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Theorem f1fveq 5794
Description: Equality of function values for a one-to-one function. (Contributed by NM, 11-Feb-1997.)
Assertion
Ref Expression
f1fveq  |-  ( ( F : A -1-1-> B  /\  ( C  e.  A  /\  D  e.  A
) )  ->  (
( F `  C
)  =  ( F `
 D )  <->  C  =  D ) )

Proof of Theorem f1fveq
StepHypRef Expression
1 f1veqaeq 5791 . 2  |-  ( ( F : A -1-1-> B  /\  ( C  e.  A  /\  D  e.  A
) )  ->  (
( F `  C
)  =  ( F `
 D )  ->  C  =  D )
)
2 fveq2 5534 . 2  |-  ( C  =  D  ->  ( F `  C )  =  ( F `  D ) )
31, 2impbid1 142 1  |-  ( ( F : A -1-1-> B  /\  ( C  e.  A  /\  D  e.  A
) )  ->  (
( F `  C
)  =  ( F `
 D )  <->  C  =  D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1364    e. wcel 2160   -1-1->wf1 5232   ` cfv 5235
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-sbc 2978  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fv 5243
This theorem is referenced by:  f1elima  5795  cocan1  5809  f1oiso  5848  2dom  6831  xpdom2  6857  en2eqpr  6935  isotilem  7035  frec2uzled  10460  hashen  10796  eulerthlemh  12263  f1ocpbllem  12787  f1ovscpbl  12789  relogef  14745  iswomninnlem  15259
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