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Theorem fsetdmprc0 8872
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 3037 . . . 4 (𝐴 ∉ V ↔ ¬ 𝐴 ∈ V)
2 vex 3467 . . . . . . 7 𝑔 ∈ V
32a1i 11 . . . . . 6 (𝑔 Fn 𝐴𝑔 ∈ V)
4 id 22 . . . . . 6 (𝑔 Fn 𝐴𝑔 Fn 𝐴)
53, 4fndmexd 7910 . . . . 5 (𝑔 Fn 𝐴𝐴 ∈ V)
65con3i 154 . . . 4 𝐴 ∈ V → ¬ 𝑔 Fn 𝐴)
71, 6sylbi 216 . . 3 (𝐴 ∉ V → ¬ 𝑔 Fn 𝐴)
87alrimiv 1922 . 2 (𝐴 ∉ V → ∀𝑔 ¬ 𝑔 Fn 𝐴)
9 fneq1 6640 . . 3 (𝑓 = 𝑔 → (𝑓 Fn 𝐴𝑔 Fn 𝐴))
109ab0w 4369 . 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 1531   = wceq 1533  wcel 2098  {cab 2702  wnel 3036  Vcvv 3463  c0 4318   Fn wfn 6538
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423  ax-un 7738
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-nel 3037  df-rab 3420  df-v 3465  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-fun 6545  df-fn 6546
This theorem is referenced by:  fsetexb  8881
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