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Theorem aovrcl 28029
Description: Reverse closure for an operation value, analogous to afvvv 27985. In contrast to ovrcl 6111, 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 27952 . . 3  |- (( A F B))  =  ( F''' <. A ,  B >. )
21eleq1i 2499 . 2  |-  ( (( A F B))  e.  C  <->  ( F''' <. A ,  B >. )  e.  C )
3 afvvdm 27981 . . 3  |-  ( ( F''' <. A ,  B >. )  e.  C  ->  <. A ,  B >.  e. 
dom  F )
4 df-br 4213 . . . 4  |-  ( A dom  F  B  <->  <. A ,  B >.  e.  dom  F
)
5 aovprc.1 . . . . 5  |-  Rel  dom  F
6 brrelex12 4915 . . . . 5  |-  ( ( Rel  dom  F  /\  A dom  F  B )  ->  ( A  e. 
_V  /\  B  e.  _V ) )
75, 6mpan 652 . . . 4  |-  ( A dom  F  B  -> 
( A  e.  _V  /\  B  e.  _V )
)
84, 7sylbir 205 . . 3  |-  ( <. A ,  B >.  e. 
dom  F  ->  ( A  e.  _V  /\  B  e.  _V ) )
93, 8syl 16 . 2  |-  ( ( F''' <. A ,  B >. )  e.  C  -> 
( A  e.  _V  /\  B  e.  _V )
)
102, 9sylbi 188 1  |-  ( (( A F B))  e.  C  ->  ( A  e.  _V  /\  B  e.  _V )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    e. wcel 1725   _Vcvv 2956   <.cop 3817   class class class wbr 4212   dom cdm 4878   Rel wrel 4883  '''cafv 27948   ((caov 27949
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-br 4213  df-opab 4267  df-xp 4884  df-rel 4885  df-fv 5462  df-dfat 27950  df-afv 27951  df-aov 27952
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