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Theorem aovvoveq 38406
Description: The alternative value of the operation on an ordered pair equals the operation's value on this ordered pair. (Contributed by Alexander van der Vekens, 26-May-2017.)
Assertion
Ref Expression
aovvoveq  |-  ( (( A F B))  e.  C  -> (( A F B))  =  ( A F B ) )

Proof of Theorem aovvoveq
StepHypRef Expression
1 df-aov 38332 . . 3  |- (( A F B))  =  ( F''' <. A ,  B >. )
21eleq1i 2499 . 2  |-  ( (( A F B))  e.  C  <->  ( F''' <. A ,  B >. )  e.  C )
3 afvvfveq 38362 . . 3  |-  ( ( F''' <. A ,  B >. )  e.  C  -> 
( F''' <. A ,  B >. )  =  ( F `
 <. A ,  B >. ) )
4 df-ov 6305 . . 3  |-  ( A F B )  =  ( F `  <. A ,  B >. )
53, 1, 43eqtr4g 2488 . 2  |-  ( ( F''' <. A ,  B >. )  e.  C  -> (( A F B))  =  ( A F B ) )
62, 5sylbi 198 1  |-  ( (( A F B))  e.  C  -> (( A F B))  =  ( A F B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1437    e. wcel 1868   <.cop 4002   ` cfv 5598  (class class class)co 6302  '''cafv 38328   ((caov 38329
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-sep 4543
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-rab 2784  df-v 3083  df-un 3441  df-if 3910  df-fv 5606  df-ov 6305  df-afv 38331  df-aov 38332
This theorem is referenced by: (None)
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