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

Step	Hyp	Ref	Expression
1		df-ov 6279	. . 3 |- ( A F B ) = ( F ` <. A , B >. )
2	1	neeq1i 2688	. 2 |- ( ( A F B ) =/= (/) <-> ( F ` <. A , B >. ) =/= (/) )
3		fvfundmfvn0 5880	. 2 |- ( ( F ` <. A , B >. ) =/= (/) -> ( <. A , B >. e. dom F /\ Fun ( F |` { <. A , B >. } ) ) )
4	2, 3	sylbi 200	1 |- ( ( A F B ) =/= (/) -> ( <. A , B >. e. dom F /\ Fun ( F |` { <. A , B >. } ) ) )

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)