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Theorem aov0nbovbi 28027
Description: The operation's value on an ordered pair is an element of a set if and only if the alternative value of the operation on this ordered pair is an element of that set, if the set does not contain the empty set. (Contributed by Alexander van der Vekens, 26-May-2017.)
Assertion
Ref Expression
aov0nbovbi  |-  ( (/)  e/  C  ->  ( (( A F B))  e.  C  <->  ( A F B )  e.  C ) )

Proof of Theorem aov0nbovbi
StepHypRef Expression
1 afv0nbfvbi 27983 . 2  |-  ( (/)  e/  C  ->  ( ( F'''
<. A ,  B >. )  e.  C  <->  ( F `  <. A ,  B >. )  e.  C ) )
2 df-aov 27944 . . 3  |- (( A F B))  =  ( F''' <. A ,  B >. )
32eleq1i 2499 . 2  |-  ( (( A F B))  e.  C  <->  ( F''' <. A ,  B >. )  e.  C )
4 df-ov 6077 . . 3  |-  ( A F B )  =  ( F `  <. A ,  B >. )
54eleq1i 2499 . 2  |-  ( ( A F B )  e.  C  <->  ( F `  <. A ,  B >. )  e.  C )
61, 3, 53bitr4g 280 1  |-  ( (/)  e/  C  ->  ( (( A F B))  e.  C  <->  ( A F B )  e.  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    e. wcel 1725    e/ wnel 2600   (/)c0 3621   <.cop 3810   ` cfv 5447  (class class class)co 6074  '''cafv 27940   ((caov 27941
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4323  ax-nul 4331  ax-pow 4370  ax-pr 4396
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2703  df-rex 2704  df-rab 2707  df-v 2951  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-nul 3622  df-if 3733  df-sn 3813  df-pr 3814  df-op 3816  df-uni 4009  df-br 4206  df-opab 4260  df-id 4491  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-res 4883  df-iota 5411  df-fun 5449  df-fv 5455  df-ov 6077  df-dfat 27942  df-afv 27943  df-aov 27944
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