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Theorem fsetsnprcnex 44229
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 ((𝑆𝑉𝐵 ∉ V) → {𝑓𝑓:{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.
StepHypRef Expression
1 eqid 2737 . . . . . . 7 {𝑦 ∣ ∃𝑏𝐵 𝑦 = {⟨𝑆, 𝑏⟩}} = {𝑦 ∣ ∃𝑏𝐵 𝑦 = {⟨𝑆, 𝑏⟩}}
2 eqid 2737 . . . . . . 7 (𝑥𝐵 ↦ {⟨𝑆, 𝑥⟩}) = (𝑥𝐵 ↦ {⟨𝑆, 𝑥⟩})
31, 2fsetsnf1o 44228 . . . . . 6 (𝑆𝑉 → (𝑥𝐵 ↦ {⟨𝑆, 𝑥⟩}):𝐵1-1-onto→{𝑦 ∣ ∃𝑏𝐵 𝑦 = {⟨𝑆, 𝑏⟩}})
4 f1ovv 7736 . . . . . 6 ((𝑥𝐵 ↦ {⟨𝑆, 𝑥⟩}):𝐵1-1-onto→{𝑦 ∣ ∃𝑏𝐵 𝑦 = {⟨𝑆, 𝑏⟩}} → (𝐵 ∈ V ↔ {𝑦 ∣ ∃𝑏𝐵 𝑦 = {⟨𝑆, 𝑏⟩}} ∈ V))
53, 4syl 17 . . . . 5 (𝑆𝑉 → (𝐵 ∈ V ↔ {𝑦 ∣ ∃𝑏𝐵 𝑦 = {⟨𝑆, 𝑏⟩}} ∈ V))
65notbid 321 . . . 4 (𝑆𝑉 → (¬ 𝐵 ∈ V ↔ ¬ {𝑦 ∣ ∃𝑏𝐵 𝑦 = {⟨𝑆, 𝑏⟩}} ∈ V))
7 df-nel 3047 . . . 4 (𝐵 ∉ V ↔ ¬ 𝐵 ∈ V)
8 df-nel 3047 . . . 4 ({𝑦 ∣ ∃𝑏𝐵 𝑦 = {⟨𝑆, 𝑏⟩}} ∉ V ↔ ¬ {𝑦 ∣ ∃𝑏𝐵 𝑦 = {⟨𝑆, 𝑏⟩}} ∈ V)
96, 7, 83bitr4g 317 . . 3 (𝑆𝑉 → (𝐵 ∉ V ↔ {𝑦 ∣ ∃𝑏𝐵 𝑦 = {⟨𝑆, 𝑏⟩}} ∉ V))
109biimpa 480 . 2 ((𝑆𝑉𝐵 ∉ V) → {𝑦 ∣ ∃𝑏𝐵 𝑦 = {⟨𝑆, 𝑏⟩}} ∉ V)
11 fsetabsnop 44224 . . . 4 (𝑆𝑉 → {𝑓𝑓:{𝑆}⟶𝐵} = {𝑦 ∣ ∃𝑏𝐵 𝑦 = {⟨𝑆, 𝑏⟩}})
1211adantr 484 . . 3 ((𝑆𝑉𝐵 ∉ V) → {𝑓𝑓:{𝑆}⟶𝐵} = {𝑦 ∣ ∃𝑏𝐵 𝑦 = {⟨𝑆, 𝑏⟩}})
13 eqidd 2738 . . 3 ((𝑆𝑉𝐵 ∉ V) → V = V)
1412, 13neleq12d 3050 . 2 ((𝑆𝑉𝐵 ∉ V) → ({𝑓𝑓:{𝑆}⟶𝐵} ∉ V ↔ {𝑦 ∣ ∃𝑏𝐵 𝑦 = {⟨𝑆, 𝑏⟩}} ∉ V))
1510, 14mpbird 260 1 ((𝑆𝑉𝐵 ∉ V) → {𝑓𝑓:{𝑆}⟶𝐵} ∉ V)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399   = wceq 1543  wcel 2110  {cab 2714  wnel 3046  wrex 3062  Vcvv 3413  {csn 4546  cop 4552  cmpt 5140  wf 6381  1-1-ontowf1o 6384
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5184  ax-sep 5197  ax-nul 5204  ax-pr 5327  ax-un 7528
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3415  df-sbc 3700  df-csb 3817  df-dif 3874  df-un 3876  df-in 3878  df-ss 3888  df-nul 4243  df-if 4445  df-sn 4547  df-pr 4549  df-op 4553  df-uni 4825  df-iun 4911  df-br 5059  df-opab 5121  df-mpt 5141  df-id 5460  df-xp 5562  df-rel 5563  df-cnv 5564  df-co 5565  df-dm 5566  df-rn 5567  df-res 5568  df-ima 5569  df-iota 6343  df-fun 6387  df-fn 6388  df-f 6389  df-f1 6390  df-fo 6391  df-f1o 6392  df-fv 6393
This theorem is referenced by:  fsetprcnexALT  44236
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