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Theorem ghomf1o 24286
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 2316 . 2  |-  ( inv `  G )  =  ( inv `  G )
81, 2, 3, 4, 5, 6, 7ghomf1olem 24285 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 1633    e. wcel 1701   A.wral 2577    X. cxp 4724   ran crn 4727    |` cres 4728   -1-1-onto->wf1o 5291   ` cfv 5292  (class class class)co 5900   GrpOpcgr 20906  GIdcgi 20907   invcgn 20908   GrpOpHom cghom 21077
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-reu 2584  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-riota 6346  df-grpo 20911  df-gid 20912  df-ginv 20913  df-subgo 21022  df-ghom 21078
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