Theorem aovovn0oveq 41274

Description: If the operation's value at an argument is not the empty set, it equals the value of the alternative operation at this argument. (Contributed by Alexander van der Vekens, 26-May-2017.)

Assertion

Ref	Expression
aovovn0oveq	|- ( ( A F B ) =/= (/) -> (( A F B)) = ( A F B ) )

Proof of Theorem aovovn0oveq

Step	Hyp	Ref	Expression
1		df-ov 6653	. . 3 |- ( A F B ) = ( F ` <. A , B >. )
2	1	neeq1i 2858	. 2 |- ( ( A F B ) =/= (/) <-> ( F ` <. A , B >. ) =/= (/) )
3		afvfvn0fveq 41230	. . 3 |- ( ( F ` <. A , B >. ) =/= (/) -> ( F''' <. A , B >. ) = ( F ` <. A , B >. ) )
4		df-aov 41198	. . 3 |- (( A F B)) = ( F''' <. A , B >. )
5	3, 4, 1	3eqtr4g 2681	. 2 |- ( ( F ` <. A , B >. ) =/= (/) -> (( A F B)) = ( A F B ) )
6	2, 5	sylbi 207	1 |- ( ( A F B ) =/= (/) -> (( A F B)) = ( A F B ) )

Syntax hints:   -> wi 4   = wceq 1483   =/= wne 2794   (/)c0 3915   <.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  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-res 5126  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-dfat 41196  df-afv 41197  df-aov 41198

This theorem is referenced by: (None)