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Theorem ndmfvg 5452
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 2029 . . . . 5 (∃!𝑥 𝐴𝐹𝑥 → ∃𝑥 𝐴𝐹𝑥)
2 eldmg 4734 . . . . 5 (𝐴 ∈ V → (𝐴 ∈ dom 𝐹 ↔ ∃𝑥 𝐴𝐹𝑥))
31, 2syl5ibr 155 . . . 4 (𝐴 ∈ V → (∃!𝑥 𝐴𝐹𝑥𝐴 ∈ dom 𝐹))
43con3d 620 . . 3 (𝐴 ∈ V → (¬ 𝐴 ∈ dom 𝐹 → ¬ ∃!𝑥 𝐴𝐹𝑥))
5 tz6.12-2 5412 . . 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 1331  wex 1468  wcel 1480  ∃!weu 1999  Vcvv 2686  c0 3363   class class class wbr 3929  dom cdm 4539  cfv 5123
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 2121
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 2002  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-dm 4549  df-iota 5088  df-fv 5131
This theorem is referenced by:  ovprc  5806  sumnul  11205
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