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Theorem aovrcl 28157
Description: Reverse closure for an operation value, analogous to afvvv 28113. In contrast to ovrcl 5904, elementhood of the operation's value in a set is required, not containing an element. (Contributed by Alexander van der Vekens, 26-May-2017.)
Hypothesis
Ref Expression
aovprc.1  |-  Rel  dom  F
Assertion
Ref Expression
aovrcl  |-  ( (( A F B))  e.  C  ->  ( A  e.  _V  /\  B  e.  _V )
)

Proof of Theorem aovrcl
StepHypRef Expression
1 df-aov 28079 . . 3  |- (( A F B))  =  ( F''' <. A ,  B >. )
21eleq1i 2359 . 2  |-  ( (( A F B))  e.  C  <->  ( F''' <. A ,  B >. )  e.  C )
3 afvvdm 28109 . . 3  |-  ( ( F''' <. A ,  B >. )  e.  C  ->  <. A ,  B >.  e. 
dom  F )
4 df-br 4040 . . . 4  |-  ( A dom  F  B  <->  <. A ,  B >.  e.  dom  F
)
5 aovprc.1 . . . . 5  |-  Rel  dom  F
6 brrelex12 4742 . . . . 5  |-  ( ( Rel  dom  F  /\  A dom  F  B )  ->  ( A  e. 
_V  /\  B  e.  _V ) )
75, 6mpan 651 . . . 4  |-  ( A dom  F  B  -> 
( A  e.  _V  /\  B  e.  _V )
)
84, 7sylbir 204 . . 3  |-  ( <. A ,  B >.  e. 
dom  F  ->  ( A  e.  _V  /\  B  e.  _V ) )
93, 8syl 15 . 2  |-  ( ( F''' <. A ,  B >. )  e.  C  -> 
( A  e.  _V  /\  B  e.  _V )
)
102, 9sylbi 187 1  |-  ( (( A F B))  e.  C  ->  ( A  e.  _V  /\  B  e.  _V )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1696   _Vcvv 2801   <.cop 3656   class class class wbr 4039   dom cdm 4705   Rel wrel 4710  '''cafv 28075   ((caov 28076
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-xp 4711  df-rel 4712  df-fv 5279  df-dfat 28077  df-afv 28078  df-aov 28079
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