Theorem ndmfvg 5706

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 2112	. . . . 5 |- ( E! x A F x -> E. x A F x )
2		eldmg 4956	. . . . 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 636	. . 3 |- ( A e. _V -> ( -. A e. dom F -> -. E! x A F x ) )
5		tz6.12-2 5666	. . 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 1398   E.wex 1541   E!weu 2082   e. wcel 2205   _Vcvv 2815   (/)c0 3512   class class class wbr 4114   dom cdm 4754   ` cfv 5357

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-dm 4764  df-iota 5317  df-fv 5365

This theorem is referenced by:  ovprc  6094  wrdsymb0  11282  lsw0  11297  pfxclz  11396  sumnul  12135  structiedg0val  16161