Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  aov0nbovbi Unicode version

Theorem aov0nbovbi 27721
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
StepHypRef Expression
1 afv0nbfvbi 27677 . 2  |-  ( (/)  e/  C  ->  ( ( F'''
<. A ,  B >. )  e.  C  <->  ( F `  <. A ,  B >. )  e.  C ) )
2 df-aov 27637 . . 3  |- (( A F B))  =  ( F''' <. A ,  B >. )
32eleq1i 2443 . 2  |-  ( (( A F B))  e.  C  <->  ( F''' <. A ,  B >. )  e.  C )
4 df-ov 6016 . . 3  |-  ( A F B )  =  ( F `  <. A ,  B >. )
54eleq1i 2443 . 2  |-  ( ( A F B )  e.  C  <->  ( F `  <. A ,  B >. )  e.  C )
61, 3, 53bitr4g 280 1  |-  ( (/)  e/  C  ->  ( (( A F B))  e.  C  <->  ( A F B )  e.  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    e. wcel 1717    e/ wnel 2544   (/)c0 3564   <.cop 3753   ` cfv 5387  (class class class)co 6013  '''cafv 27633   ((caov 27634
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-br 4147  df-opab 4201  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-res 4823  df-iota 5351  df-fun 5389  df-fv 5395  df-ov 6016  df-dfat 27635  df-afv 27636  df-aov 27637
  Copyright terms: Public domain W3C validator