Theorem aovnuoveq 38563

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 38490	. . 3 |- (( A F B)) = ( F''' <. A , B >. )
2	1	neeq1i 2705	. 2 |- ( (( A F B)) =/= _V <-> ( F''' <. A , B >. ) =/= _V )
3		afvnufveq 38519	. . 3 |- ( ( F''' <. A , B >. ) =/= _V -> ( F''' <. A , B >. ) = ( F ` <. A , B >. ) )
4		df-ov 6308	. . 3 |- ( A F B ) = ( F ` <. A , B >. )
5	3, 1, 4	3eqtr4g 2488	. 2 |- ( ( F''' <. A , B >. ) =/= _V -> (( A F B)) = ( A F B ) )
6	2, 5	sylbi 198	1 |- ( (( A F B)) =/= _V -> (( A F B)) = ( A F B ) )

Syntax hints:   -> wi 4   = wceq 1437   =/= wne 2614   _Vcvv 3080   <.cop 4004   ` cfv 5601  (class class class)co 6305  '''cafv 38486   ((caov 38487

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-rab 2780  df-v 3082  df-un 3441  df-if 3912  df-fv 5609  df-ov 6308  df-afv 38489  df-aov 38490

This theorem is referenced by: (None)