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Theorem aov0nbovbi 38697
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 38653 . 2  |-  ( (/)  e/  C  ->  ( ( F'''
<. A ,  B >. )  e.  C  <->  ( F `  <. A ,  B >. )  e.  C ) )
2 df-aov 38619 . . 3  |- (( A F B))  =  ( F''' <. A ,  B >. )
32eleq1i 2520 . 2  |-  ( (( A F B))  e.  C  <->  ( F''' <. A ,  B >. )  e.  C )
4 df-ov 6293 . . 3  |-  ( A F B )  =  ( F `  <. A ,  B >. )
54eleq1i 2520 . 2  |-  ( ( A F B )  e.  C  <->  ( F `  <. A ,  B >. )  e.  C )
61, 3, 53bitr4g 292 1  |-  ( (/)  e/  C  ->  ( (( A F B))  e.  C  <->  ( A F B )  e.  C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    e. wcel 1887    e/ wnel 2623   (/)c0 3731   <.cop 3974   ` cfv 5582  (class class class)co 6290  '''cafv 38615   ((caov 38616
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-br 4403  df-opab 4462  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-res 4846  df-iota 5546  df-fun 5584  df-fv 5590  df-ov 6293  df-dfat 38617  df-afv 38618  df-aov 38619
This theorem is referenced by: (None)
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