Theorem fsetsnf 47052

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 484	. . . 4 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵)
2		opeq2 4838	. . . . . . 7 ⊢ (𝑏 = 𝑥 → ⟨𝑆, 𝑏⟩ = ⟨𝑆, 𝑥⟩)
3	2	sneqd 4601	. . . . . 6 ⊢ (𝑏 = 𝑥 → {⟨𝑆, 𝑏⟩} = {⟨𝑆, 𝑥⟩})
4	3	eqeq2d 2740	. . . . 5 ⊢ (𝑏 = 𝑥 → ({⟨𝑆, 𝑥⟩} = {⟨𝑆, 𝑏⟩} ↔ {⟨𝑆, 𝑥⟩} = {⟨𝑆, 𝑥⟩}))
5	4	adantl 481	. . . 4 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵) ∧ 𝑏 = 𝑥) → ({⟨𝑆, 𝑥⟩} = {⟨𝑆, 𝑏⟩} ↔ {⟨𝑆, 𝑥⟩} = {⟨𝑆, 𝑥⟩}))
6		eqidd 2730	. . . 4 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵) → {⟨𝑆, 𝑥⟩} = {⟨𝑆, 𝑥⟩})
7	1, 5, 6	rspcedvd 3590	. . 3 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵) → ∃𝑏 ∈ 𝐵 {⟨𝑆, 𝑥⟩} = {⟨𝑆, 𝑏⟩})
8		snex 5391	. . . 4 ⊢ {⟨𝑆, 𝑥⟩} ∈ V
9		eqeq1 2733	. . . . 5 ⊢ (𝑦 = {⟨𝑆, 𝑥⟩} → (𝑦 = {⟨𝑆, 𝑏⟩} ↔ {⟨𝑆, 𝑥⟩} = {⟨𝑆, 𝑏⟩}))
10	9	rexbidv 3157	. . . 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 3649	. . 3 ⊢ ({⟨𝑆, 𝑥⟩} ∈ 𝐴 ↔ ∃𝑏 ∈ 𝐵 {⟨𝑆, 𝑥⟩} = {⟨𝑆, 𝑏⟩})
13	7, 12	sylibr 234	. 2 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵) → {⟨𝑆, 𝑥⟩} ∈ 𝐴)
14		fsetsnf.f	. 2 ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ {⟨𝑆, 𝑥⟩})
15	13, 14	fmptd 7086	1 ⊢ (𝑆 ∈ 𝑉 → 𝐹:𝐵⟶𝐴)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  {cab 2707  ∃wrex 3053  {csn 4589  ⟨cop 4595   ↦ cmpt 5188  ⟶wf 6507

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-fun 6513  df-fn 6514  df-f 6515

This theorem is referenced by:  fsetsnf1  47053  fsetsnfo  47054