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Theorem fnopfvb 5892
Description: Equivalence of function value and ordered pair membership. (Contributed by NM, 7-Nov-1995.)
Assertion
Ref Expression
fnopfvb  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  ( ( F `  B )  =  C  <->  <. B ,  C >.  e.  F ) )

Proof of Theorem fnopfvb
StepHypRef Expression
1 fnbrfvb 5891 . 2  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  ( ( F `  B )  =  C  <-> 
B F C ) )
2 df-br 4398 . 2  |-  ( B F C  <->  <. B ,  C >.  e.  F )
31, 2syl6bb 263 1  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  ( ( F `  B )  =  C  <->  <. B ,  C >.  e.  F ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 186    /\ wa 369    = wceq 1407    e. wcel 1844   <.cop 3980   class class class wbr 4397    Fn wfn 5566   ` cfv 5571
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-sep 4519  ax-nul 4527  ax-pr 4632
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3063  df-sbc 3280  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3741  df-if 3888  df-sn 3975  df-pr 3977  df-op 3981  df-uni 4194  df-br 4398  df-opab 4456  df-id 4740  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-iota 5535  df-fun 5573  df-fn 5574  df-fv 5579
This theorem is referenced by:  funopfvb  5894  fvopab3g  5930  f1ofveu  6275  fnotovb  6321  ovid  6402  ov  6405  ovg  6424  wfrlem14  7036  tfrlem11  7093  rdglim2  7137  tz7.48-1  7147  mdetunilem9  19416
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