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Theorem fsetsnprcnex 46063
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 2730 . . . . . . 7 {𝑦 ∣ ∃𝑏𝐵 𝑦 = {⟨𝑆, 𝑏⟩}} = {𝑦 ∣ ∃𝑏𝐵 𝑦 = {⟨𝑆, 𝑏⟩}}
2 eqid 2730 . . . . . . 7 (𝑥𝐵 ↦ {⟨𝑆, 𝑥⟩}) = (𝑥𝐵 ↦ {⟨𝑆, 𝑥⟩})
31, 2fsetsnf1o 46062 . . . . . 6 (𝑆𝑉 → (𝑥𝐵 ↦ {⟨𝑆, 𝑥⟩}):𝐵1-1-onto→{𝑦 ∣ ∃𝑏𝐵 𝑦 = {⟨𝑆, 𝑏⟩}})
4 f1ovv 7946 . . . . . 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 317 . . . 4 (𝑆𝑉 → (¬ 𝐵 ∈ V ↔ ¬ {𝑦 ∣ ∃𝑏𝐵 𝑦 = {⟨𝑆, 𝑏⟩}} ∈ V))
7 df-nel 3045 . . . 4 (𝐵 ∉ V ↔ ¬ 𝐵 ∈ V)
8 df-nel 3045 . . . 4 ({𝑦 ∣ ∃𝑏𝐵 𝑦 = {⟨𝑆, 𝑏⟩}} ∉ V ↔ ¬ {𝑦 ∣ ∃𝑏𝐵 𝑦 = {⟨𝑆, 𝑏⟩}} ∈ V)
96, 7, 83bitr4g 313 . . 3 (𝑆𝑉 → (𝐵 ∉ V ↔ {𝑦 ∣ ∃𝑏𝐵 𝑦 = {⟨𝑆, 𝑏⟩}} ∉ V))
109biimpa 475 . 2 ((𝑆𝑉𝐵 ∉ V) → {𝑦 ∣ ∃𝑏𝐵 𝑦 = {⟨𝑆, 𝑏⟩}} ∉ V)
11 fsetabsnop 46058 . . . 4 (𝑆𝑉 → {𝑓𝑓:{𝑆}⟶𝐵} = {𝑦 ∣ ∃𝑏𝐵 𝑦 = {⟨𝑆, 𝑏⟩}})
1211adantr 479 . . 3 ((𝑆𝑉𝐵 ∉ V) → {𝑓𝑓:{𝑆}⟶𝐵} = {𝑦 ∣ ∃𝑏𝐵 𝑦 = {⟨𝑆, 𝑏⟩}})
13 eqidd 2731 . . 3 ((𝑆𝑉𝐵 ∉ V) → V = V)
1412, 13neleq12d 3049 . 2 ((𝑆𝑉𝐵 ∉ V) → ({𝑓𝑓:{𝑆}⟶𝐵} ∉ V ↔ {𝑦 ∣ ∃𝑏𝐵 𝑦 = {⟨𝑆, 𝑏⟩}} ∉ V))
1510, 14mpbird 256 1 ((𝑆𝑉𝐵 ∉ V) → {𝑓𝑓:{𝑆}⟶𝐵} ∉ V)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394   = wceq 1539  wcel 2104  {cab 2707  wnel 3044  wrex 3068  Vcvv 3472  {csn 4627  cop 4633  cmpt 5230  wf 6538  1-1-ontowf1o 6541
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550
This theorem is referenced by:  fsetprcnexALT  46070
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