Theorem estrcval 16764

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 16763	. . . 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 477	. . . . 5 |- ( ( ph /\ u = U ) -> u = U )
5	4	opeq2d 4409	. . . 4 |- ( ( ph /\ u = U ) -> <. ( Base ` ndx ) , u >. = <. ( Base ` ndx ) , U >. )
6		eqidd 2623	. . . . . . 7 |- ( ( ph /\ u = U ) -> ( ( Base ` y ) ^m ( Base ` x ) ) = ( ( Base ` y ) ^m ( Base ` x ) ) )
7	4, 4, 6	mpt2eq123dv 6717	. . . . . 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 481	. . . . . 6 |- ( ( ph /\ u = U ) -> H = ( x e. U , y e. U |-> ( ( Base ` y ) ^m ( Base ` x ) ) ) )
10	7, 9	eqtr4d 2659	. . . . 5 |- ( ( ph /\ u = U ) -> ( x e. u , y e. u |-> ( ( Base ` y ) ^m ( Base ` x ) ) ) = H )
11	10	opeq2d 4409	. . . 4 |- ( ( ph /\ u = U ) -> <. ( Hom ` ndx ) , ( x e. u , y e. u |-> ( ( Base ` y ) ^m ( Base ` x ) ) ) >. = <. ( Hom ` ndx ) , H >. )
12	4	sqxpeqd 5141	. . . . . . 7 |- ( ( ph /\ u = U ) -> ( u X. u ) = ( U X. U ) )
13		eqidd 2623	. . . . . . 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 6717	. . . . . 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 481	. . . . . 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 2659	. . . . 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 4409	. . . 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 4283	. . 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 3212	. . . 4 |- ( U e. V -> U e. _V )
22	20, 21	syl 17	. . 3 |- ( ph -> U e. _V )
23		tpex 6957	. . . 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 6288	. 2 |- ( ph -> (ExtStrCat ` U ) = { <. ( Base ` ndx ) , U >. , <. ( Hom ` ndx ) , H >. , <. (comp ` ndx ) , .x. >. } )
26	1, 25	syl5eq 2668	1 |- ( ph -> C = { <. ( Base ` ndx ) , U >. , <. ( Hom ` ndx ) , H >. , <. (comp ` ndx ) , .x. >. } )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   {ctp 4181   <.cop 4183   |-> cmpt 4729   X. cxp 5112   o. ccom 5118   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   1stc1st 7166   2ndc2nd 7167   ^m cmap 7857   ndxcnx 15854   Basecbs 15857   Hom chom 15952  compcco 15953  ExtStrCatcestrc 16762

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-estrc 16763

This theorem is referenced by:  estrcbas  16765  estrchomfval  16766  estrccofval  16769  dfrngc2  41972  dfringc2  42018