Theorem fsetsnf 47001

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 4879	. . . . . . 7 ⊢ (𝑏 = 𝑥 → ⟨𝑆, 𝑏⟩ = ⟨𝑆, 𝑥⟩)
3	2	sneqd 4643	. . . . . 6 ⊢ (𝑏 = 𝑥 → {⟨𝑆, 𝑏⟩} = {⟨𝑆, 𝑥⟩})
4	3	eqeq2d 2746	. . . . 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 2736	. . . 4 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵) → {⟨𝑆, 𝑥⟩} = {⟨𝑆, 𝑥⟩})
7	1, 5, 6	rspcedvd 3624	. . 3 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵) → ∃𝑏 ∈ 𝐵 {⟨𝑆, 𝑥⟩} = {⟨𝑆, 𝑏⟩})
8		snex 5442	. . . 4 ⊢ {⟨𝑆, 𝑥⟩} ∈ V
9		eqeq1 2739	. . . . 5 ⊢ (𝑦 = {⟨𝑆, 𝑥⟩} → (𝑦 = {⟨𝑆, 𝑏⟩} ↔ {⟨𝑆, 𝑥⟩} = {⟨𝑆, 𝑏⟩}))
10	9	rexbidv 3177	. . . 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 3685	. . 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 7134	1 ⊢ (𝑆 ∈ 𝑉 → 𝐹:𝐵⟶𝐴)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2106  {cab 2712  ∃wrex 3068  {csn 4631  ⟨cop 4637   ↦ cmpt 5231  ⟶wf 6559

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-fun 6565  df-fn 6566  df-f 6567

This theorem is referenced by:  fsetsnf1  47002  fsetsnfo  47003