Theorem fsetdmprc0 8828

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 3030	. . . 4 ⊢ (𝐴 ∉ V ↔ ¬ 𝐴 ∈ V)
2		vex 3451	. . . . . . 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 7880	. . . . 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 6609	. . 3 ⊢ (𝑓 = 𝑔 → (𝑓 Fn 𝐴 ↔ 𝑔 Fn 𝐴))
10	9	ab0w 4342	. 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 2707   ∉ wnel 3029  Vcvv 3447  ∅c0 4296   Fn wfn 6506

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 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387  ax-un 7711

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 2708  df-cleq 2721  df-clel 2803  df-nel 3030  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-fun 6513  df-fn 6514

This theorem is referenced by:  fsetexb  8837