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Theorem ndmfvg 5129
 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 1912 . . . . 5 (∃!x A𝐹xx A𝐹x)
2 eldmg 4457 . . . . 5 (A V → (A dom 𝐹x A𝐹x))
31, 2syl5ibr 145 . . . 4 (A V → (∃!x A𝐹xA dom 𝐹))
43con3d 548 . . 3 (A V → (¬ A dom 𝐹 → ¬ ∃!x A𝐹x))
5 tz6.12-2 5094 . . 3 ∃!x A𝐹x → (𝐹A) = ∅)
64, 5syl6 29 . 2 (A V → (¬ A dom 𝐹 → (𝐹A) = ∅))
76imp 115 1 ((A V ¬ A dom 𝐹) → (𝐹A) = ∅)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   = wceq 1228  ∃wex 1362   ∈ wcel 1374  ∃!weu 1882  Vcvv 2535  ∅c0 3201   class class class wbr 3738  dom cdm 4272  ‘cfv 4829 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004 This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-fal 1234  df-nf 1330  df-sb 1628  df-eu 1885  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-dif 2897  df-un 2899  df-in 2901  df-ss 2908  df-nul 3202  df-sn 3356  df-pr 3357  df-op 3359  df-uni 3555  df-br 3739  df-dm 4282  df-iota 4794  df-fv 4837 This theorem is referenced by:  ovprc  5463
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