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

Theorem ovrcl 5888
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 3460 . 2  |-  ( C  e.  ( A F B )  ->  -.  ( A F B )  =  (/) )
2 ovprc1.1 . . 3  |-  Rel  dom  F
32ovprc 5885 . 2  |-  ( -.  ( A  e.  _V  /\  B  e.  _V )  ->  ( A F B )  =  (/) )
41, 3nsyl2 119 1  |-  ( C  e.  ( A F B )  ->  ( A  e.  _V  /\  B  e.  _V ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788   (/)c0 3455   dom cdm 4689   Rel wrel 4694  (class class class)co 5858
This theorem is referenced by:  cda1dif  7802
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-xp 4695  df-rel 4696  df-dm 4699  df-iota 5219  df-fv 5263  df-ov 5861
  Copyright terms: Public domain W3C validator