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Theorem rnfvprc 6883
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 6881 . . . 4 𝑋 ∈ V → (𝐹𝑋) = ∅)
31, 2eqtrid 2785 . . 3 𝑋 ∈ V → 𝑌 = ∅)
43rneqd 5936 . 2 𝑋 ∈ V → ran 𝑌 = ran ∅)
5 rn0 5924 . 2 ran ∅ = ∅
64, 5eqtrdi 2789 1 𝑋 ∈ V → ran 𝑌 = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1542  wcel 2107  Vcvv 3475  c0 4322  ran crn 5677  cfv 6541
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-cnv 5684  df-dm 5686  df-rn 5687  df-iota 6493  df-fv 6549
This theorem is referenced by:  pmtrfrn  19321  mrsubrn  34493  mrsub0  34496  mrsubf  34497  mrsubccat  34498  mrsubcn  34499  mrsubco  34501  mrsubvrs  34502  elmsubrn  34508  msubrn  34509  msubf  34512
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