Theorem ndmfvg 5721

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 2116	. . . . 5 |- ( E! x A F x -> E. x A F x )
2		eldmg 4971	. . . . 5 |- ( A e. _V -> ( A e. dom F <-> E. x A F x ) )
3	1, 2	imbitrrid 156	. . . 4 |- ( A e. _V -> ( E! x A F x -> A e. dom F ) )
4	3	con3d 640	. . 3 |- ( A e. _V -> ( -. A e. dom F -> -. E! x A F x ) )
5		tz6.12-2 5681	. . 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 1402   E.wex 1545   E!weu 2086   e. wcel 2209   _Vcvv 2821   (/)c0 3520   class class class wbr 4125   dom cdm 4769   ` cfv 5372

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-dm 4779  df-iota 5332  df-fv 5380

This theorem is referenced by:  ovprc  6111  wrdsymb0  11315  lsw0  11330  pfxclz  11429  sumnul  12169  structiedg0val  16195