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Theorem aovvdm 28016
Description: If the operation value of a class for an ordered pair is a set, the ordered pair is contained in the domain of the class. (Contributed by Alexander van der Vekens, 26-May-2017.)
Assertion
Ref Expression
aovvdm  |-  ( (( A F B))  e.  C  -> 
<. A ,  B >.  e. 
dom  F )

Proof of Theorem aovvdm
StepHypRef Expression
1 df-aov 27943 . . 3  |- (( A F B))  =  ( F''' <. A ,  B >. )
21eleq1i 2498 . 2  |-  ( (( A F B))  e.  C  <->  ( F''' <. A ,  B >. )  e.  C )
3 afvvdm 27972 . 2  |-  ( ( F''' <. A ,  B >. )  e.  C  ->  <. A ,  B >.  e. 
dom  F )
42, 3sylbi 188 1  |-  ( (( A F B))  e.  C  -> 
<. A ,  B >.  e. 
dom  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1725   <.cop 3809   dom cdm 4870  '''cafv 27939   ((caov 27940
This theorem is referenced by:  ndmaovrcl  28035  ndmaovass  28037  ndmaovdistr  28038
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 2416  ax-sep 4322
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-rab 2706  df-v 2950  df-un 3317  df-if 3732  df-fv 5454  df-dfat 27941  df-afv 27942  df-aov 27943
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