Theorem resgrprn 25696

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 5025	. . . . 5 |- dom H = dom ( G |` ( Y X. Y ) )
3		xpss12 4929	. . . . . . . 8 |- ( ( Y C_ X /\ Y C_ X ) -> ( Y X. Y ) C_ ( X X. X ) )
4	3	anidms 643	. . . . . . 7 |- ( Y C_ X -> ( Y X. Y ) C_ ( X X. X ) )
5		sseq2 3464	. . . . . . . 8 |- ( dom G = ( X X. X ) -> ( ( Y X. Y ) C_ dom G <-> ( Y X. Y ) C_ ( X X. X ) ) )
6	5	biimpar 483	. . . . . . 7 |- ( ( dom G = ( X X. X ) /\ ( Y X. Y ) C_ ( X X. X ) ) -> ( Y X. Y ) C_ dom G )
7	4, 6	sylan2 472	. . . . . 6 |- ( ( dom G = ( X X. X ) /\ Y C_ X ) -> ( Y X. Y ) C_ dom G )
8		ssdmres 5115	. . . . . 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 2455	. . . 4 |- ( ( dom G = ( X X. X ) /\ Y C_ X ) -> dom H = ( Y X. Y ) )
11	10	3adant2 1016	. . 3 |- ( ( dom G = ( X X. X ) /\ H e. GrpOp /\ Y C_ X ) -> dom H = ( Y X. Y ) )
12		eqid 2402	. . . . . 6 |- ran H = ran H
13	12	grpofo 25615	. . . . 5 |- ( H e. GrpOp -> H : ( ran H X. ran H ) -onto-> ran H )
14		fof 5778	. . . . 5 |- ( H : ( ran H X. ran H ) -onto-> ran H -> H : ( ran H X. ran H ) --> ran H )
15		fdm 5718	. . . . 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 1019	. . 3 |- ( ( dom G = ( X X. X ) /\ H e. GrpOp /\ Y C_ X ) -> dom H = ( ran H X. ran H ) )
18	11, 17	eqtr3d 2445	. 2 |- ( ( dom G = ( X X. X ) /\ H e. GrpOp /\ Y C_ X ) -> ( Y X. Y ) = ( ran H X. ran H ) )
19		xpid11 5045	. 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 367   /\ w3a 974   = wceq 1405   e. wcel 1842   C_ wss 3414   X. cxp 4821   dom cdm 4823   ran crn 4824   |` cres 4825   -->wf 5565   -onto->wfo 5567   GrpOpcgr 25602

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-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-ov 6281  df-grpo 25607

This theorem is referenced by:  ghabloOLD  25785  efghgrpOLD  25789