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Theorem fsetdmprc0 8796
Description: The set of functions with a proper class as domain is empty. (Contributed by AV, 22-Aug-2024.)
Assertion
Ref Expression
fsetdmprc0 (𝐴 ∉ V → {𝑓𝑓 Fn 𝐴} = ∅)
Distinct variable group:   𝐴,𝑓

Proof of Theorem fsetdmprc0
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 df-nel 3047 . . . 4 (𝐴 ∉ V ↔ ¬ 𝐴 ∈ V)
2 vex 3448 . . . . . . 7 𝑔 ∈ V
32a1i 11 . . . . . 6 (𝑔 Fn 𝐴𝑔 ∈ V)
4 id 22 . . . . . 6 (𝑔 Fn 𝐴𝑔 Fn 𝐴)
53, 4fndmexd 7844 . . . . 5 (𝑔 Fn 𝐴𝐴 ∈ V)
65con3i 154 . . . 4 𝐴 ∈ V → ¬ 𝑔 Fn 𝐴)
71, 6sylbi 216 . . 3 (𝐴 ∉ V → ¬ 𝑔 Fn 𝐴)
87alrimiv 1931 . 2 (𝐴 ∉ V → ∀𝑔 ¬ 𝑔 Fn 𝐴)
9 fneq1 6594 . . 3 (𝑓 = 𝑔 → (𝑓 Fn 𝐴𝑔 Fn 𝐴))
109ab0w 4334 . 2 ({𝑓𝑓 Fn 𝐴} = ∅ ↔ ∀𝑔 ¬ 𝑔 Fn 𝐴)
118, 10sylibr 233 1 (𝐴 ∉ V → {𝑓𝑓 Fn 𝐴} = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1540   = wceq 1542  wcel 2107  {cab 2710  wnel 3046  Vcvv 3444  c0 4283   Fn wfn 6492
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-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385  ax-un 7673
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-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nel 3047  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-fun 6499  df-fn 6500
This theorem is referenced by:  fsetexb  8805
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