Theorem ndmfvg 5517

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 2044	. . . . 5 |- ( E! x A F x -> E. x A F x )
2		eldmg 4799	. . . . 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 5477	. . 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 1343   E.wex 1480   E!weu 2014   e. wcel 2136   _Vcvv 2726   (/)c0 3409   class class class wbr 3982   dom cdm 4604   ` cfv 5188

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-dm 4614  df-iota 5153  df-fv 5196

This theorem is referenced by:  ovprc  5877  sumnul  11365