Theorem resgrprn 26001

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 5035	. . . . 5 |- dom H = dom ( G |` ( Y X. Y ) )
3		xpss12 4939	. . . . . . . 8 |- ( ( Y C_ X /\ Y C_ X ) -> ( Y X. Y ) C_ ( X X. X ) )
4	3	anidms 650	. . . . . . 7 |- ( Y C_ X -> ( Y X. Y ) C_ ( X X. X ) )
5		sseq2 3453	. . . . . . . 8 |- ( dom G = ( X X. X ) -> ( ( Y X. Y ) C_ dom G <-> ( Y X. Y ) C_ ( X X. X ) ) )
6	5	biimpar 488	. . . . . . 7 |- ( ( dom G = ( X X. X ) /\ ( Y X. Y ) C_ ( X X. X ) ) -> ( Y X. Y ) C_ dom G )
7	4, 6	sylan2 477	. . . . . 6 |- ( ( dom G = ( X X. X ) /\ Y C_ X ) -> ( Y X. Y ) C_ dom G )
8		ssdmres 5125	. . . . . 6 |- ( ( Y X. Y ) C_ dom G <-> dom ( G |` ( Y X. Y ) ) = ( Y X. Y ) )
9	7, 8	sylib 200	. . . . 5 |- ( ( dom G = ( X X. X ) /\ Y C_ X ) -> dom ( G |` ( Y X. Y ) ) = ( Y X. Y ) )
10	2, 9	syl5eq 2496	. . . 4 |- ( ( dom G = ( X X. X ) /\ Y C_ X ) -> dom H = ( Y X. Y ) )
11	10	3adant2 1026	. . 3 |- ( ( dom G = ( X X. X ) /\ H e. GrpOp /\ Y C_ X ) -> dom H = ( Y X. Y ) )
12		eqid 2450	. . . . . 6 |- ran H = ran H
13	12	grpofo 25920	. . . . 5 |- ( H e. GrpOp -> H : ( ran H X. ran H ) -onto-> ran H )
14		fof 5791	. . . . 5 |- ( H : ( ran H X. ran H ) -onto-> ran H -> H : ( ran H X. ran H ) --> ran H )
15		fdm 5731	. . . . 5 |- ( H : ( ran H X. ran H ) --> ran H -> dom H = ( ran H X. ran H ) )
16	13, 14, 15	3syl 18	. . . 4 |- ( H e. GrpOp -> dom H = ( ran H X. ran H ) )
17	16	3ad2ant2 1029	. . 3 |- ( ( dom G = ( X X. X ) /\ H e. GrpOp /\ Y C_ X ) -> dom H = ( ran H X. ran H ) )
18	11, 17	eqtr3d 2486	. 2 |- ( ( dom G = ( X X. X ) /\ H e. GrpOp /\ Y C_ X ) -> ( Y X. Y ) = ( ran H X. ran H ) )
19		xpid11 5055	. 2 |- ( ( Y X. Y ) = ( ran H X. ran H ) <-> Y = ran H )
20	18, 19	sylib 200	1 |- ( ( dom G = ( X X. X ) /\ H e. GrpOp /\ Y C_ X ) -> Y = ran H )

Syntax hints:   -> wi 4   /\ wa 371   /\ w3a 984   = wceq 1443   e. wcel 1886   C_ wss 3403   X. cxp 4831   dom cdm 4833   ran crn 4834   |` cres 4835   -->wf 5577   -onto->wfo 5579   GrpOpcgr 25907

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-sep 4524  ax-nul 4533  ax-pr 4638  ax-un 6580

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-ral 2741  df-rex 2742  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-nul 3731  df-if 3881  df-sn 3968  df-pr 3970  df-op 3974  df-uni 4198  df-iun 4279  df-br 4402  df-opab 4461  df-mpt 4462  df-id 4748  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-fo 5587  df-fv 5589  df-ov 6291  df-grpo 25912

This theorem is referenced by:  ghabloOLD  26090  efghgrpOLD  26094