Theorem fsetdmprc0 8447

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

Step	Hyp	Ref	Expression
1		df-nel 3056	. . . 4 ⊢ (𝐴 ∉ V ↔ ¬ 𝐴 ∈ V)
2		vex 3413	. . . . . . 7 ⊢ 𝑓 ∈ V
3	2	a1i 11	. . . . . 6 ⊢ (𝑓 Fn 𝐴 → 𝑓 ∈ V)
4		id 22	. . . . . 6 ⊢ (𝑓 Fn 𝐴 → 𝑓 Fn 𝐴)
5	3, 4	fndmexd 7621	. . . . 5 ⊢ (𝑓 Fn 𝐴 → 𝐴 ∈ V)
6	5	con3i 157	. . . 4 ⊢ (¬ 𝐴 ∈ V → ¬ 𝑓 Fn 𝐴)
7	1, 6	sylbi 220	. . 3 ⊢ (𝐴 ∉ V → ¬ 𝑓 Fn 𝐴)
8	7	alrimiv 1928	. 2 ⊢ (𝐴 ∉ V → ∀𝑓 ¬ 𝑓 Fn 𝐴)
9		ab0 4274	. 2 ⊢ ({𝑓 ∣ 𝑓 Fn 𝐴} = ∅ ↔ ∀𝑓 ¬ 𝑓 Fn 𝐴)
10	8, 9	sylibr 237	1 ⊢ (𝐴 ∉ V → {𝑓 ∣ 𝑓 Fn 𝐴} = ∅)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1536   = wceq 1538   ∈ wcel 2111  {cab 2735   ∉ wnel 3055  Vcvv 3409  ∅c0 4227   Fn wfn 6334

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5172  ax-nul 5179  ax-pr 5301  ax-un 7464

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-nel 3056  df-rab 3079  df-v 3411  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5036  df-opab 5098  df-cnv 5535  df-dm 5537  df-rn 5538  df-fn 6342

This theorem is referenced by: (None)