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Theorem fsetabsnop 44249
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 44248 . 2 (𝑆𝑉 → {𝑓𝑓:{𝑆}⟶𝐵} = 𝑏𝐵 {{⟨𝑆, 𝑏⟩}})
2 iunsn 4990 . 2 𝑏𝐵 {{⟨𝑆, 𝑏⟩}} = {𝑦 ∣ ∃𝑏𝐵 𝑦 = {⟨𝑆, 𝑏⟩}}
31, 2eqtrdi 2796 1 (𝑆𝑉 → {𝑓𝑓:{𝑆}⟶𝐵} = {𝑦 ∣ ∃𝑏𝐵 𝑦 = {⟨𝑆, 𝑏⟩}})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1543  wcel 2112  {cab 2716  wrex 3064  {csn 4557  cop 4563   ciun 4920  wf 6396
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 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2710  ax-sep 5208  ax-nul 5215  ax-pr 5338
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 2073  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2818  df-nfc 2888  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-dif 3885  df-un 3887  df-in 3889  df-ss 3899  df-nul 4254  df-if 4456  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4836  df-iun 4922  df-br 5070  df-opab 5132  df-id 5471  df-xp 5574  df-rel 5575  df-cnv 5576  df-co 5577  df-dm 5578  df-rn 5579  df-res 5580  df-ima 5581  df-iota 6358  df-fun 6402  df-fn 6403  df-f 6404  df-f1 6405  df-fo 6406  df-f1o 6407  df-fv 6408
This theorem is referenced by:  fsetsnprcnex  44254
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