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Theorem fsetsnf 47068
Description: The mapping of an element of a class to a singleton function is a function. (Contributed by AV, 13-Sep-2024.)
Hypotheses
Ref Expression
fsetsnf.a 𝐴 = {𝑦 ∣ ∃𝑏𝐵 𝑦 = {⟨𝑆, 𝑏⟩}}
fsetsnf.f 𝐹 = (𝑥𝐵 ↦ {⟨𝑆, 𝑥⟩})
Assertion
Ref Expression
fsetsnf (𝑆𝑉𝐹:𝐵𝐴)
Distinct variable groups:   𝑥,𝐴   𝐵,𝑏,𝑥,𝑦   𝑆,𝑏,𝑥,𝑦   𝑉,𝑏,𝑥
Allowed substitution hints:   𝐴(𝑦,𝑏)   𝐹(𝑥,𝑦,𝑏)   𝑉(𝑦)

Proof of Theorem fsetsnf
StepHypRef Expression
1 simpr 484 . . . 4 ((𝑆𝑉𝑥𝐵) → 𝑥𝐵)
2 opeq2 4873 . . . . . . 7 (𝑏 = 𝑥 → ⟨𝑆, 𝑏⟩ = ⟨𝑆, 𝑥⟩)
32sneqd 4637 . . . . . 6 (𝑏 = 𝑥 → {⟨𝑆, 𝑏⟩} = {⟨𝑆, 𝑥⟩})
43eqeq2d 2747 . . . . 5 (𝑏 = 𝑥 → ({⟨𝑆, 𝑥⟩} = {⟨𝑆, 𝑏⟩} ↔ {⟨𝑆, 𝑥⟩} = {⟨𝑆, 𝑥⟩}))
54adantl 481 . . . 4 (((𝑆𝑉𝑥𝐵) ∧ 𝑏 = 𝑥) → ({⟨𝑆, 𝑥⟩} = {⟨𝑆, 𝑏⟩} ↔ {⟨𝑆, 𝑥⟩} = {⟨𝑆, 𝑥⟩}))
6 eqidd 2737 . . . 4 ((𝑆𝑉𝑥𝐵) → {⟨𝑆, 𝑥⟩} = {⟨𝑆, 𝑥⟩})
71, 5, 6rspcedvd 3623 . . 3 ((𝑆𝑉𝑥𝐵) → ∃𝑏𝐵 {⟨𝑆, 𝑥⟩} = {⟨𝑆, 𝑏⟩})
8 snex 5435 . . . 4 {⟨𝑆, 𝑥⟩} ∈ V
9 eqeq1 2740 . . . . 5 (𝑦 = {⟨𝑆, 𝑥⟩} → (𝑦 = {⟨𝑆, 𝑏⟩} ↔ {⟨𝑆, 𝑥⟩} = {⟨𝑆, 𝑏⟩}))
109rexbidv 3178 . . . 4 (𝑦 = {⟨𝑆, 𝑥⟩} → (∃𝑏𝐵 𝑦 = {⟨𝑆, 𝑏⟩} ↔ ∃𝑏𝐵 {⟨𝑆, 𝑥⟩} = {⟨𝑆, 𝑏⟩}))
11 fsetsnf.a . . . 4 𝐴 = {𝑦 ∣ ∃𝑏𝐵 𝑦 = {⟨𝑆, 𝑏⟩}}
128, 10, 11elab2 3681 . . 3 ({⟨𝑆, 𝑥⟩} ∈ 𝐴 ↔ ∃𝑏𝐵 {⟨𝑆, 𝑥⟩} = {⟨𝑆, 𝑏⟩})
137, 12sylibr 234 . 2 ((𝑆𝑉𝑥𝐵) → {⟨𝑆, 𝑥⟩} ∈ 𝐴)
14 fsetsnf.f . 2 𝐹 = (𝑥𝐵 ↦ {⟨𝑆, 𝑥⟩})
1513, 14fmptd 7133 1 (𝑆𝑉𝐹:𝐵𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1539  wcel 2107  {cab 2713  wrex 3069  {csn 4625  cop 4631  cmpt 5224  wf 6556
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-fun 6562  df-fn 6563  df-f 6564
This theorem is referenced by:  fsetsnf1  47069  fsetsnfo  47070
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