MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fv2prc Structured version   Visualization version   GIF version

Theorem fv2prc 6952
Description: A function value of a function value at a proper class is the empty set. (Contributed by AV, 8-Apr-2021.)
Assertion
Ref Expression
fv2prc 𝐴 ∈ V → ((𝐹𝐴)‘𝐵) = ∅)

Proof of Theorem fv2prc
StepHypRef Expression
1 fvprc 6899 . . 3 𝐴 ∈ V → (𝐹𝐴) = ∅)
21fveq1d 6909 . 2 𝐴 ∈ V → ((𝐹𝐴)‘𝐵) = (∅‘𝐵))
3 0fv 6951 . 2 (∅‘𝐵) = ∅
42, 3eqtrdi 2791 1 𝐴 ∈ V → ((𝐹𝐴)‘𝐵) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1537  wcel 2106  Vcvv 3478  c0 4339  cfv 6563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-dm 5699  df-iota 6516  df-fv 6571
This theorem is referenced by:  elfv2ex  6953  itunitc1  10458  sralem  21193  sralemOLD  21194  srasca  21201  srascaOLD  21202  sravsca  21203  sravscaOLD  21204  sraip  21205
  Copyright terms: Public domain W3C validator