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Theorem ndmfvg 5370
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) → (𝐹𝐴) = ∅)

Proof of Theorem ndmfvg
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 euex 1985 . . . . 5 (∃!𝑥 𝐴𝐹𝑥 → ∃𝑥 𝐴𝐹𝑥)
2 eldmg 4662 . . . . 5 (𝐴 ∈ V → (𝐴 ∈ dom 𝐹 ↔ ∃𝑥 𝐴𝐹𝑥))
31, 2syl5ibr 155 . . . 4 (𝐴 ∈ V → (∃!𝑥 𝐴𝐹𝑥𝐴 ∈ dom 𝐹))
43con3d 599 . . 3 (𝐴 ∈ V → (¬ 𝐴 ∈ dom 𝐹 → ¬ ∃!𝑥 𝐴𝐹𝑥))
5 tz6.12-2 5331 . . 3 (¬ ∃!𝑥 𝐴𝐹𝑥 → (𝐹𝐴) = ∅)
64, 5syl6 33 . 2 (𝐴 ∈ V → (¬ 𝐴 ∈ dom 𝐹 → (𝐹𝐴) = ∅))
76imp 123 1 ((𝐴 ∈ V ∧ ¬ 𝐴 ∈ dom 𝐹) → (𝐹𝐴) = ∅)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103   = wceq 1296  wex 1433  wcel 1445  ∃!weu 1955  Vcvv 2633  c0 3302   class class class wbr 3867  dom cdm 4467  cfv 5049
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077
This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-fal 1302  df-nf 1402  df-sb 1700  df-eu 1958  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-v 2635  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-nul 3303  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-br 3868  df-dm 4477  df-iota 5014  df-fv 5057
This theorem is referenced by:  ovprc  5722  sumnul  10967
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