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Theorem resgrprn 21858
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 5063 . . . . 5  |-  dom  H  =  dom  ( G  |`  ( Y  X.  Y
) )
3 xpss12 4973 . . . . . . . 8  |-  ( ( Y  C_  X  /\  Y  C_  X )  -> 
( Y  X.  Y
)  C_  ( X  X.  X ) )
43anidms 627 . . . . . . 7  |-  ( Y 
C_  X  ->  ( Y  X.  Y )  C_  ( X  X.  X
) )
5 sseq2 3362 . . . . . . . 8  |-  ( dom 
G  =  ( X  X.  X )  -> 
( ( Y  X.  Y )  C_  dom  G  <-> 
( Y  X.  Y
)  C_  ( X  X.  X ) ) )
65biimpar 472 . . . . . . 7  |-  ( ( dom  G  =  ( X  X.  X )  /\  ( Y  X.  Y )  C_  ( X  X.  X ) )  ->  ( Y  X.  Y )  C_  dom  G )
74, 6sylan2 461 . . . . . 6  |-  ( ( dom  G  =  ( X  X.  X )  /\  Y  C_  X
)  ->  ( Y  X.  Y )  C_  dom  G )
8 ssdmres 5160 . . . . . 6  |-  ( ( Y  X.  Y ) 
C_  dom  G  <->  dom  ( G  |`  ( Y  X.  Y
) )  =  ( Y  X.  Y ) )
97, 8sylib 189 . . . . 5  |-  ( ( dom  G  =  ( X  X.  X )  /\  Y  C_  X
)  ->  dom  ( G  |`  ( Y  X.  Y
) )  =  ( Y  X.  Y ) )
102, 9syl5eq 2479 . . . 4  |-  ( ( dom  G  =  ( X  X.  X )  /\  Y  C_  X
)  ->  dom  H  =  ( Y  X.  Y
) )
11103adant2 976 . . 3  |-  ( ( dom  G  =  ( X  X.  X )  /\  H  e.  GrpOp  /\  Y  C_  X )  ->  dom  H  =  ( Y  X.  Y ) )
12 eqid 2435 . . . . . 6  |-  ran  H  =  ran  H
1312grpofo 21777 . . . . 5  |-  ( H  e.  GrpOp  ->  H :
( ran  H  X.  ran  H ) -onto-> ran  H
)
14 fof 5645 . . . . 5  |-  ( H : ( ran  H  X.  ran  H ) -onto-> ran 
H  ->  H :
( ran  H  X.  ran  H ) --> ran  H
)
15 fdm 5587 . . . . 5  |-  ( H : ( ran  H  X.  ran  H ) --> ran 
H  ->  dom  H  =  ( ran  H  X.  ran  H ) )
1613, 14, 153syl 19 . . . 4  |-  ( H  e.  GrpOp  ->  dom  H  =  ( ran  H  X.  ran  H ) )
17163ad2ant2 979 . . 3  |-  ( ( dom  G  =  ( X  X.  X )  /\  H  e.  GrpOp  /\  Y  C_  X )  ->  dom  H  =  ( ran  H  X.  ran  H ) )
1811, 17eqtr3d 2469 . 2  |-  ( ( dom  G  =  ( X  X.  X )  /\  H  e.  GrpOp  /\  Y  C_  X )  ->  ( Y  X.  Y
)  =  ( ran 
H  X.  ran  H
) )
19 xpid11 5083 . 2  |-  ( ( Y  X.  Y )  =  ( ran  H  X.  ran  H )  <->  Y  =  ran  H )
2018, 19sylib 189 1  |-  ( ( dom  G  =  ( X  X.  X )  /\  H  e.  GrpOp  /\  Y  C_  X )  ->  Y  =  ran  H
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    C_ wss 3312    X. cxp 4868   dom cdm 4870   ran crn 4871    |` cres 4872   -->wf 5442   -onto->wfo 5444   GrpOpcgr 21764
This theorem is referenced by:  ghablo  21947  efghgrp  21951
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fo 5452  df-fv 5454  df-ov 6076  df-grpo 21769
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