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Theorem resgrprn 20963
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 4896 . . . . 5  |-  dom  H  =  dom  ( G  |`  ( Y  X.  Y
) )
3 xpss12 4808 . . . . . . . 8  |-  ( ( Y  C_  X  /\  Y  C_  X )  -> 
( Y  X.  Y
)  C_  ( X  X.  X ) )
43anidms 626 . . . . . . 7  |-  ( Y 
C_  X  ->  ( Y  X.  Y )  C_  ( X  X.  X
) )
5 sseq2 3213 . . . . . . . 8  |-  ( dom 
G  =  ( X  X.  X )  -> 
( ( Y  X.  Y )  C_  dom  G  <-> 
( Y  X.  Y
)  C_  ( X  X.  X ) ) )
65biimpar 471 . . . . . . 7  |-  ( ( dom  G  =  ( X  X.  X )  /\  ( Y  X.  Y )  C_  ( X  X.  X ) )  ->  ( Y  X.  Y )  C_  dom  G )
74, 6sylan2 460 . . . . . 6  |-  ( ( dom  G  =  ( X  X.  X )  /\  Y  C_  X
)  ->  ( Y  X.  Y )  C_  dom  G )
8 ssdmres 4993 . . . . . 6  |-  ( ( Y  X.  Y ) 
C_  dom  G  <->  dom  ( G  |`  ( Y  X.  Y
) )  =  ( Y  X.  Y ) )
97, 8sylib 188 . . . . 5  |-  ( ( dom  G  =  ( X  X.  X )  /\  Y  C_  X
)  ->  dom  ( G  |`  ( Y  X.  Y
) )  =  ( Y  X.  Y ) )
102, 9syl5eq 2340 . . . 4  |-  ( ( dom  G  =  ( X  X.  X )  /\  Y  C_  X
)  ->  dom  H  =  ( Y  X.  Y
) )
11103adant2 974 . . 3  |-  ( ( dom  G  =  ( X  X.  X )  /\  H  e.  GrpOp  /\  Y  C_  X )  ->  dom  H  =  ( Y  X.  Y ) )
12 eqid 2296 . . . . . 6  |-  ran  H  =  ran  H
1312grpofo 20882 . . . . 5  |-  ( H  e.  GrpOp  ->  H :
( ran  H  X.  ran  H ) -onto-> ran  H
)
14 fof 5467 . . . . 5  |-  ( H : ( ran  H  X.  ran  H ) -onto-> ran 
H  ->  H :
( ran  H  X.  ran  H ) --> ran  H
)
15 fdm 5409 . . . . 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 977 . . 3  |-  ( ( dom  G  =  ( X  X.  X )  /\  H  e.  GrpOp  /\  Y  C_  X )  ->  dom  H  =  ( ran  H  X.  ran  H ) )
1811, 17eqtr3d 2330 . 2  |-  ( ( dom  G  =  ( X  X.  X )  /\  H  e.  GrpOp  /\  Y  C_  X )  ->  ( Y  X.  Y
)  =  ( ran 
H  X.  ran  H
) )
19 xpid11 4916 . 2  |-  ( ( Y  X.  Y )  =  ( ran  H  X.  ran  H )  <->  Y  =  ran  H )
2018, 19sylib 188 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 358    /\ w3a 934    = wceq 1632    e. wcel 1696    C_ wss 3165    X. cxp 4703   dom cdm 4705   ran crn 4706    |` cres 4707   -->wf 5267   -onto->wfo 5269   GrpOpcgr 20869
This theorem is referenced by:  ghablo  21052  efghgrp  21056
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fo 5277  df-fv 5279  df-ov 5877  df-grpo 20874
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