Theorem fsetdmprc0 8618

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 3052	. . . 4 ⊢ (𝐴 ∉ V ↔ ¬ 𝐴 ∈ V)
2		vex 3435	. . . . . . 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 7741	. . . . 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 1934	. 2 ⊢ (𝐴 ∉ V → ∀𝑔 ¬ 𝑔 Fn 𝐴)
9		fneq1 6521	. . 3 ⊢ (𝑓 = 𝑔 → (𝑓 Fn 𝐴 ↔ 𝑔 Fn 𝐴))
10	9	ab0w 4313	. 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 1540   = wceq 1542   ∈ wcel 2110  {cab 2717   ∉ wnel 3051  Vcvv 3431  ∅c0 4262   Fn wfn 6426

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-ext 2711  ax-sep 5227  ax-nul 5234  ax-pr 5356  ax-un 7580

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2072  df-clab 2718  df-cleq 2732  df-clel 2818  df-nel 3052  df-rab 3075  df-v 3433  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-br 5080  df-opab 5142  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-fun 6433  df-fn 6434

This theorem is referenced by:  fsetexb  8627