ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  fnopfvb Unicode version

Theorem fnopfvb 5215
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 5214 . 2  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  ( ( F `  B )  =  C  <-> 
B F C ) )
2 df-br 3765 . 2  |-  ( B F C  <->  <. B ,  C >.  e.  F )
31, 2syl6bb 185 1  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  ( ( F `  B )  =  C  <->  <. B ,  C >.  e.  F ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 97    <-> wb 98    = wceq 1243    e. wcel 1393   <.cop 3378   class class class wbr 3764    Fn wfn 4897   ` cfv 4902
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-iota 4867  df-fun 4904  df-fn 4905  df-fv 4910
This theorem is referenced by:  funopfvb  5217  fvopab3g  5245  f1ofveu  5500  fnotovb  5548  ovid  5617  ov  5620  ovg  5639
  Copyright terms: Public domain W3C validator