Theorem fsetsnprcnex 46064

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 2731	. . . . . . 7 ⊢ {𝑦 ∣ ∃𝑏 ∈ 𝐵 𝑦 = {⟨𝑆, 𝑏⟩}} = {𝑦 ∣ ∃𝑏 ∈ 𝐵 𝑦 = {⟨𝑆, 𝑏⟩}}
2		eqid 2731	. . . . . . 7 ⊢ (𝑥 ∈ 𝐵 ↦ {⟨𝑆, 𝑥⟩}) = (𝑥 ∈ 𝐵 ↦ {⟨𝑆, 𝑥⟩})
3	1, 2	fsetsnf1o 46063	. . . . . 6 ⊢ (𝑆 ∈ 𝑉 → (𝑥 ∈ 𝐵 ↦ {⟨𝑆, 𝑥⟩}):𝐵–1-1-onto→{𝑦 ∣ ∃𝑏 ∈ 𝐵 𝑦 = {⟨𝑆, 𝑏⟩}})
4		f1ovv 7948	. . . . . 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 3046	. . . 4 ⊢ (𝐵 ∉ V ↔ ¬ 𝐵 ∈ V)
8		df-nel 3046	. . . 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 46059	. . . 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 2732	. . 3 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐵 ∉ V) → V = V)
14	12, 13	neleq12d 3050	. 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 205   ∧ wa 395   = wceq 1540   ∈ wcel 2105  {cab 2708   ∉ wnel 3045  ∃wrex 3069  Vcvv 3473  {csn 4628  ⟨cop 4634   ↦ cmpt 5231  ⟶wf 6539  –1-1-onto→wf1o 6542

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7729

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551

This theorem is referenced by:  fsetprcnexALT  46071