Theorem resgrprn 24955

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 5202	. . . . 5 |- dom H = dom ( G |` ( Y X. Y ) )
3		xpss12 5106	. . . . . . . 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 3526	. . . . . . . 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 5293	. . . . . 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 2520	. . . 4 |- ( ( dom G = ( X X. X ) /\ Y C_ X ) -> dom H = ( Y X. Y ) )
11	10	3adant2 1015	. . 3 |- ( ( dom G = ( X X. X ) /\ H e. GrpOp /\ Y C_ X ) -> dom H = ( Y X. Y ) )
12		eqid 2467	. . . . . 6 |- ran H = ran H
13	12	grpofo 24874	. . . . 5 |- ( H e. GrpOp -> H : ( ran H X. ran H ) -onto-> ran H )
14		fof 5793	. . . . 5 |- ( H : ( ran H X. ran H ) -onto-> ran H -> H : ( ran H X. ran H ) --> ran H )
15		fdm 5733	. . . . 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 1018	. . 3 |- ( ( dom G = ( X X. X ) /\ H e. GrpOp /\ Y C_ X ) -> dom H = ( ran H X. ran H ) )
18	11, 17	eqtr3d 2510	. 2 |- ( ( dom G = ( X X. X ) /\ H e. GrpOp /\ Y C_ X ) -> ( Y X. Y ) = ( ran H X. ran H ) )
19		xpid11 5222	. 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 973   = wceq 1379   e. wcel 1767   C_ wss 3476   X. cxp 4997   dom cdm 4999   ran crn 5000   |` cres 5001   -->wf 5582   -onto->wfo 5584   GrpOpcgr 24861

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686  ax-un 6574

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-fo 5592  df-fv 5594  df-ov 6285  df-grpo 24866

This theorem is referenced by:  ghablo  25044  efghgrp  25048