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Theorem ndmfvg 5145
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 1927 . . . . 5 (∃!x A𝐹xx A𝐹x)
2 eldmg 4472 . . . . 5 (A V → (A dom 𝐹x A𝐹x))
31, 2syl5ibr 145 . . . 4 (A V → (∃!x A𝐹xA dom 𝐹))
43con3d 560 . . 3 (A V → (¬ A dom 𝐹 → ¬ ∃!x A𝐹x))
5 tz6.12-2 5110 . . 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 1242  wex 1378   wcel 1390  ∃!weu 1897  Vcvv 2551  c0 3218   class class class wbr 3754  dom cdm 4287  cfv 4844
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 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-sn 3372  df-pr 3373  df-op 3375  df-uni 3571  df-br 3755  df-dm 4297  df-iota 4809  df-fv 4852
This theorem is referenced by:  ovprc  5479
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