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Theorem funopfvb 5528
Description: Equivalence of function value and ordered pair membership. Theorem 4.3(ii) of [Monk1] p. 42. (Contributed by NM, 26-Jan-1997.)
Assertion
Ref Expression
funopfvb  |-  ( ( Fun  F  /\  A  e.  dom  F )  -> 
( ( F `  A )  =  B  <->  <. A ,  B >.  e.  F ) )

Proof of Theorem funopfvb
StepHypRef Expression
1 funfn 5249 . 2  |-  ( Fun 
F  <->  F  Fn  dom  F )
2 fnopfvb 5526 . 2  |-  ( ( F  Fn  dom  F  /\  A  e.  dom  F )  ->  ( ( F `  A )  =  B  <->  <. A ,  B >.  e.  F ) )
31, 2sylanb 458 1  |-  ( ( Fun  F  /\  A  e.  dom  F )  -> 
( ( F `  A )  =  B  <->  <. A ,  B >.  e.  F ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1685   <.cop 3644    dom cdm 4688   Fun wfun 5215    Fn wfn 5216   ` cfv 5221
This theorem is referenced by:  dmfco  5555  funfvop  5599  f1eqcocnv  5767
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-sbc 2993  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-br 4025  df-opab 4079  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-fv 5229
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