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Theorem ovrcl 6231
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 3751 . 2  |-  ( C  e.  ( A F B )  ->  -.  ( A F B )  =  (/) )
2 ovprc1.1 . . 3  |-  Rel  dom  F
32ovprc 6228 . 2  |-  ( -.  ( A  e.  _V  /\  B  e.  _V )  ->  ( A F B )  =  (/) )
41, 3nsyl2 127 1  |-  ( C  e.  ( A F B )  ->  ( A  e.  _V  /\  B  e.  _V ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   _Vcvv 3078   (/)c0 3746   dom cdm 4949   Rel wrel 4954  (class class class)co 6201
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-br 4402  df-opab 4460  df-xp 4955  df-rel 4956  df-dm 4959  df-iota 5490  df-fv 5535  df-ov 6204
This theorem is referenced by:  cda1dif  8457
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