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Theorem ovrcl 6303
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 3788 . 2  |-  ( C  e.  ( A F B )  ->  -.  ( A F B )  =  (/) )
2 ovprc1.1 . . 3  |-  Rel  dom  F
32ovprc 6300 . 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 367    = wceq 1398    e. wcel 1823   _Vcvv 3106   (/)c0 3783   dom cdm 4988   Rel wrel 4993  (class class class)co 6270
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-xp 4994  df-rel 4995  df-dm 4998  df-iota 5534  df-fv 5578  df-ov 6273
This theorem is referenced by:  cda1dif  8547
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