Theorem fsetsnprcnex 47930

Description: The class of all functions from a (proper) singleton into a proper class 𝐵 is not a set. (Contributed by AV, 13-Sep-2024.)

Assertion

Ref	Expression
fsetsnprcnex	⊢ ((𝑆 ∈ 𝑉 ∧ 𝐵 ∉ V) → {𝑓 ∣ 𝑓:{𝑆}⟶𝐵} ∉ V)

Distinct variable groups:   𝐵,𝑓   𝑆,𝑓

Allowed substitution hint:   𝑉(𝑓)

Proof of Theorem fsetsnprcnex

Dummy variables 𝑏 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2762	. . . . . . 7 ⊢ {𝑦 ∣ ∃𝑏 ∈ 𝐵 𝑦 = {⟨𝑆, 𝑏⟩}} = {𝑦 ∣ ∃𝑏 ∈ 𝐵 𝑦 = {⟨𝑆, 𝑏⟩}}
2		eqid 2762	. . . . . . 7 ⊢ (𝑥 ∈ 𝐵 ↦ {⟨𝑆, 𝑥⟩}) = (𝑥 ∈ 𝐵 ↦ {⟨𝑆, 𝑥⟩})
3	1, 2	fsetsnf1o 47929	. . . . . 6 ⊢ (𝑆 ∈ 𝑉 → (𝑥 ∈ 𝐵 ↦ {⟨𝑆, 𝑥⟩}):𝐵–1-1-onto→{𝑦 ∣ ∃𝑏 ∈ 𝐵 𝑦 = {⟨𝑆, 𝑏⟩}})
4		f1ovv 7958	. . . . . 6 ⊢ ((𝑥 ∈ 𝐵 ↦ {⟨𝑆, 𝑥⟩}):𝐵–1-1-onto→{𝑦 ∣ ∃𝑏 ∈ 𝐵 𝑦 = {⟨𝑆, 𝑏⟩}} → (𝐵 ∈ V ↔ {𝑦 ∣ ∃𝑏 ∈ 𝐵 𝑦 = {⟨𝑆, 𝑏⟩}} ∈ V))
5	3, 4	syl 18	. . . . 5 ⊢ (𝑆 ∈ 𝑉 → (𝐵 ∈ V ↔ {𝑦 ∣ ∃𝑏 ∈ 𝐵 𝑦 = {⟨𝑆, 𝑏⟩}} ∈ V))
6	5	notbid 321	. . . 4 ⊢ (𝑆 ∈ 𝑉 → (¬ 𝐵 ∈ V ↔ ¬ {𝑦 ∣ ∃𝑏 ∈ 𝐵 𝑦 = {⟨𝑆, 𝑏⟩}} ∈ V))
7		df-nel 3064	. . . 4 ⊢ (𝐵 ∉ V ↔ ¬ 𝐵 ∈ V)
8		df-nel 3064	. . . 4 ⊢ ({𝑦 ∣ ∃𝑏 ∈ 𝐵 𝑦 = {⟨𝑆, 𝑏⟩}} ∉ V ↔ ¬ {𝑦 ∣ ∃𝑏 ∈ 𝐵 𝑦 = {⟨𝑆, 𝑏⟩}} ∈ V)
9	6, 7, 8	3bitr4g 317	. . 3 ⊢ (𝑆 ∈ 𝑉 → (𝐵 ∉ V ↔ {𝑦 ∣ ∃𝑏 ∈ 𝐵 𝑦 = {⟨𝑆, 𝑏⟩}} ∉ V))
10	9	biimpa 482	. 2 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐵 ∉ V) → {𝑦 ∣ ∃𝑏 ∈ 𝐵 𝑦 = {⟨𝑆, 𝑏⟩}} ∉ V)
11		fsetabsnop 47925	. . . 4 ⊢ (𝑆 ∈ 𝑉 → {𝑓 ∣ 𝑓:{𝑆}⟶𝐵} = {𝑦 ∣ ∃𝑏 ∈ 𝐵 𝑦 = {⟨𝑆, 𝑏⟩}})
12	11	adantr 486	. . 3 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐵 ∉ V) → {𝑓 ∣ 𝑓:{𝑆}⟶𝐵} = {𝑦 ∣ ∃𝑏 ∈ 𝐵 𝑦 = {⟨𝑆, 𝑏⟩}})
13		eqidd 2763	. . 3 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐵 ∉ V) → V = V)
14	12, 13	neleq12d 3068	. 2 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐵 ∉ V) → ({𝑓 ∣ 𝑓:{𝑆}⟶𝐵} ∉ V ↔ {𝑦 ∣ ∃𝑏 ∈ 𝐵 𝑦 = {⟨𝑆, 𝑏⟩}} ∉ V))
15	10, 14	mpbird 260	1 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐵 ∉ V) → {𝑓 ∣ 𝑓:{𝑆}⟶𝐵} ∉ V)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  {cab 2740   ∉ wnel 3063  ∃wrex 3088  Vcvv 3453  {csn 4587  ⟨cop 4593   ↦ cmpt 5190  ⟶wf 6533  –1-1-onto→wf1o 6536

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pr 5402  ax-un 7739

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3064  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545

This theorem is used by:  fsetprcnexALT  47937