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Theorem aovovn0oveq 27729
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 6025 . . 3  |-  ( A F B )  =  ( F `  <. A ,  B >. )
21neeq1i 2562 . 2  |-  ( ( A F B )  =/=  (/)  <->  ( F `  <. A ,  B >. )  =/=  (/) )
3 afvfvn0fveq 27685 . . 3  |-  ( ( F `  <. A ,  B >. )  =/=  (/)  ->  ( F'''
<. A ,  B >. )  =  ( F `  <. A ,  B >. ) )
4 df-aov 27646 . . 3  |- (( A F B))  =  ( F''' <. A ,  B >. )
53, 4, 13eqtr4g 2446 . 2  |-  ( ( F `  <. A ,  B >. )  =/=  (/)  -> (( A F B))  =  ( A F B ) )
62, 5sylbi 188 1  |-  ( ( A F B )  =/=  (/)  -> (( A F B))  =  ( A F B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    =/= wne 2552   (/)c0 3573   <.cop 3762   ` cfv 5396  (class class class)co 6022  '''cafv 27642   ((caov 27643
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-br 4156  df-opab 4210  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-res 4832  df-iota 5360  df-fun 5398  df-fv 5404  df-ov 6025  df-dfat 27644  df-afv 27645  df-aov 27646
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