MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ovrcl Structured version   Unicode version

Theorem ovrcl 6113
Description: Reverse closure for an operation value. (Contributed by Mario Carneiro, 5-May-2015.)
Hypothesis
Ref Expression
ovprc1.1  |-  Rel  dom  F
Assertion
Ref Expression
ovrcl  |-  ( C  e.  ( A F B )  ->  ( A  e.  _V  /\  B  e.  _V ) )

Proof of Theorem ovrcl
StepHypRef Expression
1 n0i 3635 . 2  |-  ( C  e.  ( A F B )  ->  -.  ( A F B )  =  (/) )
2 ovprc1.1 . . 3  |-  Rel  dom  F
32ovprc 6110 . 2  |-  ( -.  ( A  e.  _V  /\  B  e.  _V )  ->  ( A F B )  =  (/) )
41, 3nsyl2 122 1  |-  ( C  e.  ( A F B )  ->  ( A  e.  _V  /\  B  e.  _V ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   _Vcvv 2958   (/)c0 3630   dom cdm 4880   Rel wrel 4885  (class class class)co 6083
This theorem is referenced by:  cda1dif  8058
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-xp 4886  df-rel 4887  df-dm 4890  df-iota 5420  df-fv 5464  df-ov 6086
  Copyright terms: Public domain W3C validator