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Theorem f1imaeq 6013
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 6012 . . 3  |-  ( ( F : A -1-1-> B  /\  ( C  C_  A  /\  D  C_  A ) )  ->  ( ( F " C )  C_  ( F " D )  <-> 
C  C_  D )
)
2 f1imass 6012 . . . 4  |-  ( ( F : A -1-1-> B  /\  ( D  C_  A  /\  C  C_  A ) )  ->  ( ( F " D )  C_  ( F " C )  <-> 
D  C_  C )
)
32ancom2s 779 . . 3  |-  ( ( F : A -1-1-> B  /\  ( C  C_  A  /\  D  C_  A ) )  ->  ( ( F " D )  C_  ( F " C )  <-> 
D  C_  C )
)
41, 3anbi12d 693 . 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 3365 . 2  |-  ( ( F " C )  =  ( F " D )  <->  ( ( F " C )  C_  ( F " D )  /\  ( F " D )  C_  ( F " C ) ) )
6 eqss 3365 . 2  |-  ( C  =  D  <->  ( C  C_  D  /\  D  C_  C ) )
74, 5, 63bitr4g 281 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    <-> wb 178    /\ wa 360    = wceq 1653    C_ wss 3322   "cima 4883   -1-1->wf1 5453
This theorem is referenced by:  f1imapss  6014  dfac12lem2  8026  hmeoimaf1o  17804  imasf1oxms  18521
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fv 5464
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