Theorem fsetdmprc0 8804

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 3446	. . . . . . 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 7856	. . . . 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 1929	. 2 ⊢ (𝐴 ∉ V → ∀𝑔 ¬ 𝑔 Fn 𝐴)
9		fneq1 6591	. . 3 ⊢ (𝑓 = 𝑔 → (𝑓 Fn 𝐴 ↔ 𝑔 Fn 𝐴))
10	9	ab0w 4333	. 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 1540   = wceq 1542   ∈ wcel 2114  {cab 2715   ∉ wnel 3037  Vcvv 3442  ∅c0 4287   Fn wfn 6495

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5243  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-nel 3038  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-fun 6502  df-fn 6503

This theorem is referenced by:  fsetexb  8813