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Theorem ndmfvg 5279
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 1973 . . . . 5 (∃!𝑥 𝐴𝐹𝑥 → ∃𝑥 𝐴𝐹𝑥)
2 eldmg 4588 . . . . 5 (𝐴 ∈ V → (𝐴 ∈ dom 𝐹 ↔ ∃𝑥 𝐴𝐹𝑥))
31, 2syl5ibr 154 . . . 4 (𝐴 ∈ V → (∃!𝑥 𝐴𝐹𝑥𝐴 ∈ dom 𝐹))
43con3d 594 . . 3 (𝐴 ∈ V → (¬ 𝐴 ∈ dom 𝐹 → ¬ ∃!𝑥 𝐴𝐹𝑥))
5 tz6.12-2 5243 . . 3 (¬ ∃!𝑥 𝐴𝐹𝑥 → (𝐹𝐴) = ∅)
64, 5syl6 33 . 2 (𝐴 ∈ V → (¬ 𝐴 ∈ dom 𝐹 → (𝐹𝐴) = ∅))
76imp 122 1 ((𝐴 ∈ V ∧ ¬ 𝐴 ∈ dom 𝐹) → (𝐹𝐴) = ∅)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102   = wceq 1285  wex 1422  wcel 1434  ∃!weu 1943  Vcvv 2612  c0 3269   class class class wbr 3811  dom cdm 4400  cfv 4968
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2614  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-nul 3270  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-br 3812  df-dm 4410  df-iota 4933  df-fv 4976
This theorem is referenced by:  ovprc  5618
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