Theorem ndmfvg 5538

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 2054	. . . . 5 |- ( E! x A F x -> E. x A F x )
2		eldmg 4815	. . . . 5 |- ( A e. _V -> ( A e. dom F <-> E. x A F x ) )
3	1, 2	syl5ibr 156	. . . 4 |- ( A e. _V -> ( E! x A F x -> A e. dom F ) )
4	3	con3d 631	. . 3 |- ( A e. _V -> ( -. A e. dom F -> -. E! x A F x ) )
5		tz6.12-2 5498	. . 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 124	1 |- ( ( A e. _V /\ -. A e. dom F ) -> ( F ` A ) = (/) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   = wceq 1353   E.wex 1490   E!weu 2024   e. wcel 2146   _Vcvv 2735   (/)c0 3420   class class class wbr 3998   dom cdm 4620   ` cfv 5208

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-rex 2459  df-v 2737  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-nul 3421  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-br 3999  df-dm 4630  df-iota 5170  df-fv 5216

This theorem is referenced by:  ovprc  5900  sumnul  11398