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Theorem aov0nbovbi 38453
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 38409 . 2  |-  ( (/)  e/  C  ->  ( ( F'''
<. A ,  B >. )  e.  C  <->  ( F `  <. A ,  B >. )  e.  C ) )
2 df-aov 38376 . . 3  |- (( A F B))  =  ( F''' <. A ,  B >. )
32eleq1i 2500 . 2  |-  ( (( A F B))  e.  C  <->  ( F''' <. A ,  B >. )  e.  C )
4 df-ov 6306 . . 3  |-  ( A F B )  =  ( F `  <. A ,  B >. )
54eleq1i 2500 . 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 1869    e/ wnel 2620   (/)c0 3762   <.cop 4003   ` cfv 5599  (class class class)co 6303  '''cafv 38372   ((caov 38373
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-nel 2622  df-ral 2781  df-rex 2782  df-rab 2785  df-v 3084  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3763  df-if 3911  df-sn 3998  df-pr 4000  df-op 4004  df-uni 4218  df-br 4422  df-opab 4481  df-id 4766  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-res 4863  df-iota 5563  df-fun 5601  df-fv 5607  df-ov 6306  df-dfat 38374  df-afv 38375  df-aov 38376
This theorem is referenced by: (None)
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