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Theorem resgrprn 23899
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 5136 . . . . 5  |-  dom  H  =  dom  ( G  |`  ( Y  X.  Y
) )
3 xpss12 5040 . . . . . . . 8  |-  ( ( Y  C_  X  /\  Y  C_  X )  -> 
( Y  X.  Y
)  C_  ( X  X.  X ) )
43anidms 645 . . . . . . 7  |-  ( Y 
C_  X  ->  ( Y  X.  Y )  C_  ( X  X.  X
) )
5 sseq2 3473 . . . . . . . 8  |-  ( dom 
G  =  ( X  X.  X )  -> 
( ( Y  X.  Y )  C_  dom  G  <-> 
( Y  X.  Y
)  C_  ( X  X.  X ) ) )
65biimpar 485 . . . . . . 7  |-  ( ( dom  G  =  ( X  X.  X )  /\  ( Y  X.  Y )  C_  ( X  X.  X ) )  ->  ( Y  X.  Y )  C_  dom  G )
74, 6sylan2 474 . . . . . 6  |-  ( ( dom  G  =  ( X  X.  X )  /\  Y  C_  X
)  ->  ( Y  X.  Y )  C_  dom  G )
8 ssdmres 5227 . . . . . 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 2503 . . . 4  |-  ( ( dom  G  =  ( X  X.  X )  /\  Y  C_  X
)  ->  dom  H  =  ( Y  X.  Y
) )
11103adant2 1007 . . 3  |-  ( ( dom  G  =  ( X  X.  X )  /\  H  e.  GrpOp  /\  Y  C_  X )  ->  dom  H  =  ( Y  X.  Y ) )
12 eqid 2451 . . . . . 6  |-  ran  H  =  ran  H
1312grpofo 23818 . . . . 5  |-  ( H  e.  GrpOp  ->  H :
( ran  H  X.  ran  H ) -onto-> ran  H
)
14 fof 5715 . . . . 5  |-  ( H : ( ran  H  X.  ran  H ) -onto-> ran 
H  ->  H :
( ran  H  X.  ran  H ) --> ran  H
)
15 fdm 5658 . . . . 5  |-  ( H : ( ran  H  X.  ran  H ) --> ran 
H  ->  dom  H  =  ( ran  H  X.  ran  H ) )
1613, 14, 153syl 20 . . . 4  |-  ( H  e.  GrpOp  ->  dom  H  =  ( ran  H  X.  ran  H ) )
17163ad2ant2 1010 . . 3  |-  ( ( dom  G  =  ( X  X.  X )  /\  H  e.  GrpOp  /\  Y  C_  X )  ->  dom  H  =  ( ran  H  X.  ran  H ) )
1811, 17eqtr3d 2493 . 2  |-  ( ( dom  G  =  ( X  X.  X )  /\  H  e.  GrpOp  /\  Y  C_  X )  ->  ( Y  X.  Y
)  =  ( ran 
H  X.  ran  H
) )
19 xpid11 5156 . 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 369    /\ w3a 965    = wceq 1370    e. wcel 1758    C_ wss 3423    X. cxp 4933   dom cdm 4935   ran crn 4936    |` cres 4937   -->wf 5509   -onto->wfo 5511   GrpOpcgr 23805
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4508  ax-nul 4516  ax-pr 4626  ax-un 6469
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-ne 2644  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3067  df-sbc 3282  df-csb 3384  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-nul 3733  df-if 3887  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4187  df-iun 4268  df-br 4388  df-opab 4446  df-mpt 4447  df-id 4731  df-xp 4941  df-rel 4942  df-cnv 4943  df-co 4944  df-dm 4945  df-rn 4946  df-res 4947  df-iota 5476  df-fun 5515  df-fn 5516  df-f 5517  df-fo 5519  df-fv 5521  df-ov 6190  df-grpo 23810
This theorem is referenced by:  ghablo  23988  efghgrp  23992
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