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Theorem f1imaeq 5683
Description: Taking images under a one-to-one function preserves equality. (Contributed by Stefan O'Rear, 30-Oct-2014.)
Assertion
Ref Expression
f1imaeq  |-  ( ( F : A -1-1-> B  /\  ( C  C_  A  /\  D  C_  A ) )  ->  ( ( F " C )  =  ( F " D
)  <->  C  =  D
) )

Proof of Theorem f1imaeq
StepHypRef Expression
1 f1imass 5682 . . 3  |-  ( ( F : A -1-1-> B  /\  ( C  C_  A  /\  D  C_  A ) )  ->  ( ( F " C )  C_  ( F " D )  <-> 
C  C_  D )
)
2 f1imass 5682 . . . 4  |-  ( ( F : A -1-1-> B  /\  ( D  C_  A  /\  C  C_  A ) )  ->  ( ( F " D )  C_  ( F " C )  <-> 
D  C_  C )
)
32ancom2s 556 . . 3  |-  ( ( F : A -1-1-> B  /\  ( C  C_  A  /\  D  C_  A ) )  ->  ( ( F " D )  C_  ( F " C )  <-> 
D  C_  C )
)
41, 3anbi12d 465 . 2  |-  ( ( F : A -1-1-> B  /\  ( C  C_  A  /\  D  C_  A ) )  ->  ( (
( F " C
)  C_  ( F " D )  /\  ( F " D )  C_  ( F " C ) )  <->  ( C  C_  D  /\  D  C_  C
) ) )
5 eqss 3116 . 2  |-  ( ( F " C )  =  ( F " D )  <->  ( ( F " C )  C_  ( F " D )  /\  ( F " D )  C_  ( F " C ) ) )
6 eqss 3116 . 2  |-  ( C  =  D  <->  ( C  C_  D  /\  D  C_  C ) )
74, 5, 63bitr4g 222 1  |-  ( ( F : A -1-1-> B  /\  ( C  C_  A  /\  D  C_  A ) )  ->  ( ( F " C )  =  ( F " D
)  <->  C  =  D
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1332    C_ wss 3075   "cima 4549   -1-1->wf1 5127
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-pow 4105  ax-pr 4138
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-sbc 2913  df-un 3079  df-in 3081  df-ss 3088  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-br 3937  df-opab 3997  df-id 4222  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-rn 4557  df-res 4558  df-ima 4559  df-iota 5095  df-fun 5132  df-fn 5133  df-f 5134  df-f1 5135  df-fv 5138
This theorem is referenced by:  hmeoimaf1o  12520
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