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Theorem f1imaeq 5805
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 5804 . . 3  |-  ( ( F : A -1-1-> B  /\  ( C  C_  A  /\  D  C_  A ) )  ->  ( ( F " C )  C_  ( F " D )  <-> 
C  C_  D )
)
2 f1imass 5804 . . . 4  |-  ( ( F : A -1-1-> B  /\  ( D  C_  A  /\  C  C_  A ) )  ->  ( ( F " D )  C_  ( F " C )  <-> 
D  C_  C )
)
32ancom2s 777 . . 3  |-  ( ( F : A -1-1-> B  /\  ( C  C_  A  /\  D  C_  A ) )  ->  ( ( F " D )  C_  ( F " C )  <-> 
D  C_  C )
)
41, 3anbi12d 691 . 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 3207 . 2  |-  ( ( F " C )  =  ( F " D )  <->  ( ( F " C )  C_  ( F " D )  /\  ( F " D )  C_  ( F " C ) ) )
6 eqss 3207 . 2  |-  ( C  =  D  <->  ( C  C_  D  /\  D  C_  C ) )
74, 5, 63bitr4g 279 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 176    /\ wa 358    = wceq 1632    C_ wss 3165   "cima 4708   -1-1->wf1 5268
This theorem is referenced by:  f1imapss  5806  dfac12lem2  7786  hmeoimaf1o  17477  imasf1oxms  18051
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fv 5279
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