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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
StepHypRef Expression
1 resgrprn.1 . . . . . 6  |-  H  =  ( G  |`  ( Y  X.  Y ) )
21dmeqi 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 ) )
43anidms 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 ) ) )
65biimpar 488 . . . . . . 7  |-  ( ( dom  G  =  ( X  X.  X )  /\  ( Y  X.  Y )  C_  ( X  X.  X ) )  ->  ( Y  X.  Y )  C_  dom  G )
74, 6sylan2 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 ) )
97, 8sylib 200 . . . . 5  |-  ( ( dom  G  =  ( X  X.  X )  /\  Y  C_  X
)  ->  dom  ( G  |`  ( Y  X.  Y
) )  =  ( Y  X.  Y ) )
102, 9syl5eq 2496 . . . 4  |-  ( ( dom  G  =  ( X  X.  X )  /\  Y  C_  X
)  ->  dom  H  =  ( Y  X.  Y
) )
11103adant2 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
1312grpofo 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 ) )
1613, 14, 153syl 18 . . . 4  |-  ( H  e.  GrpOp  ->  dom  H  =  ( ran  H  X.  ran  H ) )
17163ad2ant2 1029 . . 3  |-  ( ( dom  G  =  ( X  X.  X )  /\  H  e.  GrpOp  /\  Y  C_  X )  ->  dom  H  =  ( ran  H  X.  ran  H ) )
1811, 17eqtr3d 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 )
2018, 19sylib 200 1  |-  ( ( dom  G  =  ( X  X.  X )  /\  H  e.  GrpOp  /\  Y  C_  X )  ->  Y  =  ran  H
)
Colors of variables: wff setvar class
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
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