Theorem exidresid 29795

Description: The restriction of a binary operation with identity to a subset containing the identity has the same identity element. (Contributed by Jeff Madsen, 8-Jun-2010.) (Revised by Mario Carneiro, 23-Dec-2013.)

Hypotheses

Ref	Expression
exidres.1	|- X = ran G
exidres.2	|- U = (GId ` G )
exidres.3	|- H = ( G |` ( Y X. Y ) )

Assertion

Ref	Expression
exidresid	|- ( ( ( G e. ( Magma i^i ExId ) /\ Y C_ X /\ U e. Y ) /\ H e. Magma ) -> (GId ` H ) = U )

Proof of Theorem exidresid

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

Step	Hyp	Ref	Expression
1		exidres.3	. . . . . 6 |- H = ( G |` ( Y X. Y ) )
2		resexg 5307	. . . . . 6 |- ( G e. ( Magma i^i ExId ) -> ( G |` ( Y X. Y ) ) e. _V )
3	1, 2	syl5eqel 2552	. . . . 5 |- ( G e. ( Magma i^i ExId ) -> H e. _V )
4		eqid 2460	. . . . . 6 |- ran H = ran H
5	4	gidval 24741	. . . . 5 |- ( H e. _V -> (GId ` H ) = ( iota_ u e. ran H A. x e. ran H ( ( u H x ) = x /\ ( x H u ) = x ) ) )
6	3, 5	syl 16	. . . 4 |- ( G e. ( Magma i^i ExId ) -> (GId ` H ) = ( iota_ u e. ran H A. x e. ran H ( ( u H x ) = x /\ ( x H u ) = x ) ) )
7	6	3ad2ant1 1012	. . 3 |- ( ( G e. ( Magma i^i ExId ) /\ Y C_ X /\ U e. Y ) -> (GId ` H ) = ( iota_ u e. ran H A. x e. ran H ( ( u H x ) = x /\ ( x H u ) = x ) ) )
8	7	adantr 465	. 2 |- ( ( ( G e. ( Magma i^i ExId ) /\ Y C_ X /\ U e. Y ) /\ H e. Magma ) -> (GId ` H ) = ( iota_ u e. ran H A. x e. ran H ( ( u H x ) = x /\ ( x H u ) = x ) ) )
9		exidres.1	. . . . . . 7 |- X = ran G
10		exidres.2	. . . . . . 7 |- U = (GId ` G )
11	9, 10, 1	exidreslem 29793	. . . . . 6 |- ( ( G e. ( Magma i^i ExId ) /\ Y C_ X /\ U e. Y ) -> ( U e. dom dom H /\ A. x e. dom dom H ( ( U H x ) = x /\ ( x H U ) = x ) ) )
12	11	simprd 463	. . . . 5 |- ( ( G e. ( Magma i^i ExId ) /\ Y C_ X /\ U e. Y ) -> A. x e. dom dom H ( ( U H x ) = x /\ ( x H U ) = x ) )
13	12	adantr 465	. . . 4 |- ( ( ( G e. ( Magma i^i ExId ) /\ Y C_ X /\ U e. Y ) /\ H e. Magma ) -> A. x e. dom dom H ( ( U H x ) = x /\ ( x H U ) = x ) )
14	9, 10, 1	exidres 29794	. . . . . 6 |- ( ( G e. ( Magma i^i ExId ) /\ Y C_ X /\ U e. Y ) -> H e. ExId )
15		elin 3680	. . . . . . . 8 |- ( H e. ( Magma i^i ExId ) <-> ( H e. Magma /\ H e. ExId ) )
16		rngopid 24851	. . . . . . . 8 |- ( H e. ( Magma i^i ExId ) -> ran H = dom dom H )
17	15, 16	sylbir 213	. . . . . . 7 |- ( ( H e. Magma /\ H e. ExId ) -> ran H = dom dom H )
18	17	ancoms 453	. . . . . 6 |- ( ( H e. ExId /\ H e. Magma ) -> ran H = dom dom H )
19	14, 18	sylan 471	. . . . 5 |- ( ( ( G e. ( Magma i^i ExId ) /\ Y C_ X /\ U e. Y ) /\ H e. Magma ) -> ran H = dom dom H )
20	19	raleqdv 3057	. . . 4 |- ( ( ( G e. ( Magma i^i ExId ) /\ Y C_ X /\ U e. Y ) /\ H e. Magma ) -> ( A. x e. ran H ( ( U H x ) = x /\ ( x H U ) = x ) <-> A. x e. dom dom H ( ( U H x ) = x /\ ( x H U ) = x ) ) )
21	13, 20	mpbird 232	. . 3 |- ( ( ( G e. ( Magma i^i ExId ) /\ Y C_ X /\ U e. Y ) /\ H e. Magma ) -> A. x e. ran H ( ( U H x ) = x /\ ( x H U ) = x ) )
22	11	simpld 459	. . . . . 6 |- ( ( G e. ( Magma i^i ExId ) /\ Y C_ X /\ U e. Y ) -> U e. dom dom H )
23	22	adantr 465	. . . . 5 |- ( ( ( G e. ( Magma i^i ExId ) /\ Y C_ X /\ U e. Y ) /\ H e. Magma ) -> U e. dom dom H )
24	23, 19	eleqtrrd 2551	. . . 4 |- ( ( ( G e. ( Magma i^i ExId ) /\ Y C_ X /\ U e. Y ) /\ H e. Magma ) -> U e. ran H )
25	4	exidu1 24854	. . . . . . 7 |- ( H e. ( Magma i^i ExId ) -> E! u e. ran H A. x e. ran H ( ( u H x ) = x /\ ( x H u ) = x ) )
26	15, 25	sylbir 213	. . . . . 6 |- ( ( H e. Magma /\ H e. ExId ) -> E! u e. ran H A. x e. ran H ( ( u H x ) = x /\ ( x H u ) = x ) )
27	26	ancoms 453	. . . . 5 |- ( ( H e. ExId /\ H e. Magma ) -> E! u e. ran H A. x e. ran H ( ( u H x ) = x /\ ( x H u ) = x ) )
28	14, 27	sylan 471	. . . 4 |- ( ( ( G e. ( Magma i^i ExId ) /\ Y C_ X /\ U e. Y ) /\ H e. Magma ) -> E! u e. ran H A. x e. ran H ( ( u H x ) = x /\ ( x H u ) = x ) )
29		oveq1 6282	. . . . . . . 8 |- ( u = U -> ( u H x ) = ( U H x ) )
30	29	eqeq1d 2462	. . . . . . 7 |- ( u = U -> ( ( u H x ) = x <-> ( U H x ) = x ) )
31		oveq2 6283	. . . . . . . 8 |- ( u = U -> ( x H u ) = ( x H U ) )
32	31	eqeq1d 2462	. . . . . . 7 |- ( u = U -> ( ( x H u ) = x <-> ( x H U ) = x ) )
33	30, 32	anbi12d 710	. . . . . 6 |- ( u = U -> ( ( ( u H x ) = x /\ ( x H u ) = x ) <-> ( ( U H x ) = x /\ ( x H U ) = x ) ) )
34	33	ralbidv 2896	. . . . 5 |- ( u = U -> ( A. x e. ran H ( ( u H x ) = x /\ ( x H u ) = x ) <-> A. x e. ran H ( ( U H x ) = x /\ ( x H U ) = x ) ) )
35	34	riota2 6259	. . . 4 |- ( ( U e. ran H /\ E! u e. ran H A. x e. ran H ( ( u H x ) = x /\ ( x H u ) = x ) ) -> ( A. x e. ran H ( ( U H x ) = x /\ ( x H U ) = x ) <-> ( iota_ u e. ran H A. x e. ran H ( ( u H x ) = x /\ ( x H u ) = x ) ) = U ) )
36	24, 28, 35	syl2anc 661	. . 3 |- ( ( ( G e. ( Magma i^i ExId ) /\ Y C_ X /\ U e. Y ) /\ H e. Magma ) -> ( A. x e. ran H ( ( U H x ) = x /\ ( x H U ) = x ) <-> ( iota_ u e. ran H A. x e. ran H ( ( u H x ) = x /\ ( x H u ) = x ) ) = U ) )
37	21, 36	mpbid 210	. 2 |- ( ( ( G e. ( Magma i^i ExId ) /\ Y C_ X /\ U e. Y ) /\ H e. Magma ) -> ( iota_ u e. ran H A. x e. ran H ( ( u H x ) = x /\ ( x H u ) = x ) ) = U )
38	8, 37	eqtrd 2501	1 |- ( ( ( G e. ( Magma i^i ExId ) /\ Y C_ X /\ U e. Y ) /\ H e. Magma ) -> (GId ` H ) = U )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 968   = wceq 1374   e. wcel 1762   A.wral 2807   E!wreu 2809   _Vcvv 3106   i^i cin 3468   C_ wss 3469   X. cxp 4990   dom cdm 4992   ran crn 4993   |` cres 4994   ` cfv 5579   iota_crio 6235  (class class class)co 6275  GIdcgi 24715   ExId cexid 24842   Magmacmagm 24846

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pr 4679  ax-un 6567

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-fo 5585  df-fv 5587  df-riota 6236  df-ov 6278  df-gid 24720  df-exid 24843  df-mgm 24847

This theorem is referenced by:  isdrngo2  29815