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Theorem f1imaeq 5554
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 5553 . . 3  |-  ( ( F : A -1-1-> B  /\  ( C  C_  A  /\  D  C_  A ) )  ->  ( ( F " C )  C_  ( F " D )  <-> 
C  C_  D )
)
2 f1imass 5553 . . . 4  |-  ( ( F : A -1-1-> B  /\  ( D  C_  A  /\  C  C_  A ) )  ->  ( ( F " D )  C_  ( F " C )  <-> 
D  C_  C )
)
32ancom2s 533 . . 3  |-  ( ( F : A -1-1-> B  /\  ( C  C_  A  /\  D  C_  A ) )  ->  ( ( F " D )  C_  ( F " C )  <-> 
D  C_  C )
)
41, 3anbi12d 457 . 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 3040 . 2  |-  ( ( F " C )  =  ( F " D )  <->  ( ( F " C )  C_  ( F " D )  /\  ( F " D )  C_  ( F " C ) ) )
6 eqss 3040 . 2  |-  ( C  =  D  <->  ( C  C_  D  /\  D  C_  C ) )
74, 5, 63bitr4g 221 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 102    <-> wb 103    = wceq 1289    C_ wss 2999   "cima 4441   -1-1->wf1 5012
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2841  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fv 5023
This theorem is referenced by: (None)
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