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Theorem f1imaeq 6090
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 6089 . . 3  |-  ( ( F : A -1-1-> B  /\  ( C  C_  A  /\  D  C_  A ) )  ->  ( ( F " C )  C_  ( F " D )  <-> 
C  C_  D )
)
2 f1imass 6089 . . . 4  |-  ( ( F : A -1-1-> B  /\  ( D  C_  A  /\  C  C_  A ) )  ->  ( ( F " D )  C_  ( F " C )  <-> 
D  C_  C )
)
32ancom2s 800 . . 3  |-  ( ( F : A -1-1-> B  /\  ( C  C_  A  /\  D  C_  A ) )  ->  ( ( F " D )  C_  ( F " C )  <-> 
D  C_  C )
)
41, 3anbi12d 710 . 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 3482 . 2  |-  ( ( F " C )  =  ( F " D )  <->  ( ( F " C )  C_  ( F " D )  /\  ( F " D )  C_  ( F " C ) ) )
6 eqss 3482 . 2  |-  ( C  =  D  <->  ( C  C_  D  /\  D  C_  C ) )
74, 5, 63bitr4g 288 1  |-  ( ( F : A -1-1-> B  /\  ( C  C_  A  /\  D  C_  A ) )  ->  ( ( F " C )  =  ( F " D
)  <->  C  =  D
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    C_ wss 3439   "cima 4954   -1-1->wf1 5526
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pr 4642
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fv 5537
This theorem is referenced by:  f1imapss  6091  dfac12lem2  8428  hmeoimaf1o  19485  imasf1oxms  20206
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