Theorem fsetsniunop 44430

Description: The class of all functions from a (proper) singleton into 𝐵 is the union of all the singletons of (proper) ordered pairs over the elements of 𝐵 as second component. (Contributed by AV, 13-Sep-2024.)

Assertion

Ref	Expression
fsetsniunop	⊢ (𝑆 ∈ 𝑉 → {𝑓 ∣ 𝑓:{𝑆}⟶𝐵} = ∪ 𝑏 ∈ 𝐵 {{⟨𝑆, 𝑏⟩}})

Distinct variable groups:   𝐵,𝑏,𝑓   𝑆,𝑏,𝑓   𝑉,𝑏

Allowed substitution hint:   𝑉(𝑓)

Proof of Theorem fsetsniunop

Dummy variable 𝑔 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fsn2g 6992	. . . . . 6 ⊢ (𝑆 ∈ 𝑉 → (𝑔:{𝑆}⟶𝐵 ↔ ((𝑔‘𝑆) ∈ 𝐵 ∧ 𝑔 = {⟨𝑆, (𝑔‘𝑆)⟩})))
2		simpl 482	. . . . . . 7 ⊢ (((𝑔‘𝑆) ∈ 𝐵 ∧ 𝑔 = {⟨𝑆, (𝑔‘𝑆)⟩}) → (𝑔‘𝑆) ∈ 𝐵)
3		opeq2 4802	. . . . . . . . . 10 ⊢ (𝑏 = (𝑔‘𝑆) → ⟨𝑆, 𝑏⟩ = ⟨𝑆, (𝑔‘𝑆)⟩)
4	3	sneqd 4570	. . . . . . . . 9 ⊢ (𝑏 = (𝑔‘𝑆) → {⟨𝑆, 𝑏⟩} = {⟨𝑆, (𝑔‘𝑆)⟩})
5	4	eqeq2d 2749	. . . . . . . 8 ⊢ (𝑏 = (𝑔‘𝑆) → (𝑔 = {⟨𝑆, 𝑏⟩} ↔ 𝑔 = {⟨𝑆, (𝑔‘𝑆)⟩}))
6	5	adantl 481	. . . . . . 7 ⊢ ((((𝑔‘𝑆) ∈ 𝐵 ∧ 𝑔 = {⟨𝑆, (𝑔‘𝑆)⟩}) ∧ 𝑏 = (𝑔‘𝑆)) → (𝑔 = {⟨𝑆, 𝑏⟩} ↔ 𝑔 = {⟨𝑆, (𝑔‘𝑆)⟩}))
7		simpr 484	. . . . . . 7 ⊢ (((𝑔‘𝑆) ∈ 𝐵 ∧ 𝑔 = {⟨𝑆, (𝑔‘𝑆)⟩}) → 𝑔 = {⟨𝑆, (𝑔‘𝑆)⟩})
8	2, 6, 7	rspcedvd 3555	. . . . . 6 ⊢ (((𝑔‘𝑆) ∈ 𝐵 ∧ 𝑔 = {⟨𝑆, (𝑔‘𝑆)⟩}) → ∃𝑏 ∈ 𝐵 𝑔 = {⟨𝑆, 𝑏⟩})
9	1, 8	syl6bi 252	. . . . 5 ⊢ (𝑆 ∈ 𝑉 → (𝑔:{𝑆}⟶𝐵 → ∃𝑏 ∈ 𝐵 𝑔 = {⟨𝑆, 𝑏⟩}))
10		simpl 482	. . . . . . . . . 10 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑏 ∈ 𝐵) → 𝑆 ∈ 𝑉)
11		simpr 484	. . . . . . . . . 10 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑏 ∈ 𝐵) → 𝑏 ∈ 𝐵)
12	10, 11	fsnd 6742	. . . . . . . . 9 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑏 ∈ 𝐵) → {⟨𝑆, 𝑏⟩}:{𝑆}⟶𝐵)
13	12	adantr 480	. . . . . . . 8 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑏 ∈ 𝐵) ∧ 𝑔 = {⟨𝑆, 𝑏⟩}) → {⟨𝑆, 𝑏⟩}:{𝑆}⟶𝐵)
14		simpr 484	. . . . . . . . 9 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑏 ∈ 𝐵) ∧ 𝑔 = {⟨𝑆, 𝑏⟩}) → 𝑔 = {⟨𝑆, 𝑏⟩})
15	14	feq1d 6569	. . . . . . . 8 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑏 ∈ 𝐵) ∧ 𝑔 = {⟨𝑆, 𝑏⟩}) → (𝑔:{𝑆}⟶𝐵 ↔ {⟨𝑆, 𝑏⟩}:{𝑆}⟶𝐵))
16	13, 15	mpbird 256	. . . . . . 7 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑏 ∈ 𝐵) ∧ 𝑔 = {⟨𝑆, 𝑏⟩}) → 𝑔:{𝑆}⟶𝐵)
17	16	ex 412	. . . . . 6 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑏 ∈ 𝐵) → (𝑔 = {⟨𝑆, 𝑏⟩} → 𝑔:{𝑆}⟶𝐵))
18	17	rexlimdva 3212	. . . . 5 ⊢ (𝑆 ∈ 𝑉 → (∃𝑏 ∈ 𝐵 𝑔 = {⟨𝑆, 𝑏⟩} → 𝑔:{𝑆}⟶𝐵))
19	9, 18	impbid 211	. . . 4 ⊢ (𝑆 ∈ 𝑉 → (𝑔:{𝑆}⟶𝐵 ↔ ∃𝑏 ∈ 𝐵 𝑔 = {⟨𝑆, 𝑏⟩}))
20		velsn 4574	. . . . . 6 ⊢ (𝑔 ∈ {{⟨𝑆, 𝑏⟩}} ↔ 𝑔 = {⟨𝑆, 𝑏⟩})
21	20	bicomi 223	. . . . 5 ⊢ (𝑔 = {⟨𝑆, 𝑏⟩} ↔ 𝑔 ∈ {{⟨𝑆, 𝑏⟩}})
22	21	rexbii 3177	. . . 4 ⊢ (∃𝑏 ∈ 𝐵 𝑔 = {⟨𝑆, 𝑏⟩} ↔ ∃𝑏 ∈ 𝐵 𝑔 ∈ {{⟨𝑆, 𝑏⟩}})
23	19, 22	bitrdi 286	. . 3 ⊢ (𝑆 ∈ 𝑉 → (𝑔:{𝑆}⟶𝐵 ↔ ∃𝑏 ∈ 𝐵 𝑔 ∈ {{⟨𝑆, 𝑏⟩}}))
24		vex 3426	. . . 4 ⊢ 𝑔 ∈ V
25		feq1 6565	. . . 4 ⊢ (𝑓 = 𝑔 → (𝑓:{𝑆}⟶𝐵 ↔ 𝑔:{𝑆}⟶𝐵))
26	24, 25	elab 3602	. . 3 ⊢ (𝑔 ∈ {𝑓 ∣ 𝑓:{𝑆}⟶𝐵} ↔ 𝑔:{𝑆}⟶𝐵)
27		eliun 4925	. . 3 ⊢ (𝑔 ∈ ∪ 𝑏 ∈ 𝐵 {{⟨𝑆, 𝑏⟩}} ↔ ∃𝑏 ∈ 𝐵 𝑔 ∈ {{⟨𝑆, 𝑏⟩}})
28	23, 26, 27	3bitr4g 313	. 2 ⊢ (𝑆 ∈ 𝑉 → (𝑔 ∈ {𝑓 ∣ 𝑓:{𝑆}⟶𝐵} ↔ 𝑔 ∈ ∪ 𝑏 ∈ 𝐵 {{⟨𝑆, 𝑏⟩}}))
29	28	eqrdv 2736	1 ⊢ (𝑆 ∈ 𝑉 → {𝑓 ∣ 𝑓:{𝑆}⟶𝐵} = ∪ 𝑏 ∈ 𝐵 {{⟨𝑆, 𝑏⟩}})

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108  {cab 2715  ∃wrex 3064  {csn 4558  ⟨cop 4564  ∪ ciun 4921  ⟶wf 6414  ‘cfv 6418

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426

This theorem is referenced by:  fsetabsnop  44431