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Theorem fsetsniunop 46354
Description: The class of all functions from a (proper) singleton into 𝐵 is the union of all the singletons of (proper) ordered pairs over the elements of 𝐵 as second component. (Contributed by AV, 13-Sep-2024.)
Assertion
Ref Expression
fsetsniunop (𝑆𝑉 → {𝑓𝑓:{𝑆}⟶𝐵} = 𝑏𝐵 {{⟨𝑆, 𝑏⟩}})
Distinct variable groups:   𝐵,𝑏,𝑓   𝑆,𝑏,𝑓   𝑉,𝑏
Allowed substitution hint:   𝑉(𝑓)

Proof of Theorem fsetsniunop
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 fsn2g 7141 . . . . . 6 (𝑆𝑉 → (𝑔:{𝑆}⟶𝐵 ↔ ((𝑔𝑆) ∈ 𝐵𝑔 = {⟨𝑆, (𝑔𝑆)⟩})))
2 simpl 482 . . . . . . 7 (((𝑔𝑆) ∈ 𝐵𝑔 = {⟨𝑆, (𝑔𝑆)⟩}) → (𝑔𝑆) ∈ 𝐵)
3 opeq2 4870 . . . . . . . . . 10 (𝑏 = (𝑔𝑆) → ⟨𝑆, 𝑏⟩ = ⟨𝑆, (𝑔𝑆)⟩)
43sneqd 4636 . . . . . . . . 9 (𝑏 = (𝑔𝑆) → {⟨𝑆, 𝑏⟩} = {⟨𝑆, (𝑔𝑆)⟩})
54eqeq2d 2738 . . . . . . . 8 (𝑏 = (𝑔𝑆) → (𝑔 = {⟨𝑆, 𝑏⟩} ↔ 𝑔 = {⟨𝑆, (𝑔𝑆)⟩}))
65adantl 481 . . . . . . 7 ((((𝑔𝑆) ∈ 𝐵𝑔 = {⟨𝑆, (𝑔𝑆)⟩}) ∧ 𝑏 = (𝑔𝑆)) → (𝑔 = {⟨𝑆, 𝑏⟩} ↔ 𝑔 = {⟨𝑆, (𝑔𝑆)⟩}))
7 simpr 484 . . . . . . 7 (((𝑔𝑆) ∈ 𝐵𝑔 = {⟨𝑆, (𝑔𝑆)⟩}) → 𝑔 = {⟨𝑆, (𝑔𝑆)⟩})
82, 6, 7rspcedvd 3609 . . . . . 6 (((𝑔𝑆) ∈ 𝐵𝑔 = {⟨𝑆, (𝑔𝑆)⟩}) → ∃𝑏𝐵 𝑔 = {⟨𝑆, 𝑏⟩})
91, 8syl6bi 253 . . . . 5 (𝑆𝑉 → (𝑔:{𝑆}⟶𝐵 → ∃𝑏𝐵 𝑔 = {⟨𝑆, 𝑏⟩}))
10 simpl 482 . . . . . . . . . 10 ((𝑆𝑉𝑏𝐵) → 𝑆𝑉)
11 simpr 484 . . . . . . . . . 10 ((𝑆𝑉𝑏𝐵) → 𝑏𝐵)
1210, 11fsnd 6876 . . . . . . . . 9 ((𝑆𝑉𝑏𝐵) → {⟨𝑆, 𝑏⟩}:{𝑆}⟶𝐵)
1312adantr 480 . . . . . . . 8 (((𝑆𝑉𝑏𝐵) ∧ 𝑔 = {⟨𝑆, 𝑏⟩}) → {⟨𝑆, 𝑏⟩}:{𝑆}⟶𝐵)
14 simpr 484 . . . . . . . . 9 (((𝑆𝑉𝑏𝐵) ∧ 𝑔 = {⟨𝑆, 𝑏⟩}) → 𝑔 = {⟨𝑆, 𝑏⟩})
1514feq1d 6701 . . . . . . . 8 (((𝑆𝑉𝑏𝐵) ∧ 𝑔 = {⟨𝑆, 𝑏⟩}) → (𝑔:{𝑆}⟶𝐵 ↔ {⟨𝑆, 𝑏⟩}:{𝑆}⟶𝐵))
1613, 15mpbird 257 . . . . . . 7 (((𝑆𝑉𝑏𝐵) ∧ 𝑔 = {⟨𝑆, 𝑏⟩}) → 𝑔:{𝑆}⟶𝐵)
1716ex 412 . . . . . 6 ((𝑆𝑉𝑏𝐵) → (𝑔 = {⟨𝑆, 𝑏⟩} → 𝑔:{𝑆}⟶𝐵))
1817rexlimdva 3150 . . . . 5 (𝑆𝑉 → (∃𝑏𝐵 𝑔 = {⟨𝑆, 𝑏⟩} → 𝑔:{𝑆}⟶𝐵))
199, 18impbid 211 . . . 4 (𝑆𝑉 → (𝑔:{𝑆}⟶𝐵 ↔ ∃𝑏𝐵 𝑔 = {⟨𝑆, 𝑏⟩}))
20 velsn 4640 . . . . . 6 (𝑔 ∈ {{⟨𝑆, 𝑏⟩}} ↔ 𝑔 = {⟨𝑆, 𝑏⟩})
2120bicomi 223 . . . . 5 (𝑔 = {⟨𝑆, 𝑏⟩} ↔ 𝑔 ∈ {{⟨𝑆, 𝑏⟩}})
2221rexbii 3089 . . . 4 (∃𝑏𝐵 𝑔 = {⟨𝑆, 𝑏⟩} ↔ ∃𝑏𝐵 𝑔 ∈ {{⟨𝑆, 𝑏⟩}})
2319, 22bitrdi 287 . . 3 (𝑆𝑉 → (𝑔:{𝑆}⟶𝐵 ↔ ∃𝑏𝐵 𝑔 ∈ {{⟨𝑆, 𝑏⟩}}))
24 vex 3473 . . . 4 𝑔 ∈ V
25 feq1 6697 . . . 4 (𝑓 = 𝑔 → (𝑓:{𝑆}⟶𝐵𝑔:{𝑆}⟶𝐵))
2624, 25elab 3665 . . 3 (𝑔 ∈ {𝑓𝑓:{𝑆}⟶𝐵} ↔ 𝑔:{𝑆}⟶𝐵)
27 eliun 4995 . . 3 (𝑔 𝑏𝐵 {{⟨𝑆, 𝑏⟩}} ↔ ∃𝑏𝐵 𝑔 ∈ {{⟨𝑆, 𝑏⟩}})
2823, 26, 273bitr4g 314 . 2 (𝑆𝑉 → (𝑔 ∈ {𝑓𝑓:{𝑆}⟶𝐵} ↔ 𝑔 𝑏𝐵 {{⟨𝑆, 𝑏⟩}}))
2928eqrdv 2725 1 (𝑆𝑉 → {𝑓𝑓:{𝑆}⟶𝐵} = 𝑏𝐵 {{⟨𝑆, 𝑏⟩}})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1534  wcel 2099  {cab 2704  wrex 3065  {csn 4624  cop 4630   ciun 4991  wf 6538  cfv 6542
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-ne 2936  df-ral 3057  df-rex 3066  df-reu 3372  df-rab 3428  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  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:  fsetabsnop  46355
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