Theorem fsetdmprc0 8874

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 → {𝑓 ∣ 𝑓 Fn 𝐴} = ∅)

Distinct variable group:   𝐴,𝑓

Proof of Theorem fsetdmprc0

Dummy variable 𝑔 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-nel 3038	. . . 4 ⊢ (𝐴 ∉ V ↔ ¬ 𝐴 ∈ V)
2		vex 3468	. . . . . . 7 ⊢ 𝑔 ∈ V
3	2	a1i 11	. . . . . 6 ⊢ (𝑔 Fn 𝐴 → 𝑔 ∈ V)
4		id 22	. . . . . 6 ⊢ (𝑔 Fn 𝐴 → 𝑔 Fn 𝐴)
5	3, 4	fndmexd 7905	. . . . 5 ⊢ (𝑔 Fn 𝐴 → 𝐴 ∈ V)
6	5	con3i 154	. . . 4 ⊢ (¬ 𝐴 ∈ V → ¬ 𝑔 Fn 𝐴)
7	1, 6	sylbi 217	. . 3 ⊢ (𝐴 ∉ V → ¬ 𝑔 Fn 𝐴)
8	7	alrimiv 1927	. 2 ⊢ (𝐴 ∉ V → ∀𝑔 ¬ 𝑔 Fn 𝐴)
9		fneq1 6634	. . 3 ⊢ (𝑓 = 𝑔 → (𝑓 Fn 𝐴 ↔ 𝑔 Fn 𝐴))
10	9	ab0w 4359	. 2 ⊢ ({𝑓 ∣ 𝑓 Fn 𝐴} = ∅ ↔ ∀𝑔 ¬ 𝑔 Fn 𝐴)
11	8, 10	sylibr 234	1 ⊢ (𝐴 ∉ V → {𝑓 ∣ 𝑓 Fn 𝐴} = ∅)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1538   = wceq 1540   ∈ wcel 2109  {cab 2714   ∉ wnel 3037  Vcvv 3464  ∅c0 4313   Fn wfn 6531

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407  ax-un 7734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-nel 3038  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-br 5125  df-opab 5187  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-fun 6538  df-fn 6539

This theorem is referenced by:  fsetexb  8883