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Theorem rnfvprc 6657
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 6656 . . . 4 𝑋 ∈ V → (𝐹𝑋) = ∅)
31, 2syl5eq 2865 . . 3 𝑋 ∈ V → 𝑌 = ∅)
43rneqd 5801 . 2 𝑋 ∈ V → ran 𝑌 = ran ∅)
5 rn0 5789 . 2 ran ∅ = ∅
64, 5syl6eq 2869 1 𝑋 ∈ V → ran 𝑌 = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1528  wcel 2105  Vcvv 3492  c0 4288  ran crn 5549  cfv 6348
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-cnv 5556  df-dm 5558  df-rn 5559  df-iota 6307  df-fv 6356
This theorem is referenced by:  pmtrfrn  18515  mrsubrn  32657  mrsub0  32660  mrsubf  32661  mrsubccat  32662  mrsubcn  32663  mrsubco  32665  mrsubvrs  32666  elmsubrn  32672  msubrn  32673  msubf  32676
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