Theorem exidresid 28879

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 5244	. . . . . 6 |- ( G e. ( Magma i^i ExId ) -> ( G |` ( Y X. Y ) ) e. _V )
3	1, 2	syl5eqel 2541	. . . . 5 |- ( G e. ( Magma i^i ExId ) -> H e. _V )
4		eqid 2451	. . . . . 6 |- ran H = ran H
5	4	gidval 23832	. . . . 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 1009	. . 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 28877	. . . . . 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 28878	. . . . . 6 |- ( ( G e. ( Magma i^i ExId ) /\ Y C_ X /\ U e. Y ) -> H e. ExId )
15		elin 3634	. . . . . . . 8 |- ( H e. ( Magma i^i ExId ) <-> ( H e. Magma /\ H e. ExId ) )
16		rngopid 23942	. . . . . . . 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 3016	. . . 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 2540	. . . 4 |- ( ( ( G e. ( Magma i^i ExId ) /\ Y C_ X /\ U e. Y ) /\ H e. Magma ) -> U e. ran H )
25	4	exidu1 23945	. . . . . . 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 6194	. . . . . . . 8 |- ( u = U -> ( u H x ) = ( U H x ) )
30	29	eqeq1d 2453	. . . . . . 7 |- ( u = U -> ( ( u H x ) = x <-> ( U H x ) = x ) )
31		oveq2 6195	. . . . . . . 8 |- ( u = U -> ( x H u ) = ( x H U ) )
32	31	eqeq1d 2453	. . . . . . 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 2839	. . . . 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 6171	. . . 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 2491	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 965   = wceq 1370   e. wcel 1758   A.wral 2793   E!wreu 2795   _Vcvv 3065   i^i cin 3422   C_ wss 3423   X. cxp 4933   dom cdm 4935   ran crn 4936   |` cres 4937   ` cfv 5513   iota_crio 6147  (class class class)co 6187  GIdcgi 23806   ExId cexid 23933   Magmacmagm 23937

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4508  ax-nul 4516  ax-pr 4626  ax-un 6469

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-ne 2644  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3067  df-sbc 3282  df-csb 3384  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-nul 3733  df-if 3887  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4187  df-iun 4268  df-br 4388  df-opab 4446  df-mpt 4447  df-id 4731  df-xp 4941  df-rel 4942  df-cnv 4943  df-co 4944  df-dm 4945  df-rn 4946  df-res 4947  df-iota 5476  df-fun 5515  df-fn 5516  df-f 5517  df-fo 5519  df-fv 5521  df-riota 6148  df-ov 6190  df-gid 23811  df-exid 23934  df-mgm 23938

This theorem is referenced by:  isdrngo2  28899