Theorem fsetdmprc0 8851

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 3041	. . . 4 ⊢ (𝐴 ∉ V ↔ ¬ 𝐴 ∈ V)
2		vex 3472	. . . . . . 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 7894	. . . . 5 ⊢ (𝑔 Fn 𝐴 → 𝐴 ∈ V)
6	5	con3i 154	. . . 4 ⊢ (¬ 𝐴 ∈ V → ¬ 𝑔 Fn 𝐴)
7	1, 6	sylbi 216	. . 3 ⊢ (𝐴 ∉ V → ¬ 𝑔 Fn 𝐴)
8	7	alrimiv 1922	. 2 ⊢ (𝐴 ∉ V → ∀𝑔 ¬ 𝑔 Fn 𝐴)
9		fneq1 6634	. . 3 ⊢ (𝑓 = 𝑔 → (𝑓 Fn 𝐴 ↔ 𝑔 Fn 𝐴))
10	9	ab0w 4368	. 2 ⊢ ({𝑓 ∣ 𝑓 Fn 𝐴} = ∅ ↔ ∀𝑔 ¬ 𝑔 Fn 𝐴)
11	8, 10	sylibr 233	1 ⊢ (𝐴 ∉ V → {𝑓 ∣ 𝑓 Fn 𝐴} = ∅)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1531   = wceq 1533   ∈ wcel 2098  {cab 2703   ∉ wnel 3040  Vcvv 3468  ∅c0 4317   Fn wfn 6532

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 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7722

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-nel 3041  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-fun 6539  df-fn 6540

This theorem is referenced by:  fsetexb  8860