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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
Assertion
Ref Expression
aovrcl (())

Proof of Theorem aovrcl
StepHypRef Expression
1 df-aov 27952 . . 3 (()) '''
21eleq1i 2499 . 2 (()) '''
3 afvvdm 27981 . . 3 '''
4 df-br 4213 . . . 4
5 aovprc.1 . . . . 5
6 brrelex12 4915 . . . . 5
75, 6mpan 652 . . . 4
84, 7sylbir 205 . . 3
93, 8syl 16 . 2 '''
102, 9sylbi 188 1 (())
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359   wcel 1725  cvv 2956  cop 3817   class class class wbr 4212   cdm 4878   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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