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Theorem rnfvprc 6729
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 6727 . . . 4 𝑋 ∈ V → (𝐹𝑋) = ∅)
31, 2eqtrid 2790 . . 3 𝑋 ∈ V → 𝑌 = ∅)
43rneqd 5821 . 2 𝑋 ∈ V → ran 𝑌 = ran ∅)
5 rn0 5809 . 2 ran ∅ = ∅
64, 5eqtrdi 2795 1 𝑋 ∈ V → ran 𝑌 = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1543  wcel 2111  Vcvv 3420  c0 4251  ran crn 5566  cfv 6397
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2159  ax-12 2176  ax-ext 2709  ax-sep 5206  ax-nul 5213  ax-pr 5336
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2072  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-ne 2942  df-ral 3067  df-rex 3068  df-rab 3071  df-v 3422  df-dif 3883  df-un 3885  df-in 3887  df-ss 3897  df-nul 4252  df-if 4454  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4834  df-br 5068  df-opab 5130  df-cnv 5573  df-dm 5575  df-rn 5576  df-iota 6355  df-fv 6405
This theorem is referenced by:  pmtrfrn  18874  mrsubrn  33211  mrsub0  33214  mrsubf  33215  mrsubccat  33216  mrsubcn  33217  mrsubco  33219  mrsubvrs  33220  elmsubrn  33226  msubrn  33227  msubf  33230
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