Theorem ndmfvg 5460

Description: The value of a class outside its domain is the empty set. (Contributed by Jim Kingdon, 15-Jan-2019.)

Assertion

Ref	Expression
ndmfvg	|- ( ( A e. _V /\ -. A e. dom F ) -> ( F ` A ) = (/) )

Proof of Theorem ndmfvg

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		euex 2030	. . . . 5 |- ( E! x A F x -> E. x A F x )
2		eldmg 4742	. . . . 5 |- ( A e. _V -> ( A e. dom F <-> E. x A F x ) )
3	1, 2	syl5ibr 155	. . . 4 |- ( A e. _V -> ( E! x A F x -> A e. dom F ) )
4	3	con3d 621	. . 3 |- ( A e. _V -> ( -. A e. dom F -> -. E! x A F x ) )
5		tz6.12-2 5420	. . 3 |- ( -. E! x A F x -> ( F ` A ) = (/) )
6	4, 5	syl6 33	. 2 |- ( A e. _V -> ( -. A e. dom F -> ( F ` A ) = (/) ) )
7	6	imp 123	1 |- ( ( A e. _V /\ -. A e. dom F ) -> ( F ` A ) = (/) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   = wceq 1332   E.wex 1469   e. wcel 1481   E!weu 2000   _Vcvv 2689   (/)c0 3368   class class class wbr 3937   dom cdm 4547   ` cfv 5131

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-br 3938  df-dm 4557  df-iota 5096  df-fv 5139

This theorem is referenced by:  ovprc  5814  sumnul  11225