Theorem ndmfvg 5129

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 ∈ V ∧ ¬ A ∈ dom 𝐹) → (𝐹‘A) = ∅)

Proof of Theorem ndmfvg

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		euex 1912	. . . . 5 ⊢ (∃!x A𝐹x → ∃x A𝐹x)
2		eldmg 4457	. . . . 5 ⊢ (A ∈ V → (A ∈ dom 𝐹 ↔ ∃x A𝐹x))
3	1, 2	syl5ibr 145	. . . 4 ⊢ (A ∈ V → (∃!x A𝐹x → A ∈ dom 𝐹))
4	3	con3d 548	. . 3 ⊢ (A ∈ V → (¬ A ∈ dom 𝐹 → ¬ ∃!x A𝐹x))
5		tz6.12-2 5094	. . 3 ⊢ (¬ ∃!x A𝐹x → (𝐹‘A) = ∅)
6	4, 5	syl6 29	. 2 ⊢ (A ∈ V → (¬ A ∈ dom 𝐹 → (𝐹‘A) = ∅))
7	6	imp 115	1 ⊢ ((A ∈ V ∧ ¬ A ∈ dom 𝐹) → (𝐹‘A) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   = wceq 1228  ∃wex 1362   ∈ wcel 1374  ∃!weu 1882  Vcvv 2535  ∅c0 3201   class class class wbr 3738  dom cdm 4272  ‘cfv 4829

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004

This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-fal 1234  df-nf 1330  df-sb 1628  df-eu 1885  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-dif 2897  df-un 2899  df-in 2901  df-ss 2908  df-nul 3202  df-sn 3356  df-pr 3357  df-op 3359  df-uni 3555  df-br 3739  df-dm 4282  df-iota 4794  df-fv 4837

This theorem is referenced by:  ovprc  5463