Theorem fsetsnprcnex 47519

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 2737	. . . . . . 7 ⊢ {𝑦 ∣ ∃𝑏 ∈ 𝐵 𝑦 = {⟨𝑆, 𝑏⟩}} = {𝑦 ∣ ∃𝑏 ∈ 𝐵 𝑦 = {⟨𝑆, 𝑏⟩}}
2		eqid 2737	. . . . . . 7 ⊢ (𝑥 ∈ 𝐵 ↦ {⟨𝑆, 𝑥⟩}) = (𝑥 ∈ 𝐵 ↦ {⟨𝑆, 𝑥⟩})
3	1, 2	fsetsnf1o 47518	. . . . . 6 ⊢ (𝑆 ∈ 𝑉 → (𝑥 ∈ 𝐵 ↦ {⟨𝑆, 𝑥⟩}):𝐵–1-1-onto→{𝑦 ∣ ∃𝑏 ∈ 𝐵 𝑦 = {⟨𝑆, 𝑏⟩}})
4		f1ovv 7906	. . . . . 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 17	. . . . 5 ⊢ (𝑆 ∈ 𝑉 → (𝐵 ∈ V ↔ {𝑦 ∣ ∃𝑏 ∈ 𝐵 𝑦 = {⟨𝑆, 𝑏⟩}} ∈ V))
6	5	notbid 318	. . . 4 ⊢ (𝑆 ∈ 𝑉 → (¬ 𝐵 ∈ V ↔ ¬ {𝑦 ∣ ∃𝑏 ∈ 𝐵 𝑦 = {⟨𝑆, 𝑏⟩}} ∈ V))
7		df-nel 3038	. . . 4 ⊢ (𝐵 ∉ V ↔ ¬ 𝐵 ∈ V)
8		df-nel 3038	. . . 4 ⊢ ({𝑦 ∣ ∃𝑏 ∈ 𝐵 𝑦 = {⟨𝑆, 𝑏⟩}} ∉ V ↔ ¬ {𝑦 ∣ ∃𝑏 ∈ 𝐵 𝑦 = {⟨𝑆, 𝑏⟩}} ∈ V)
9	6, 7, 8	3bitr4g 314	. . 3 ⊢ (𝑆 ∈ 𝑉 → (𝐵 ∉ V ↔ {𝑦 ∣ ∃𝑏 ∈ 𝐵 𝑦 = {⟨𝑆, 𝑏⟩}} ∉ V))
10	9	biimpa 476	. 2 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐵 ∉ V) → {𝑦 ∣ ∃𝑏 ∈ 𝐵 𝑦 = {⟨𝑆, 𝑏⟩}} ∉ V)
11		fsetabsnop 47514	. . . 4 ⊢ (𝑆 ∈ 𝑉 → {𝑓 ∣ 𝑓:{𝑆}⟶𝐵} = {𝑦 ∣ ∃𝑏 ∈ 𝐵 𝑦 = {⟨𝑆, 𝑏⟩}})
12	11	adantr 480	. . 3 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐵 ∉ V) → {𝑓 ∣ 𝑓:{𝑆}⟶𝐵} = {𝑦 ∣ ∃𝑏 ∈ 𝐵 𝑦 = {⟨𝑆, 𝑏⟩}})
13		eqidd 2738	. . 3 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐵 ∉ V) → V = V)
14	12, 13	neleq12d 3042	. 2 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐵 ∉ V) → ({𝑓 ∣ 𝑓:{𝑆}⟶𝐵} ∉ V ↔ {𝑦 ∣ ∃𝑏 ∈ 𝐵 𝑦 = {⟨𝑆, 𝑏⟩}} ∉ V))
15	10, 14	mpbird 257	1 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐵 ∉ V) → {𝑓 ∣ 𝑓:{𝑆}⟶𝐵} ∉ V)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  {cab 2715   ∉ wnel 3037  ∃wrex 3062  Vcvv 3430  {csn 4568  ⟨cop 4574   ↦ cmpt 5167  ⟶wf 6490  –1-1-onto→wf1o 6493

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-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pr 5372  ax-un 7684

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-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5521  df-xp 5632  df-rel 5633  df-cnv 5634  df-co 5635  df-dm 5636  df-rn 5637  df-res 5638  df-ima 5639  df-iota 6450  df-fun 6496  df-fn 6497  df-f 6498  df-f1 6499  df-fo 6500  df-f1o 6501  df-fv 6502

This theorem is referenced by:  fsetprcnexALT  47526