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Theorem fsetsniunop 44246
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 6972 . . . . . 6 (𝑆𝑉 → (𝑔:{𝑆}⟶𝐵 ↔ ((𝑔𝑆) ∈ 𝐵𝑔 = {⟨𝑆, (𝑔𝑆)⟩})))
2 simpl 486 . . . . . . 7 (((𝑔𝑆) ∈ 𝐵𝑔 = {⟨𝑆, (𝑔𝑆)⟩}) → (𝑔𝑆) ∈ 𝐵)
3 opeq2 4800 . . . . . . . . . 10 (𝑏 = (𝑔𝑆) → ⟨𝑆, 𝑏⟩ = ⟨𝑆, (𝑔𝑆)⟩)
43sneqd 4568 . . . . . . . . 9 (𝑏 = (𝑔𝑆) → {⟨𝑆, 𝑏⟩} = {⟨𝑆, (𝑔𝑆)⟩})
54eqeq2d 2749 . . . . . . . 8 (𝑏 = (𝑔𝑆) → (𝑔 = {⟨𝑆, 𝑏⟩} ↔ 𝑔 = {⟨𝑆, (𝑔𝑆)⟩}))
65adantl 485 . . . . . . 7 ((((𝑔𝑆) ∈ 𝐵𝑔 = {⟨𝑆, (𝑔𝑆)⟩}) ∧ 𝑏 = (𝑔𝑆)) → (𝑔 = {⟨𝑆, 𝑏⟩} ↔ 𝑔 = {⟨𝑆, (𝑔𝑆)⟩}))
7 simpr 488 . . . . . . 7 (((𝑔𝑆) ∈ 𝐵𝑔 = {⟨𝑆, (𝑔𝑆)⟩}) → 𝑔 = {⟨𝑆, (𝑔𝑆)⟩})
82, 6, 7rspcedvd 3553 . . . . . 6 (((𝑔𝑆) ∈ 𝐵𝑔 = {⟨𝑆, (𝑔𝑆)⟩}) → ∃𝑏𝐵 𝑔 = {⟨𝑆, 𝑏⟩})
91, 8syl6bi 256 . . . . 5 (𝑆𝑉 → (𝑔:{𝑆}⟶𝐵 → ∃𝑏𝐵 𝑔 = {⟨𝑆, 𝑏⟩}))
10 simpl 486 . . . . . . . . . 10 ((𝑆𝑉𝑏𝐵) → 𝑆𝑉)
11 simpr 488 . . . . . . . . . 10 ((𝑆𝑉𝑏𝐵) → 𝑏𝐵)
1210, 11fsnd 6722 . . . . . . . . 9 ((𝑆𝑉𝑏𝐵) → {⟨𝑆, 𝑏⟩}:{𝑆}⟶𝐵)
1312adantr 484 . . . . . . . 8 (((𝑆𝑉𝑏𝐵) ∧ 𝑔 = {⟨𝑆, 𝑏⟩}) → {⟨𝑆, 𝑏⟩}:{𝑆}⟶𝐵)
14 simpr 488 . . . . . . . . 9 (((𝑆𝑉𝑏𝐵) ∧ 𝑔 = {⟨𝑆, 𝑏⟩}) → 𝑔 = {⟨𝑆, 𝑏⟩})
1514feq1d 6549 . . . . . . . 8 (((𝑆𝑉𝑏𝐵) ∧ 𝑔 = {⟨𝑆, 𝑏⟩}) → (𝑔:{𝑆}⟶𝐵 ↔ {⟨𝑆, 𝑏⟩}:{𝑆}⟶𝐵))
1613, 15mpbird 260 . . . . . . 7 (((𝑆𝑉𝑏𝐵) ∧ 𝑔 = {⟨𝑆, 𝑏⟩}) → 𝑔:{𝑆}⟶𝐵)
1716ex 416 . . . . . 6 ((𝑆𝑉𝑏𝐵) → (𝑔 = {⟨𝑆, 𝑏⟩} → 𝑔:{𝑆}⟶𝐵))
1817rexlimdva 3211 . . . . 5 (𝑆𝑉 → (∃𝑏𝐵 𝑔 = {⟨𝑆, 𝑏⟩} → 𝑔:{𝑆}⟶𝐵))
199, 18impbid 215 . . . 4 (𝑆𝑉 → (𝑔:{𝑆}⟶𝐵 ↔ ∃𝑏𝐵 𝑔 = {⟨𝑆, 𝑏⟩}))
20 velsn 4572 . . . . . 6 (𝑔 ∈ {{⟨𝑆, 𝑏⟩}} ↔ 𝑔 = {⟨𝑆, 𝑏⟩})
2120bicomi 227 . . . . 5 (𝑔 = {⟨𝑆, 𝑏⟩} ↔ 𝑔 ∈ {{⟨𝑆, 𝑏⟩}})
2221rexbii 3176 . . . 4 (∃𝑏𝐵 𝑔 = {⟨𝑆, 𝑏⟩} ↔ ∃𝑏𝐵 𝑔 ∈ {{⟨𝑆, 𝑏⟩}})
2319, 22bitrdi 290 . . 3 (𝑆𝑉 → (𝑔:{𝑆}⟶𝐵 ↔ ∃𝑏𝐵 𝑔 ∈ {{⟨𝑆, 𝑏⟩}}))
24 vex 3425 . . . 4 𝑔 ∈ V
25 feq1 6545 . . . 4 (𝑓 = 𝑔 → (𝑓:{𝑆}⟶𝐵𝑔:{𝑆}⟶𝐵))
2624, 25elab 3600 . . 3 (𝑔 ∈ {𝑓𝑓:{𝑆}⟶𝐵} ↔ 𝑔:{𝑆}⟶𝐵)
27 eliun 4923 . . 3 (𝑔 𝑏𝐵 {{⟨𝑆, 𝑏⟩}} ↔ ∃𝑏𝐵 𝑔 ∈ {{⟨𝑆, 𝑏⟩}})
2823, 26, 273bitr4g 317 . 2 (𝑆𝑉 → (𝑔 ∈ {𝑓𝑓:{𝑆}⟶𝐵} ↔ 𝑔 𝑏𝐵 {{⟨𝑆, 𝑏⟩}}))
2928eqrdv 2736 1 (𝑆𝑉 → {𝑓𝑓:{𝑆}⟶𝐵} = 𝑏𝐵 {{⟨𝑆, 𝑏⟩}})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1543  wcel 2111  {cab 2715  wrex 3063  {csn 4556  cop 4562   ciun 4919  wf 6394  cfv 6398
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 2113  ax-9 2121  ax-10 2142  ax-11 2159  ax-12 2176  ax-ext 2709  ax-sep 5207  ax-nul 5214  ax-pr 5337
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 2072  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2887  df-ral 3067  df-rex 3068  df-reu 3069  df-rab 3071  df-v 3423  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4253  df-if 4455  df-sn 4557  df-pr 4559  df-op 4563  df-uni 4835  df-iun 4921  df-br 5069  df-opab 5131  df-id 5470  df-xp 5572  df-rel 5573  df-cnv 5574  df-co 5575  df-dm 5576  df-rn 5577  df-res 5578  df-ima 5579  df-iota 6356  df-fun 6400  df-fn 6401  df-f 6402  df-f1 6403  df-fo 6404  df-f1o 6405  df-fv 6406
This theorem is referenced by:  fsetabsnop  44247
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