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Theorem rnfvprc 6900
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 6898 . . . 4 𝑋 ∈ V → (𝐹𝑋) = ∅)
31, 2eqtrid 2786 . . 3 𝑋 ∈ V → 𝑌 = ∅)
43rneqd 5951 . 2 𝑋 ∈ V → ran 𝑌 = ran ∅)
5 rn0 5938 . 2 ran ∅ = ∅
64, 5eqtrdi 2790 1 𝑋 ∈ V → ran 𝑌 = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1536  wcel 2105  Vcvv 3477  c0 4338  ran crn 5689  cfv 6562
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-cnv 5696  df-dm 5698  df-rn 5699  df-iota 6515  df-fv 6570
This theorem is referenced by:  pmtrfrn  19490  mrsubrn  35497  mrsub0  35500  mrsubf  35501  mrsubccat  35502  mrsubcn  35503  mrsubco  35505  mrsubvrs  35506  elmsubrn  35512  msubrn  35513  msubf  35516
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