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Theorem fsetabsnop 47062
Description: The class of all functions from a (proper) singleton into 𝐵 is the class 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
fsetabsnop (𝑆𝑉 → {𝑓𝑓:{𝑆}⟶𝐵} = {𝑦 ∣ ∃𝑏𝐵 𝑦 = {⟨𝑆, 𝑏⟩}})
Distinct variable groups:   𝐵,𝑏,𝑓   𝑆,𝑏,𝑓   𝑉,𝑏   𝑦,𝐵,𝑏   𝑦,𝑆
Allowed substitution hints:   𝑉(𝑦,𝑓)

Proof of Theorem fsetabsnop
StepHypRef Expression
1 fsetsniunop 47061 . 2 (𝑆𝑉 → {𝑓𝑓:{𝑆}⟶𝐵} = 𝑏𝐵 {{⟨𝑆, 𝑏⟩}})
2 iunsn 5066 . 2 𝑏𝐵 {{⟨𝑆, 𝑏⟩}} = {𝑦 ∣ ∃𝑏𝐵 𝑦 = {⟨𝑆, 𝑏⟩}}
31, 2eqtrdi 2793 1 (𝑆𝑉 → {𝑓𝑓:{𝑆}⟶𝐵} = {𝑦 ∣ ∃𝑏𝐵 𝑦 = {⟨𝑆, 𝑏⟩}})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  {cab 2714  wrex 3070  {csn 4626  cop 4632   ciun 4991  wf 6557
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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569
This theorem is referenced by:  fsetsnprcnex  47067
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