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Theorem aovovn0oveq 28054
Description: If the operation's value at an argument is not the empty set, it equals the value of the alternative operation at this argument. (Contributed by Alexander van der Vekens, 26-May-2017.)
Assertion
Ref Expression
aovovn0oveq  |-  ( ( A F B )  =/=  (/)  -> (( A F B))  =  ( A F B ) )

Proof of Theorem aovovn0oveq
StepHypRef Expression
1 df-ov 5861 . . 3  |-  ( A F B )  =  ( F `  <. A ,  B >. )
21neeq1i 2456 . 2  |-  ( ( A F B )  =/=  (/)  <->  ( F `  <. A ,  B >. )  =/=  (/) )
3 afvfvn0fveq 28013 . . 3  |-  ( ( F `  <. A ,  B >. )  =/=  (/)  ->  ( F'''
<. A ,  B >. )  =  ( F `  <. A ,  B >. ) )
4 df-aov 27976 . . 3  |- (( A F B))  =  ( F''' <. A ,  B >. )
53, 4, 13eqtr4g 2340 . 2  |-  ( ( F `  <. A ,  B >. )  =/=  (/)  -> (( A F B))  =  ( A F B ) )
62, 5sylbi 187 1  |-  ( ( A F B )  =/=  (/)  -> (( A F B))  =  ( A F B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    =/= wne 2446   (/)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-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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