Theorem aovnuoveq 31743

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 31670	. . 3 |- (( A F B)) = ( F''' <. A , B >. )
2	1	neeq1i 2752	. 2 |- ( (( A F B)) =/= _V <-> ( F''' <. A , B >. ) =/= _V )
3		afvnufveq 31699	. . 3 |- ( ( F''' <. A , B >. ) =/= _V -> ( F''' <. A , B >. ) = ( F ` <. A , B >. ) )
4		df-ov 6285	. . 3 |- ( A F B ) = ( F ` <. A , B >. )
5	3, 1, 4	3eqtr4g 2533	. 2 |- ( ( F''' <. A , B >. ) =/= _V -> (( A F B)) = ( A F B ) )
6	2, 5	sylbi 195	1 |- ( (( A F B)) =/= _V -> (( A F B)) = ( A F B ) )

Syntax hints:   -> wi 4   = wceq 1379   =/= wne 2662   _Vcvv 3113   <.cop 4033   ` cfv 5586  (class class class)co 6282  '''cafv 31666   ((caov 31667

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-rab 2823  df-v 3115  df-un 3481  df-if 3940  df-fv 5594  df-ov 6285  df-afv 31669  df-aov 31670

This theorem is referenced by: (None)