Theorem aov0nbovbi 41275

Description: The operation's value on an ordered pair is an element of a set if and only if the alternative value of the operation on this ordered pair is an element of that set, if the set does not contain the empty set. (Contributed by Alexander van der Vekens, 26-May-2017.)

Assertion

Ref	Expression
aov0nbovbi	|- ( (/) e/ C -> ( (( A F B)) e. C <-> ( A F B ) e. C ) )

Proof of Theorem aov0nbovbi

Step	Hyp	Ref	Expression
1		afv0nbfvbi 41231	. 2 |- ( (/) e/ C -> ( ( F''' <. A , B >. ) e. C <-> ( F ` <. A , B >. ) e. C ) )
2		df-aov 41198	. . 3 |- (( A F B)) = ( F''' <. A , B >. )
3	2	eleq1i 2692	. 2 |- ( (( A F B)) e. C <-> ( F''' <. A , B >. ) e. C )
4		df-ov 6653	. . 3 |- ( A F B ) = ( F ` <. A , B >. )
5	4	eleq1i 2692	. 2 |- ( ( A F B ) e. C <-> ( F ` <. A , B >. ) e. C )
6	1, 3, 5	3bitr4g 303	1 |- ( (/) e/ C -> ( (( A F B)) e. C <-> ( A F B ) e. C ) )

Syntax hints:   -> wi 4   <-> wb 196   e. wcel 1990   e/ wnel 2897   (/)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-nel 2898  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)