Theorem exidresid 33678

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 5442	. . . . . 6 |- ( G e. ( Magma i^i ExId ) -> ( G |` ( Y X. Y ) ) e. _V )
3	1, 2	syl5eqel 2705	. . . . 5 |- ( G e. ( Magma i^i ExId ) -> H e. _V )
4		eqid 2622	. . . . . 6 |- ran H = ran H
5	4	gidval 27366	. . . . 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 1082	. . 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 481	. 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 33676	. . . . . 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 479	. . . . 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 481	. . . 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 33677	. . . . . 6 |- ( ( G e. ( Magma i^i ExId ) /\ Y C_ X /\ U e. Y ) -> H e. ExId )
15		elin 3796	. . . . . . . 8 |- ( H e. ( Magma i^i ExId ) <-> ( H e. Magma /\ H e. ExId ) )
16		rngopidOLD 33652	. . . . . . . 8 |- ( H e. ( Magma i^i ExId ) -> ran H = dom dom H )
17	15, 16	sylbir 225	. . . . . . 7 |- ( ( H e. Magma /\ H e. ExId ) -> ran H = dom dom H )
18	17	ancoms 469	. . . . . 6 |- ( ( H e. ExId /\ H e. Magma ) -> ran H = dom dom H )
19	14, 18	sylan 488	. . . . 5 |- ( ( ( G e. ( Magma i^i ExId ) /\ Y C_ X /\ U e. Y ) /\ H e. Magma ) -> ran H = dom dom H )
20	19	raleqdv 3144	. . . 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 247	. . 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 475	. . . . . 6 |- ( ( G e. ( Magma i^i ExId ) /\ Y C_ X /\ U e. Y ) -> U e. dom dom H )
23	22	adantr 481	. . . . 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 2704	. . . 4 |- ( ( ( G e. ( Magma i^i ExId ) /\ Y C_ X /\ U e. Y ) /\ H e. Magma ) -> U e. ran H )
25	4	exidu1 33655	. . . . . . 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 225	. . . . . 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 469	. . . . 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 488	. . . 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 6657	. . . . . . . 8 |- ( u = U -> ( u H x ) = ( U H x ) )
30	29	eqeq1d 2624	. . . . . . 7 |- ( u = U -> ( ( u H x ) = x <-> ( U H x ) = x ) )
31		oveq2 6658	. . . . . . . 8 |- ( u = U -> ( x H u ) = ( x H U ) )
32	31	eqeq1d 2624	. . . . . . 7 |- ( u = U -> ( ( x H u ) = x <-> ( x H U ) = x ) )
33	30, 32	anbi12d 747	. . . . . 6 |- ( u = U -> ( ( ( u H x ) = x /\ ( x H u ) = x ) <-> ( ( U H x ) = x /\ ( x H U ) = x ) ) )
34	33	ralbidv 2986	. . . . 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 6633	. . . 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 693	. . 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 222	. 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 2656	1 |- ( ( ( G e. ( Magma i^i ExId ) /\ Y C_ X /\ U e. Y ) /\ H e. Magma ) -> (GId ` H ) = U )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   E!wreu 2914   _Vcvv 3200   i^i cin 3573   C_ wss 3574   X. cxp 5112   dom cdm 5114   ran crn 5115   |` cres 5116   ` cfv 5888   iota_crio 6610  (class class class)co 6650  GIdcgi 27344   ExId cexid 33643   Magmacmagm 33647

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-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  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-op 4184  df-uni 4437  df-iun 4522  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-rn 5125  df-res 5126  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fo 5894  df-fv 5896  df-riota 6611  df-ov 6653  df-gid 27348  df-exid 33644  df-mgmOLD 33648

This theorem is referenced by:  isdrngo2  33757