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Theorem ovn0dmfun 40089
Description: If a class' operation value for two operands is not the empty set, the operands are contained in the domain of the class, and the class restricted to the operands is a function, analogous to fvfundmfvn0 5880. (Contributed by AV, 27-Jan-2020.)
Assertion
Ref Expression
ovn0dmfun  |-  ( ( A F B )  =/=  (/)  ->  ( <. A ,  B >.  e.  dom  F  /\  Fun  ( F  |`  { <. A ,  B >. } ) ) )

Proof of Theorem ovn0dmfun
StepHypRef Expression
1 df-ov 6279 . . 3  |-  ( A F B )  =  ( F `  <. A ,  B >. )
21neeq1i 2688 . 2  |-  ( ( A F B )  =/=  (/)  <->  ( F `  <. A ,  B >. )  =/=  (/) )
3 fvfundmfvn0 5880 . 2  |-  ( ( F `  <. A ,  B >. )  =/=  (/)  ->  ( <. A ,  B >.  e. 
dom  F  /\  Fun  ( F  |`  { <. A ,  B >. } ) ) )
42, 3sylbi 200 1  |-  ( ( A F B )  =/=  (/)  ->  ( <. A ,  B >.  e.  dom  F  /\  Fun  ( F  |`  { <. A ,  B >. } ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 375    e. wcel 1891    =/= wne 2622   (/)c0 3699   {csn 3936   <.cop 3942   dom cdm 4812    |` cres 4814   Fun wfun 5555   ` cfv 5561  (class class class)co 6276
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1673  ax-4 1686  ax-5 1762  ax-6 1809  ax-7 1855  ax-8 1893  ax-9 1900  ax-10 1919  ax-11 1924  ax-12 1937  ax-13 2092  ax-ext 2432  ax-sep 4497  ax-nul 4506  ax-pow 4554  ax-pr 4612
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 988  df-tru 1451  df-ex 1668  df-nf 1672  df-sb 1802  df-eu 2304  df-mo 2305  df-clab 2439  df-cleq 2445  df-clel 2448  df-nfc 2582  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3015  df-dif 3375  df-un 3377  df-in 3379  df-ss 3386  df-nul 3700  df-if 3850  df-sn 3937  df-pr 3939  df-op 3943  df-uni 4169  df-br 4375  df-opab 4434  df-id 4727  df-xp 4818  df-rel 4819  df-cnv 4820  df-co 4821  df-dm 4822  df-res 4824  df-iota 5525  df-fun 5563  df-fv 5569  df-ov 6279
This theorem is referenced by: (None)
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