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

Theorem fvprc 5113
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 5112 . 2 A V → ¬ ∃!x A𝐹x)
2 tz6.12-2 5110 . 2 ∃!x A𝐹x → (𝐹A) = ∅)
31, 2syl 14 1 A V → (𝐹A) = ∅)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1242   wcel 1390  ∃!weu 1897  Vcvv 2551  c0 3218   class class class wbr 3754  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-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3865  ax-pow 3917  ax-setind 4219
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-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  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-pw 3352  df-sn 3372  df-pr 3373  df-op 3375  df-uni 3571  df-br 3755  df-iota 4809  df-fv 4852
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator