Theorem ovn0dmfun 32757

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 5904. (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 6299	. . 3 |- ( A F B ) = ( F ` <. A , B >. )
2	1	neeq1i 2742	. 2 |- ( ( A F B ) =/= (/) <-> ( F ` <. A , B >. ) =/= (/) )
3		fvfundmfvn0 5904	. 2 |- ( ( F ` <. A , B >. ) =/= (/) -> ( <. A , B >. e. dom F /\ Fun ( F |` { <. A , B >. } ) ) )
4	2, 3	sylbi 195	1 |- ( ( A F B ) =/= (/) -> ( <. A , B >. e. dom F /\ Fun ( F |` { <. A , B >. } ) ) )

Syntax hints:   -> wi 4   /\ wa 369   e. wcel 1819   =/= wne 2652   (/)c0 3793   {csn 4032   <.cop 4038   dom cdm 5008   |` cres 5010   Fun wfun 5588   ` cfv 5594  (class class class)co 6296

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-res 5020  df-iota 5557  df-fun 5596  df-fv 5602  df-ov 6299

This theorem is referenced by: (None)