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

Theorem fv2prc 6391
Description: A function's value at a function's 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 6348 . . 3 𝐴 ∈ V → (𝐹𝐴) = ∅)
21fveq1d 6356 . 2 𝐴 ∈ V → ((𝐹𝐴)‘𝐵) = (∅‘𝐵))
3 0fv 6390 . 2 (∅‘𝐵) = ∅
42, 3syl6eq 2811 1 𝐴 ∈ V → ((𝐹𝐴)‘𝐵) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1632  wcel 2140  Vcvv 3341  c0 4059  cfv 6050
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-8 2142  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-nul 4942  ax-pow 4993
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3343  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-nul 4060  df-if 4232  df-sn 4323  df-pr 4325  df-op 4329  df-uni 4590  df-br 4806  df-dm 5277  df-iota 6013  df-fv 6058
This theorem is referenced by:  elfv2ex  6392
  Copyright terms: Public domain W3C validator