Theorem aovnuoveq 38784

Description: The alternative value of the operation on an ordered pair equals the operation's value at this ordered pair. (Contributed by Alexander van der Vekens, 26-May-2017.)

Assertion

Ref	Expression
aovnuoveq	|- ( (( A F B)) =/= _V -> (( A F B)) = ( A F B ) )

Proof of Theorem aovnuoveq

Step	Hyp	Ref	Expression
1		df-aov 38710	. . 3 |- (( A F B)) = ( F''' <. A , B >. )
2	1	neeq1i 2688	. 2 |- ( (( A F B)) =/= _V <-> ( F''' <. A , B >. ) =/= _V )
3		afvnufveq 38740	. . 3 |- ( ( F''' <. A , B >. ) =/= _V -> ( F''' <. A , B >. ) = ( F ` <. A , B >. ) )
4		df-ov 6279	. . 3 |- ( A F B ) = ( F ` <. A , B >. )
5	3, 1, 4	3eqtr4g 2511	. 2 |- ( ( F''' <. A , B >. ) =/= _V -> (( A F B)) = ( A F B ) )
6	2, 5	sylbi 200	1 |- ( (( A F B)) =/= _V -> (( A F B)) = ( A F B ) )

Syntax hints:   -> wi 4   = wceq 1448   =/= wne 2622   _Vcvv 3013   <.cop 3942   ` cfv 5561  (class class class)co 6276  '''cafv 38706   ((caov 38707

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1673  ax-4 1686  ax-5 1762  ax-6 1809  ax-7 1855  ax-10 1919  ax-11 1924  ax-12 1937  ax-13 2092  ax-ext 2432

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-tru 1451  df-ex 1668  df-nf 1672  df-sb 1802  df-clab 2439  df-cleq 2445  df-clel 2448  df-nfc 2582  df-ne 2624  df-rab 2746  df-v 3015  df-un 3377  df-if 3850  df-fv 5569  df-ov 6279  df-afv 38709  df-aov 38710

This theorem is referenced by: (None)