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Theorem fsetsnf 46315
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 4869 . . . . . . 7 (𝑏 = 𝑥 → ⟨𝑆, 𝑏⟩ = ⟨𝑆, 𝑥⟩)
32sneqd 4635 . . . . . 6 (𝑏 = 𝑥 → {⟨𝑆, 𝑏⟩} = {⟨𝑆, 𝑥⟩})
43eqeq2d 2737 . . . . 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 2727 . . . 4 ((𝑆𝑉𝑥𝐵) → {⟨𝑆, 𝑥⟩} = {⟨𝑆, 𝑥⟩})
71, 5, 6rspcedvd 3608 . . 3 ((𝑆𝑉𝑥𝐵) → ∃𝑏𝐵 {⟨𝑆, 𝑥⟩} = {⟨𝑆, 𝑏⟩})
8 snex 5424 . . . 4 {⟨𝑆, 𝑥⟩} ∈ V
9 eqeq1 2730 . . . . 5 (𝑦 = {⟨𝑆, 𝑥⟩} → (𝑦 = {⟨𝑆, 𝑏⟩} ↔ {⟨𝑆, 𝑥⟩} = {⟨𝑆, 𝑏⟩}))
109rexbidv 3172 . . . 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 3667 . . 3 ({⟨𝑆, 𝑥⟩} ∈ 𝐴 ↔ ∃𝑏𝐵 {⟨𝑆, 𝑥⟩} = {⟨𝑆, 𝑏⟩})
137, 12sylibr 233 . 2 ((𝑆𝑉𝑥𝐵) → {⟨𝑆, 𝑥⟩} ∈ 𝐴)
14 fsetsnf.f . 2 𝐹 = (𝑥𝐵 ↦ {⟨𝑆, 𝑥⟩})
1513, 14fmptd 7108 1 (𝑆𝑉𝐹:𝐵𝐴)
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  cmpt 5224  wf 6532
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-11 2146  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-nfc 2879  df-ral 3056  df-rex 3065  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-br 5142  df-opab 5204  df-mpt 5225  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-fun 6538  df-fn 6539  df-f 6540
This theorem is referenced by:  fsetsnf1  46316  fsetsnfo  46317
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