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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
StepHypRef Expression
1 resgrprn.1 . . . . . 6  |-  H  =  ( G  |`  ( Y  X.  Y ) )
21dmeqi 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 ) )
43anidms 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 ) ) )
65biimpar 483 . . . . . . 7  |-  ( ( dom  G  =  ( X  X.  X )  /\  ( Y  X.  Y )  C_  ( X  X.  X ) )  ->  ( Y  X.  Y )  C_  dom  G )
74, 6sylan2 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 ) )
97, 8sylib 196 . . . . 5  |-  ( ( dom  G  =  ( X  X.  X )  /\  Y  C_  X
)  ->  dom  ( G  |`  ( Y  X.  Y
) )  =  ( Y  X.  Y ) )
102, 9syl5eq 2455 . . . 4  |-  ( ( dom  G  =  ( X  X.  X )  /\  Y  C_  X
)  ->  dom  H  =  ( Y  X.  Y
) )
11103adant2 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
1312grpofo 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 ) )
1613, 14, 153syl 18 . . . 4  |-  ( H  e.  GrpOp  ->  dom  H  =  ( ran  H  X.  ran  H ) )
17163ad2ant2 1019 . . 3  |-  ( ( dom  G  =  ( X  X.  X )  /\  H  e.  GrpOp  /\  Y  C_  X )  ->  dom  H  =  ( ran  H  X.  ran  H ) )
1811, 17eqtr3d 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 )
2018, 19sylib 196 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 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
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