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Theorem fsetsniunop 46336
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 7132 . . . . . 6 (𝑆𝑉 → (𝑔:{𝑆}⟶𝐵 ↔ ((𝑔𝑆) ∈ 𝐵𝑔 = {⟨𝑆, (𝑔𝑆)⟩})))
2 simpl 482 . . . . . . 7 (((𝑔𝑆) ∈ 𝐵𝑔 = {⟨𝑆, (𝑔𝑆)⟩}) → (𝑔𝑆) ∈ 𝐵)
3 opeq2 4869 . . . . . . . . . 10 (𝑏 = (𝑔𝑆) → ⟨𝑆, 𝑏⟩ = ⟨𝑆, (𝑔𝑆)⟩)
43sneqd 4635 . . . . . . . . 9 (𝑏 = (𝑔𝑆) → {⟨𝑆, 𝑏⟩} = {⟨𝑆, (𝑔𝑆)⟩})
54eqeq2d 2737 . . . . . . . 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 3608 . . . . . 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 6870 . . . . . . . . 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 6696 . . . . . . . 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 3149 . . . . 5 (𝑆𝑉 → (∃𝑏𝐵 𝑔 = {⟨𝑆, 𝑏⟩} → 𝑔:{𝑆}⟶𝐵))
199, 18impbid 211 . . . 4 (𝑆𝑉 → (𝑔:{𝑆}⟶𝐵 ↔ ∃𝑏𝐵 𝑔 = {⟨𝑆, 𝑏⟩}))
20 velsn 4639 . . . . . 6 (𝑔 ∈ {{⟨𝑆, 𝑏⟩}} ↔ 𝑔 = {⟨𝑆, 𝑏⟩})
2120bicomi 223 . . . . 5 (𝑔 = {⟨𝑆, 𝑏⟩} ↔ 𝑔 ∈ {{⟨𝑆, 𝑏⟩}})
2221rexbii 3088 . . . 4 (∃𝑏𝐵 𝑔 = {⟨𝑆, 𝑏⟩} ↔ ∃𝑏𝐵 𝑔 ∈ {{⟨𝑆, 𝑏⟩}})
2319, 22bitrdi 287 . . 3 (𝑆𝑉 → (𝑔:{𝑆}⟶𝐵 ↔ ∃𝑏𝐵 𝑔 ∈ {{⟨𝑆, 𝑏⟩}}))
24 vex 3472 . . . 4 𝑔 ∈ V
25 feq1 6692 . . . 4 (𝑓 = 𝑔 → (𝑓:{𝑆}⟶𝐵𝑔:{𝑆}⟶𝐵))
2624, 25elab 3663 . . 3 (𝑔 ∈ {𝑓𝑓:{𝑆}⟶𝐵} ↔ 𝑔:{𝑆}⟶𝐵)
27 eliun 4994 . . 3 (𝑔 𝑏𝐵 {{⟨𝑆, 𝑏⟩}} ↔ ∃𝑏𝐵 𝑔 ∈ {{⟨𝑆, 𝑏⟩}})
2823, 26, 273bitr4g 314 . 2 (𝑆𝑉 → (𝑔 ∈ {𝑓𝑓:{𝑆}⟶𝐵} ↔ 𝑔 𝑏𝐵 {{⟨𝑆, 𝑏⟩}}))
2928eqrdv 2724 1 (𝑆𝑉 → {𝑓𝑓:{𝑆}⟶𝐵} = 𝑏𝐵 {{⟨𝑆, 𝑏⟩}})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1533  wcel 2098  {cab 2703  wrex 3064  {csn 4623  cop 4629   ciun 4990  wf 6533  cfv 6537
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545
This theorem is referenced by:  fsetabsnop  46337
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