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Theorem fsetabsnop 44431
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 44430 . 2 (𝑆𝑉 → {𝑓𝑓:{𝑆}⟶𝐵} = 𝑏𝐵 {{⟨𝑆, 𝑏⟩}})
2 iunsn 4991 . 2 𝑏𝐵 {{⟨𝑆, 𝑏⟩}} = {𝑦 ∣ ∃𝑏𝐵 𝑦 = {⟨𝑆, 𝑏⟩}}
31, 2eqtrdi 2795 1 (𝑆𝑉 → {𝑓𝑓:{𝑆}⟶𝐵} = {𝑦 ∣ ∃𝑏𝐵 𝑦 = {⟨𝑆, 𝑏⟩}})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  wcel 2108  {cab 2715  wrex 3064  {csn 4558  cop 4564   ciun 4921  wf 6414
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426
This theorem is referenced by:  fsetsnprcnex  44436
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