ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  fvprc GIF version

Theorem fvprc 5159
Description: A function's value at a proper class is the empty set. (Contributed by NM, 20-May-1998.)
Assertion
Ref Expression
fvprc 𝐴 ∈ V → (𝐹𝐴) = ∅)

Proof of Theorem fvprc
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 brprcneu 5158 . 2 𝐴 ∈ V → ¬ ∃!𝑥 𝐴𝐹𝑥)
2 tz6.12-2 5156 . 2 (¬ ∃!𝑥 𝐴𝐹𝑥 → (𝐹𝐴) = ∅)
31, 2syl 14 1 𝐴 ∈ V → (𝐹𝐴) = ∅)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1243  wcel 1393  ∃!weu 1900  Vcvv 2554  c0 3221   class class class wbr 3761  cfv 4889
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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3872  ax-pow 3924  ax-setind 4256
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2308  df-rex 2309  df-v 2556  df-dif 2917  df-un 2919  df-in 2921  df-ss 2928  df-nul 3222  df-pw 3358  df-sn 3378  df-pr 3379  df-op 3381  df-uni 3578  df-br 3762  df-iota 4854  df-fv 4897
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator