Theorem exidresid 32170

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 5146	. . . . . 6 |- ( G e. ( Magma i^i ExId ) -> ( G |` ( Y X. Y ) ) e. _V )
3	1, 2	syl5eqel 2532	. . . . 5 |- ( G e. ( Magma i^i ExId ) -> H e. _V )
4		eqid 2450	. . . . . 6 |- ran H = ran H
5	4	gidval 25934	. . . . 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 17	. . . 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 1028	. . 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 467	. 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 32168	. . . . . 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 465	. . . . 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 467	. . . 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 32169	. . . . . 6 |- ( ( G e. ( Magma i^i ExId ) /\ Y C_ X /\ U e. Y ) -> H e. ExId )
15		elin 3616	. . . . . . . 8 |- ( H e. ( Magma i^i ExId ) <-> ( H e. Magma /\ H e. ExId ) )
16		rngopidOLD 26044	. . . . . . . 8 |- ( H e. ( Magma i^i ExId ) -> ran H = dom dom H )
17	15, 16	sylbir 217	. . . . . . 7 |- ( ( H e. Magma /\ H e. ExId ) -> ran H = dom dom H )
18	17	ancoms 455	. . . . . 6 |- ( ( H e. ExId /\ H e. Magma ) -> ran H = dom dom H )
19	14, 18	sylan 474	. . . . 5 |- ( ( ( G e. ( Magma i^i ExId ) /\ Y C_ X /\ U e. Y ) /\ H e. Magma ) -> ran H = dom dom H )
20	19	raleqdv 2992	. . . 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 236	. . 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 461	. . . . . 6 |- ( ( G e. ( Magma i^i ExId ) /\ Y C_ X /\ U e. Y ) -> U e. dom dom H )
23	22	adantr 467	. . . . 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 2531	. . . 4 |- ( ( ( G e. ( Magma i^i ExId ) /\ Y C_ X /\ U e. Y ) /\ H e. Magma ) -> U e. ran H )
25	4	exidu1 26047	. . . . . . 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 217	. . . . . 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 455	. . . . 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 474	. . . 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 6295	. . . . . . . 8 |- ( u = U -> ( u H x ) = ( U H x ) )
30	29	eqeq1d 2452	. . . . . . 7 |- ( u = U -> ( ( u H x ) = x <-> ( U H x ) = x ) )
31		oveq2 6296	. . . . . . . 8 |- ( u = U -> ( x H u ) = ( x H U ) )
32	31	eqeq1d 2452	. . . . . . 7 |- ( u = U -> ( ( x H u ) = x <-> ( x H U ) = x ) )
33	30, 32	anbi12d 716	. . . . . 6 |- ( u = U -> ( ( ( u H x ) = x /\ ( x H u ) = x ) <-> ( ( U H x ) = x /\ ( x H U ) = x ) ) )
34	33	ralbidv 2826	. . . . 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 6272	. . . 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 666	. . 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 214	. 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 2484	1 |- ( ( ( G e. ( Magma i^i ExId ) /\ Y C_ X /\ U e. Y ) /\ H e. Magma ) -> (GId ` H ) = U )

Syntax hints:   -> wi 4   <-> wb 188   /\ wa 371   /\ w3a 984   = wceq 1443   e. wcel 1886   A.wral 2736   E!wreu 2738   _Vcvv 3044   i^i cin 3402   C_ wss 3403   X. cxp 4831   dom cdm 4833   ran crn 4834   |` cres 4835   ` cfv 5581   iota_crio 6249  (class class class)co 6288  GIdcgi 25908   ExId cexid 26035   Magmacmagm 26039

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-sep 4524  ax-nul 4533  ax-pr 4638  ax-un 6580

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-nul 3731  df-if 3881  df-sn 3968  df-pr 3970  df-op 3974  df-uni 4198  df-iun 4279  df-br 4402  df-opab 4461  df-mpt 4462  df-id 4748  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-fo 5587  df-fv 5589  df-riota 6250  df-ov 6291  df-gid 25913  df-exid 26036  df-mgmOLD 26040

This theorem is referenced by:  isdrngo2  32190