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Theorem fnopfvb 5902
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 5901 . 2  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  ( ( F `  B )  =  C  <-> 
B F C ) )
2 df-br 4443 . 2  |-  ( B F C  <->  <. B ,  C >.  e.  F )
31, 2syl6bb 261 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 184    /\ wa 369    = wceq 1374    e. wcel 1762   <.cop 4028   class class class wbr 4442    Fn wfn 5576   ` cfv 5581
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pr 4681
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-sbc 3327  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-br 4443  df-opab 4501  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-iota 5544  df-fun 5583  df-fn 5584  df-fv 5589
This theorem is referenced by:  funopfvb  5904  fvopab3g  5939  f1ofveu  6272  fnotovb  6316  ovid  6396  ov  6399  ovg  6418  tfrlem11  7049  rdglim2  7090  tz7.48-1  7100  mdetunilem9  18884  wfrlem14  28921
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