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Theorem ghomf1o 24002
Description: Two ways of saying a group homomorphism is 1-1-onto its image. (Contributed by Paul Chapman, 3-Mar-2008.)
Hypotheses
Ref Expression
ghomf1olem.1  |-  X  =  ran  G
ghomf1olem.2  |-  Y  =  ran  F
ghomf1olem.3  |-  S  =  ( H  |`  ( Y  X.  Y ) )
ghomf1olem.4  |-  Z  =  ran  S
ghomf1olem.5  |-  U  =  (GId `  G )
ghomf1olem.6  |-  T  =  (GId `  H )
Assertion
Ref Expression
ghomf1o  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  ( F : X
-1-1-onto-> Z 
<-> 
A. x  e.  X  ( ( F `  x )  =  T  ->  x  =  U ) ) )
Distinct variable groups:    x, F    x, G    x, H    x, T    x, U    x, X    x, Z
Allowed substitution hints:    S( x)    Y( x)

Proof of Theorem ghomf1o
StepHypRef Expression
1 ghomf1olem.1 . 2  |-  X  =  ran  G
2 ghomf1olem.2 . 2  |-  Y  =  ran  F
3 ghomf1olem.3 . 2  |-  S  =  ( H  |`  ( Y  X.  Y ) )
4 ghomf1olem.4 . 2  |-  Z  =  ran  S
5 ghomf1olem.5 . 2  |-  U  =  (GId `  G )
6 ghomf1olem.6 . 2  |-  T  =  (GId `  H )
7 eqid 2283 . 2  |-  ( inv `  G )  =  ( inv `  G )
81, 2, 3, 4, 5, 6, 7ghomf1olem 24001 1  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  ( F : X
-1-1-onto-> Z 
<-> 
A. x  e.  X  ( ( F `  x )  =  T  ->  x  =  U ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543    X. cxp 4687   ran crn 4690    |` cres 4691   -1-1-onto->wf1o 5254   ` cfv 5255  (class class class)co 5858   GrpOpcgr 20853  GIdcgi 20854   invcgn 20855   GrpOpHom cghom 21024
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-grpo 20858  df-gid 20859  df-ginv 20860  df-subgo 20969  df-ghom 21025
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