Theorem aov0ov0 41273

Description: If the alternative value of the operation on an ordered pair is the empty set, the operation's value at this ordered pair is the empty set. (Contributed by Alexander van der Vekens, 26-May-2017.)

Assertion

Ref	Expression
aov0ov0	|- ( (( A F B)) = (/) -> ( A F B ) = (/) )

Proof of Theorem aov0ov0

Step	Hyp	Ref	Expression
1		afv0fv0 41229	. 2 |- ( ( F''' <. A , B >. ) = (/) -> ( F ` <. A , B >. ) = (/) )
2		df-aov 41198	. . 3 |- (( A F B)) = ( F''' <. A , B >. )
3	2	eqeq1i 2627	. 2 |- ( (( A F B)) = (/) <-> ( F''' <. A , B >. ) = (/) )
4		df-ov 6653	. . 3 |- ( A F B ) = ( F ` <. A , B >. )
5	4	eqeq1i 2627	. 2 |- ( ( A F B ) = (/) <-> ( F ` <. A , B >. ) = (/) )
6	1, 3, 5	3imtr4i 281	1 |- ( (( A F B)) = (/) -> ( A F B ) = (/) )

Syntax hints:   -> wi 4   = wceq 1483   (/)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

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-dif 3577  df-un 3579  df-nul 3916  df-if 4087  df-fv 5896  df-ov 6653  df-afv 41197  df-aov 41198

This theorem is referenced by: (None)