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

Theorem rnfvprc 6637
Description: The range of a function value at a proper class is empty. (Contributed by AV, 20-Aug-2022.)
Hypothesis
Ref Expression
rnfvprc.y 𝑌 = (𝐹𝑋)
Assertion
Ref Expression
rnfvprc 𝑋 ∈ V → ran 𝑌 = ∅)

Proof of Theorem rnfvprc
StepHypRef Expression
1 rnfvprc.y . . . 4 𝑌 = (𝐹𝑋)
2 fvprc 6636 . . . 4 𝑋 ∈ V → (𝐹𝑋) = ∅)
31, 2syl5eq 2868 . . 3 𝑋 ∈ V → 𝑌 = ∅)
43rneqd 5781 . 2 𝑋 ∈ V → ran 𝑌 = ran ∅)
5 rn0 5769 . 2 ran ∅ = ∅
64, 5syl6eq 2872 1 𝑋 ∈ V → ran 𝑌 = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1538  wcel 2115  Vcvv 3471  c0 4266  ran crn 5529  cfv 6328
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ral 3131  df-rex 3132  df-rab 3135  df-v 3473  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-br 5040  df-opab 5102  df-cnv 5536  df-dm 5538  df-rn 5539  df-iota 6287  df-fv 6336
This theorem is referenced by:  pmtrfrn  18564  mrsubrn  32767  mrsub0  32770  mrsubf  32771  mrsubccat  32772  mrsubcn  32773  mrsubco  32775  mrsubvrs  32776  elmsubrn  32782  msubrn  32783  msubf  32786
  Copyright terms: Public domain W3C validator