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Theorem aov0nbovbi 28055
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 28014 . 2  |-  ( (/)  e/  C  ->  ( ( F'''
<. A ,  B >. )  e.  C  <->  ( F `  <. A ,  B >. )  e.  C ) )
2 df-aov 27976 . . 3  |- (( A F B))  =  ( F''' <. A ,  B >. )
32eleq1i 2346 . 2  |-  ( (( A F B))  e.  C  <->  ( F''' <. A ,  B >. )  e.  C )
4 df-ov 5861 . . 3  |-  ( A F B )  =  ( F `  <. A ,  B >. )
54eleq1i 2346 . 2  |-  ( ( A F B )  e.  C  <->  ( F `  <. A ,  B >. )  e.  C )
61, 3, 53bitr4g 279 1  |-  ( (/)  e/  C  ->  ( (( A F B))  e.  C  <->  ( A F B )  e.  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    e. wcel 1684    e/ wnel 2447   (/)c0 3455   <.cop 3643   ` cfv 5255  (class class class)co 5858  '''cafv 27972   ((caov 27973
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-dfat 27974  df-afv 27975  df-aov 27976
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