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Theorem aovvdm 28153
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 28079 . . 3  |- (( A F B))  =  ( F''' <. A ,  B >. )
21eleq1i 2359 . 2  |-  ( (( A F B))  e.  C  <->  ( F''' <. A ,  B >. )  e.  C )
3 afvvdm 28109 . 2  |-  ( ( F''' <. A ,  B >. )  e.  C  ->  <. A ,  B >.  e. 
dom  F )
42, 3sylbi 187 1  |-  ( (( A F B))  e.  C  -> 
<. A ,  B >.  e. 
dom  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1696   <.cop 3656   dom cdm 4705  '''cafv 28075   ((caov 28076
This theorem is referenced by:  ndmaovrcl  28172  ndmaovass  28174  ndmaovdistr  28175
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
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-rab 2565  df-v 2803  df-un 3170  df-if 3579  df-fv 5279  df-dfat 28077  df-afv 28078  df-aov 28079
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