Theorem ndmfvg 5589

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 2075	. . . . 5 ⊢ (∃!𝑥 𝐴𝐹𝑥 → ∃𝑥 𝐴𝐹𝑥)
2		eldmg 4861	. . . . 5 ⊢ (𝐴 ∈ V → (𝐴 ∈ dom 𝐹 ↔ ∃𝑥 𝐴𝐹𝑥))
3	1, 2	imbitrrid 156	. . . 4 ⊢ (𝐴 ∈ V → (∃!𝑥 𝐴𝐹𝑥 → 𝐴 ∈ dom 𝐹))
4	3	con3d 632	. . 3 ⊢ (𝐴 ∈ V → (¬ 𝐴 ∈ dom 𝐹 → ¬ ∃!𝑥 𝐴𝐹𝑥))
5		tz6.12-2 5549	. . 3 ⊢ (¬ ∃!𝑥 𝐴𝐹𝑥 → (𝐹‘𝐴) = ∅)
6	4, 5	syl6 33	. 2 ⊢ (𝐴 ∈ V → (¬ 𝐴 ∈ dom 𝐹 → (𝐹‘𝐴) = ∅))
7	6	imp 124	1 ⊢ ((𝐴 ∈ V ∧ ¬ 𝐴 ∈ dom 𝐹) → (𝐹‘𝐴) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   = wceq 1364  ∃wex 1506  ∃!weu 2045   ∈ wcel 2167  Vcvv 2763  ∅c0 3450   class class class wbr 4033  dom cdm 4663  ‘cfv 5258

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-dm 4673  df-iota 5219  df-fv 5266

This theorem is referenced by:  ovprc  5957  wrdsymb0  10967  sumnul  11589