Theorem exidresid 31623

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 5136	. . . . . 6 |- ( G e. ( Magma i^i ExId ) -> ( G |` ( Y X. Y ) ) e. _V )
3	1, 2	syl5eqel 2494	. . . . 5 |- ( G e. ( Magma i^i ExId ) -> H e. _V )
4		eqid 2402	. . . . . 6 |- ran H = ran H
5	4	gidval 25629	. . . . 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 1018	. . 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 463	. 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 31621	. . . . . 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 461	. . . . 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 463	. . . 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 31622	. . . . . 6 |- ( ( G e. ( Magma i^i ExId ) /\ Y C_ X /\ U e. Y ) -> H e. ExId )
15		elin 3626	. . . . . . . 8 |- ( H e. ( Magma i^i ExId ) <-> ( H e. Magma /\ H e. ExId ) )
16		rngopidOLD 25739	. . . . . . . 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 451	. . . . . 6 |- ( ( H e. ExId /\ H e. Magma ) -> ran H = dom dom H )
19	14, 18	sylan 469	. . . . 5 |- ( ( ( G e. ( Magma i^i ExId ) /\ Y C_ X /\ U e. Y ) /\ H e. Magma ) -> ran H = dom dom H )
20	19	raleqdv 3010	. . . 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 457	. . . . . 6 |- ( ( G e. ( Magma i^i ExId ) /\ Y C_ X /\ U e. Y ) -> U e. dom dom H )
23	22	adantr 463	. . . . 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 2493	. . . 4 |- ( ( ( G e. ( Magma i^i ExId ) /\ Y C_ X /\ U e. Y ) /\ H e. Magma ) -> U e. ran H )
25	4	exidu1 25742	. . . . . . 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 451	. . . . 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 469	. . . 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 6285	. . . . . . . 8 |- ( u = U -> ( u H x ) = ( U H x ) )
30	29	eqeq1d 2404	. . . . . . 7 |- ( u = U -> ( ( u H x ) = x <-> ( U H x ) = x ) )
31		oveq2 6286	. . . . . . . 8 |- ( u = U -> ( x H u ) = ( x H U ) )
32	31	eqeq1d 2404	. . . . . . 7 |- ( u = U -> ( ( x H u ) = x <-> ( x H U ) = x ) )
33	30, 32	anbi12d 709	. . . . . 6 |- ( u = U -> ( ( ( u H x ) = x /\ ( x H u ) = x ) <-> ( ( U H x ) = x /\ ( x H U ) = x ) ) )
34	33	ralbidv 2843	. . . . 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 6262	. . . 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 659	. . 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 2443	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 367   /\ w3a 974   = wceq 1405   e. wcel 1842   A.wral 2754   E!wreu 2756   _Vcvv 3059   i^i cin 3413   C_ wss 3414   X. cxp 4821   dom cdm 4823   ran crn 4824   |` cres 4825   ` cfv 5569   iota_crio 6239  (class class class)co 6278  GIdcgi 25603   ExId cexid 25730   Magmacmagm 25734

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pr 4630  ax-un 6574

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-fo 5575  df-fv 5577  df-riota 6240  df-ov 6281  df-gid 25608  df-exid 25731  df-mgmOLD 25735

This theorem is referenced by:  isdrngo2  31643