /\ u = U ) -> ( v e. ( u X. u ) , z e. u |-> ( g e. ( ( Base ` z ) ^m ( Base ` ( 2nd ` v ) ) ) , f e. ( ( Base ` ( 2nd ` v ) ) ^m ( Base ` ( 1st ` v ) ) ) |-> ( g o. f ) ) ) = ( v e. ( U X. U ) , z e. U |-> ( g e. ( ( Base ` z ) ^m ( Base ` ( 2nd ` v ) ) ) , f e. ( ( Base ` ( 2nd ` v ) ) ^m ( Base ` ( 1st ` v ) ) ) |-> ( g o. f ) ) ) )
15		estrcval.o	. . . . . . 7 |- ( ph -> .x. = ( v e. ( U X. U ) , z e. U |-> ( g e. ( ( Base ` z ) ^m ( Base ` ( 2nd ` v ) ) ) , f e. ( ( Base ` ( 2nd ` v ) ) ^m ( Base ` ( 1st ` v ) ) ) |-> ( g o. f ) ) ) )
16	15	adantr 471	. . . . . 6 |- ( ( ph /\ u = U ) -> .x. = ( v e. ( U X. U ) , z e. U |-> ( g e. ( ( Base ` z ) ^m ( Base ` ( 2nd ` v ) ) ) , f e. ( ( Base ` ( 2nd ` v ) ) ^m ( Base ` ( 1st ` v ) ) ) |-> ( g o. f ) ) ) )
17	14, 16	eqtr4d 2489	. . . . 5 |- ( ( ph /\ u = U ) -> ( v e. ( u X. u ) , z e. u |-> ( g e. ( ( Base ` z ) ^m ( Base ` ( 2nd ` v ) ) ) , f e. ( ( Base ` ( 2nd ` v ) ) ^m ( Base ` ( 1st ` v ) ) ) |-> ( g o. f ) ) ) = .x. )
18	17	opeq2d 4143	. . . 4 |- ( ( ph /\ u = U ) -> <. (comp ` ndx ) , ( v e. ( u X. u ) , z e. u |-> ( g e. ( ( Base ` z ) ^m ( Base ` ( 2nd ` v ) ) ) , f e. ( ( Base ` ( 2nd ` v ) ) ^m ( Base ` ( 1st ` v ) ) ) |-> ( g o. f ) ) ) >. = <. (comp ` ndx ) , .x. >. )
19	5, 11, 18	tpeq123d 4035	. . 3 |- ( ( ph /\ u = U ) -> { <. ( Base ` ndx ) , u >. , <. ( Hom ` ndx ) , ( x e. u , y e. u |-> ( ( Base ` y ) ^m ( Base ` x ) ) ) >. , <. (comp ` ndx ) , ( v e. ( u X. u ) , z e. u |-> ( g e. ( ( Base ` z ) ^m ( Base ` ( 2nd ` v ) ) ) , f e. ( ( Base ` ( 2nd ` v ) ) ^m ( Base ` ( 1st ` v ) ) ) |-> ( g o. f ) ) ) >. } = { <. ( Base ` ndx ) , U >. , <. ( Hom ` ndx ) , H >. , <. (comp ` ndx ) , .x. >. } )
20		estrcval.u	. . . 4 |- ( ph -> U e. V )
21		elex 3022	. . . 4 |- ( U e. V -> U e. _V )
22	20, 21	syl 17	. . 3 |- ( ph -> U e. _V )
23		tpex 6578	. . . 4 |- { <. ( Base ` ndx ) , U >. , <. ( Hom ` ndx ) , H >. , <. (comp ` ndx ) , .x. >. } e. _V
24	23	a1i 11	. . 3 |- ( ph -> { <. ( Base ` ndx ) , U >. , <. ( Hom ` ndx ) , H >. , <. (comp ` ndx ) , .x. >. } e. _V )
25	3, 19, 22, 24	fvmptd 5938	. 2 |- ( ph -> (ExtStrCat ` U ) = { <. ( Base ` ndx ) , U >. , <. ( Hom ` ndx ) , H >. , <. (comp ` ndx ) , .x. >. } )
26	1, 25	syl5eq 2498	1 |- ( ph -> C = { <. ( Base ` ndx ) , U >. , <. ( Hom ` ndx ) , H >. , <. (comp ` ndx ) , .x. >. } )

Syntax hints:   -> wi 4   /\ wa 375   = wceq 1448   e. wcel 1891   _Vcvv 3013   {ctp 3940   <.cop 3942   |-> cmpt 4433   X. cxp 4810   o. ccom 4816   ` cfv 5561  (class class class)co 6276   |-> cmpt2 6278   1stc1st 6779   2ndc2nd 6780   ^m cmap 7459   ndxcnx 15129   Basecbs 15132   Hom chom 15212  compcco 15213  ExtStrCatcestrc 16018

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1673  ax-4 1686  ax-5 1762  ax-6 1809  ax-7 1855  ax-8 1893  ax-9 1900  ax-10 1919  ax-11 1924  ax-12 1937  ax-13 2092  ax-ext 2432  ax-sep 4497  ax-nul 4506  ax-pr 4612  ax-un 6571

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 988  df-tru 1451  df-ex 1668  df-nf 1672  df-sb 1802  df-eu 2304  df-mo 2305  df-clab 2439  df-cleq 2445  df-clel 2448  df-nfc 2582  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3015  df-sbc 3236  df-csb 3332  df-dif 3375  df-un 3377  df-in 3379  df-ss 3386  df-nul 3700  df-if 3850  df-sn 3937  df-pr 3939  df-tp 3941  df-op 3943  df-uni 4169  df-br 4375  df-opab 4434  df-mpt 4435  df-id 4727  df-xp 4818  df-rel 4819  df-cnv 4820  df-co 4821  df-dm 4822  df-iota 5525  df-fun 5563  df-fv 5569  df-oprab 6280  df-mpt2 6281  df-estrc 16019

This theorem is referenced by:  estrcbas  16021  estrchomfval  16022  estrccofval  16025  dfrngc2  40299  dfringc2  40345