Theorem fsetsnf 47926

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 ∣ ∃𝑏 ∈ 𝐵 𝑦 = {⟨𝑆, 𝑏⟩}}
fsetsnf.f	⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ {⟨𝑆, 𝑥⟩})

Assertion

Ref	Expression
fsetsnf	⊢ (𝑆 ∈ 𝑉 → 𝐹:𝐵⟶𝐴)

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

Step	Hyp	Ref	Expression
1		simpr 490	. . . 4 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵)
2		opeq2 4837	. . . . . . 7 ⊢ (𝑏 = 𝑥 → ⟨𝑆, 𝑏⟩ = ⟨𝑆, 𝑥⟩)
3	2	sneqd 4599	. . . . . 6 ⊢ (𝑏 = 𝑥 → {⟨𝑆, 𝑏⟩} = {⟨𝑆, 𝑥⟩})
4	3	eqeq2d 2773	. . . . 5 ⊢ (𝑏 = 𝑥 → ({⟨𝑆, 𝑥⟩} = {⟨𝑆, 𝑏⟩} ↔ {⟨𝑆, 𝑥⟩} = {⟨𝑆, 𝑥⟩}))
5	4	adantl 487	. . . 4 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵) ∧ 𝑏 = 𝑥) → ({⟨𝑆, 𝑥⟩} = {⟨𝑆, 𝑏⟩} ↔ {⟨𝑆, 𝑥⟩} = {⟨𝑆, 𝑥⟩}))
6		eqidd 2763	. . . 4 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵) → {⟨𝑆, 𝑥⟩} = {⟨𝑆, 𝑥⟩})
7	1, 5, 6	rspcedvd 3581	. . 3 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵) → ∃𝑏 ∈ 𝐵 {⟨𝑆, 𝑥⟩} = {⟨𝑆, 𝑏⟩})
8		snex 5408	. . . 4 ⊢ {⟨𝑆, 𝑥⟩} ∈ V
9		eqeq1 2766	. . . . 5 ⊢ (𝑦 = {⟨𝑆, 𝑥⟩} → (𝑦 = {⟨𝑆, 𝑏⟩} ↔ {⟨𝑆, 𝑥⟩} = {⟨𝑆, 𝑏⟩}))
10	9	rexbidv 3188	. . . 4 ⊢ (𝑦 = {⟨𝑆, 𝑥⟩} → (∃𝑏 ∈ 𝐵 𝑦 = {⟨𝑆, 𝑏⟩} ↔ ∃𝑏 ∈ 𝐵 {⟨𝑆, 𝑥⟩} = {⟨𝑆, 𝑏⟩}))
11		fsetsnf.a	. . . 4 ⊢ 𝐴 = {𝑦 ∣ ∃𝑏 ∈ 𝐵 𝑦 = {⟨𝑆, 𝑏⟩}}
12	8, 10, 11	elab2 3639	. . 3 ⊢ ({⟨𝑆, 𝑥⟩} ∈ 𝐴 ↔ ∃𝑏 ∈ 𝐵 {⟨𝑆, 𝑥⟩} = {⟨𝑆, 𝑏⟩})
13	7, 12	sylibr 237	. 2 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵) → {⟨𝑆, 𝑥⟩} ∈ 𝐴)
14		fsetsnf.f	. 2 ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ {⟨𝑆, 𝑥⟩})
15	13, 14	fmptd 7110	1 ⊢ (𝑆 ∈ 𝑉 → 𝐹:𝐵⟶𝐴)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  {cab 2740  ∃wrex 3088  {csn 4587  ⟨cop 4593   ↦ cmpt 5190  ⟶wf 6533

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-fun 6539  df-fn 6540  df-f 6541

This theorem is used by:  fsetsnf1  47927  fsetsnfo  47928