Theorem ndmfvg 5445

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 2027	. . . . 5 ⊢ (∃!𝑥 𝐴𝐹𝑥 → ∃𝑥 𝐴𝐹𝑥)
2		eldmg 4729	. . . . 5 ⊢ (𝐴 ∈ V → (𝐴 ∈ dom 𝐹 ↔ ∃𝑥 𝐴𝐹𝑥))
3	1, 2	syl5ibr 155	. . . 4 ⊢ (𝐴 ∈ V → (∃!𝑥 𝐴𝐹𝑥 → 𝐴 ∈ dom 𝐹))
4	3	con3d 620	. . 3 ⊢ (𝐴 ∈ V → (¬ 𝐴 ∈ dom 𝐹 → ¬ ∃!𝑥 𝐴𝐹𝑥))
5		tz6.12-2 5405	. . 3 ⊢ (¬ ∃!𝑥 𝐴𝐹𝑥 → (𝐹‘𝐴) = ∅)
6	4, 5	syl6 33	. 2 ⊢ (𝐴 ∈ V → (¬ 𝐴 ∈ dom 𝐹 → (𝐹‘𝐴) = ∅))
7	6	imp 123	1 ⊢ ((𝐴 ∈ V ∧ ¬ 𝐴 ∈ dom 𝐹) → (𝐹‘𝐴) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   = wceq 1331  ∃wex 1468   ∈ wcel 1480  ∃!weu 1997  Vcvv 2681  ∅c0 3358   class class class wbr 3924  dom cdm 4534  ‘cfv 5118

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-dm 4544  df-iota 5083  df-fv 5126

This theorem is referenced by:  ovprc  5799  sumnul  11186