Theorem resgrprn 23899

Description: The underlying set of a group operation which is a restriction of a mapping. (Contributed by Paul Chapman, 25-Mar-2008.) (New usage is discouraged.)

Hypothesis

Ref	Expression
resgrprn.1	|- H = ( G |` ( Y X. Y ) )

Assertion

Ref	Expression
resgrprn	|- ( ( dom G = ( X X. X ) /\ H e. GrpOp /\ Y C_ X ) -> Y = ran H )

Proof of Theorem resgrprn

Step	Hyp	Ref	Expression
1		resgrprn.1	. . . . . 6 |- H = ( G |` ( Y X. Y ) )
2	1	dmeqi 5136	. . . . 5 |- dom H = dom ( G |` ( Y X. Y ) )
3		xpss12 5040	. . . . . . . 8 |- ( ( Y C_ X /\ Y C_ X ) -> ( Y X. Y ) C_ ( X X. X ) )
4	3	anidms 645	. . . . . . 7 |- ( Y C_ X -> ( Y X. Y ) C_ ( X X. X ) )
5		sseq2 3473	. . . . . . . 8 |- ( dom G = ( X X. X ) -> ( ( Y X. Y ) C_ dom G <-> ( Y X. Y ) C_ ( X X. X ) ) )
6	5	biimpar 485	. . . . . . 7 |- ( ( dom G = ( X X. X ) /\ ( Y X. Y ) C_ ( X X. X ) ) -> ( Y X. Y ) C_ dom G )
7	4, 6	sylan2 474	. . . . . 6 |- ( ( dom G = ( X X. X ) /\ Y C_ X ) -> ( Y X. Y ) C_ dom G )
8		ssdmres 5227	. . . . . 6 |- ( ( Y X. Y ) C_ dom G <-> dom ( G |` ( Y X. Y ) ) = ( Y X. Y ) )
9	7, 8	sylib 196	. . . . 5 |- ( ( dom G = ( X X. X ) /\ Y C_ X ) -> dom ( G |` ( Y X. Y ) ) = ( Y X. Y ) )
10	2, 9	syl5eq 2503	. . . 4 |- ( ( dom G = ( X X. X ) /\ Y C_ X ) -> dom H = ( Y X. Y ) )
11	10	3adant2 1007	. . 3 |- ( ( dom G = ( X X. X ) /\ H e. GrpOp /\ Y C_ X ) -> dom H = ( Y X. Y ) )
12		eqid 2451	. . . . . 6 |- ran H = ran H
13	12	grpofo 23818	. . . . 5 |- ( H e. GrpOp -> H : ( ran H X. ran H ) -onto-> ran H )
14		fof 5715	. . . . 5 |- ( H : ( ran H X. ran H ) -onto-> ran H -> H : ( ran H X. ran H ) --> ran H )
15		fdm 5658	. . . . 5 |- ( H : ( ran H X. ran H ) --> ran H -> dom H = ( ran H X. ran H ) )
16	13, 14, 15	3syl 20	. . . 4 |- ( H e. GrpOp -> dom H = ( ran H X. ran H ) )
17	16	3ad2ant2 1010	. . 3 |- ( ( dom G = ( X X. X ) /\ H e. GrpOp /\ Y C_ X ) -> dom H = ( ran H X. ran H ) )
18	11, 17	eqtr3d 2493	. 2 |- ( ( dom G = ( X X. X ) /\ H e. GrpOp /\ Y C_ X ) -> ( Y X. Y ) = ( ran H X. ran H ) )
19		xpid11 5156	. 2 |- ( ( Y X. Y ) = ( ran H X. ran H ) <-> Y = ran H )
20	18, 19	sylib 196	1 |- ( ( dom G = ( X X. X ) /\ H e. GrpOp /\ Y C_ X ) -> Y = ran H )

Syntax hints:   -> wi 4   /\ wa 369   /\ w3a 965   = wceq 1370   e. wcel 1758   C_ wss 3423   X. cxp 4933   dom cdm 4935   ran crn 4936   |` cres 4937   -->wf 5509   -onto->wfo 5511   GrpOpcgr 23805

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-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-ov 6190  df-grpo 23810

This theorem is referenced by:  ghablo  23988  efghgrp  23992