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Theorem fsetsniunop 46998
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 7157 . . . . . 6 (𝑆𝑉 → (𝑔:{𝑆}⟶𝐵 ↔ ((𝑔𝑆) ∈ 𝐵𝑔 = {⟨𝑆, (𝑔𝑆)⟩})))
2 simpl 482 . . . . . . 7 (((𝑔𝑆) ∈ 𝐵𝑔 = {⟨𝑆, (𝑔𝑆)⟩}) → (𝑔𝑆) ∈ 𝐵)
3 opeq2 4878 . . . . . . . . . 10 (𝑏 = (𝑔𝑆) → ⟨𝑆, 𝑏⟩ = ⟨𝑆, (𝑔𝑆)⟩)
43sneqd 4642 . . . . . . . . 9 (𝑏 = (𝑔𝑆) → {⟨𝑆, 𝑏⟩} = {⟨𝑆, (𝑔𝑆)⟩})
54eqeq2d 2745 . . . . . . . 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 3623 . . . . . 6 (((𝑔𝑆) ∈ 𝐵𝑔 = {⟨𝑆, (𝑔𝑆)⟩}) → ∃𝑏𝐵 𝑔 = {⟨𝑆, 𝑏⟩})
91, 8biimtrdi 253 . . . . 5 (𝑆𝑉 → (𝑔:{𝑆}⟶𝐵 → ∃𝑏𝐵 𝑔 = {⟨𝑆, 𝑏⟩}))
10 simpl 482 . . . . . . . . . 10 ((𝑆𝑉𝑏𝐵) → 𝑆𝑉)
11 simpr 484 . . . . . . . . . 10 ((𝑆𝑉𝑏𝐵) → 𝑏𝐵)
1210, 11fsnd 6891 . . . . . . . . 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 6720 . . . . . . . 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 3152 . . . . 5 (𝑆𝑉 → (∃𝑏𝐵 𝑔 = {⟨𝑆, 𝑏⟩} → 𝑔:{𝑆}⟶𝐵))
199, 18impbid 212 . . . 4 (𝑆𝑉 → (𝑔:{𝑆}⟶𝐵 ↔ ∃𝑏𝐵 𝑔 = {⟨𝑆, 𝑏⟩}))
20 velsn 4646 . . . . . 6 (𝑔 ∈ {{⟨𝑆, 𝑏⟩}} ↔ 𝑔 = {⟨𝑆, 𝑏⟩})
2120bicomi 224 . . . . 5 (𝑔 = {⟨𝑆, 𝑏⟩} ↔ 𝑔 ∈ {{⟨𝑆, 𝑏⟩}})
2221rexbii 3091 . . . 4 (∃𝑏𝐵 𝑔 = {⟨𝑆, 𝑏⟩} ↔ ∃𝑏𝐵 𝑔 ∈ {{⟨𝑆, 𝑏⟩}})
2319, 22bitrdi 287 . . 3 (𝑆𝑉 → (𝑔:{𝑆}⟶𝐵 ↔ ∃𝑏𝐵 𝑔 ∈ {{⟨𝑆, 𝑏⟩}}))
24 vex 3481 . . . 4 𝑔 ∈ V
25 feq1 6716 . . . 4 (𝑓 = 𝑔 → (𝑓:{𝑆}⟶𝐵𝑔:{𝑆}⟶𝐵))
2624, 25elab 3680 . . 3 (𝑔 ∈ {𝑓𝑓:{𝑆}⟶𝐵} ↔ 𝑔:{𝑆}⟶𝐵)
27 eliun 4999 . . 3 (𝑔 𝑏𝐵 {{⟨𝑆, 𝑏⟩}} ↔ ∃𝑏𝐵 𝑔 ∈ {{⟨𝑆, 𝑏⟩}})
2823, 26, 273bitr4g 314 . 2 (𝑆𝑉 → (𝑔 ∈ {𝑓𝑓:{𝑆}⟶𝐵} ↔ 𝑔 𝑏𝐵 {{⟨𝑆, 𝑏⟩}}))
2928eqrdv 2732 1 (𝑆𝑉 → {𝑓𝑓:{𝑆}⟶𝐵} = 𝑏𝐵 {{⟨𝑆, 𝑏⟩}})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1536  wcel 2105  {cab 2711  wrex 3067  {csn 4630  cop 4636   ciun 4995  wf 6558  cfv 6562
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570
This theorem is referenced by:  fsetabsnop  46999
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