Theorem aovvoveq 41272

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

Step	Hyp	Ref	Expression
1		df-aov 41198	. . 3 |- (( A F B)) = ( F''' <. A , B >. )
2	1	eleq1i 2692	. 2 |- ( (( A F B)) e. C <-> ( F''' <. A , B >. ) e. C )
3		afvvfveq 41228	. . 3 |- ( ( F''' <. A , B >. ) e. C -> ( F''' <. A , B >. ) = ( F ` <. A , B >. ) )
4		df-ov 6653	. . 3 |- ( A F B ) = ( F ` <. A , B >. )
5	3, 1, 4	3eqtr4g 2681	. 2 |- ( ( F''' <. A , B >. ) e. C -> (( A F B)) = ( A F B ) )
6	2, 5	sylbi 207	1 |- ( (( A F B)) e. C -> (( A F B)) = ( A F B ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   <.cop 4183   ` cfv 5888  (class class class)co 6650  '''cafv 41194   ((caov 41195

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-rab 2921  df-v 3202  df-un 3579  df-if 4087  df-fv 5896  df-ov 6653  df-afv 41197  df-aov 41198

This theorem is referenced by: (None)