Theorem fsetsniunop 47924

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 7135	. . . . . 6 ⊢ (𝑆 ∈ 𝑉 → (𝑔:{𝑆}⟶𝐵 ↔ ((𝑔‘𝑆) ∈ 𝐵 ∧ 𝑔 = {⟨𝑆, (𝑔‘𝑆)⟩})))
2		simpl 488	. . . . . . 7 ⊢ (((𝑔‘𝑆) ∈ 𝐵 ∧ 𝑔 = {⟨𝑆, (𝑔‘𝑆)⟩}) → (𝑔‘𝑆) ∈ 𝐵)
3		opeq2 4837	. . . . . . . . . 10 ⊢ (𝑏 = (𝑔‘𝑆) → ⟨𝑆, 𝑏⟩ = ⟨𝑆, (𝑔‘𝑆)⟩)
4	3	sneqd 4599	. . . . . . . . 9 ⊢ (𝑏 = (𝑔‘𝑆) → {⟨𝑆, 𝑏⟩} = {⟨𝑆, (𝑔‘𝑆)⟩})
5	4	eqeq2d 2773	. . . . . . . 8 ⊢ (𝑏 = (𝑔‘𝑆) → (𝑔 = {⟨𝑆, 𝑏⟩} ↔ 𝑔 = {⟨𝑆, (𝑔‘𝑆)⟩}))
6	5	adantl 487	. . . . . . 7 ⊢ ((((𝑔‘𝑆) ∈ 𝐵 ∧ 𝑔 = {⟨𝑆, (𝑔‘𝑆)⟩}) ∧ 𝑏 = (𝑔‘𝑆)) → (𝑔 = {⟨𝑆, 𝑏⟩} ↔ 𝑔 = {⟨𝑆, (𝑔‘𝑆)⟩}))
7		simpr 490	. . . . . . 7 ⊢ (((𝑔‘𝑆) ∈ 𝐵 ∧ 𝑔 = {⟨𝑆, (𝑔‘𝑆)⟩}) → 𝑔 = {⟨𝑆, (𝑔‘𝑆)⟩})
8	2, 6, 7	rspcedvd 3581	. . . . . 6 ⊢ (((𝑔‘𝑆) ∈ 𝐵 ∧ 𝑔 = {⟨𝑆, (𝑔‘𝑆)⟩}) → ∃𝑏 ∈ 𝐵 𝑔 = {⟨𝑆, 𝑏⟩})
9	1, 8	biimtrdi 256	. . . . 5 ⊢ (𝑆 ∈ 𝑉 → (𝑔:{𝑆}⟶𝐵 → ∃𝑏 ∈ 𝐵 𝑔 = {⟨𝑆, 𝑏⟩}))
10		simpl 488	. . . . . . . . . 10 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑏 ∈ 𝐵) → 𝑆 ∈ 𝑉)
11		simpr 490	. . . . . . . . . 10 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑏 ∈ 𝐵) → 𝑏 ∈ 𝐵)
12	10, 11	fsnd 6866	. . . . . . . . 9 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑏 ∈ 𝐵) → {⟨𝑆, 𝑏⟩}:{𝑆}⟶𝐵)
13	12	adantr 486	. . . . . . . 8 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑏 ∈ 𝐵) ∧ 𝑔 = {⟨𝑆, 𝑏⟩}) → {⟨𝑆, 𝑏⟩}:{𝑆}⟶𝐵)
14		simpr 490	. . . . . . . . 9 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑏 ∈ 𝐵) ∧ 𝑔 = {⟨𝑆, 𝑏⟩}) → 𝑔 = {⟨𝑆, 𝑏⟩})
15	14	feq1d 6688	. . . . . . . 8 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑏 ∈ 𝐵) ∧ 𝑔 = {⟨𝑆, 𝑏⟩}) → (𝑔:{𝑆}⟶𝐵 ↔ {⟨𝑆, 𝑏⟩}:{𝑆}⟶𝐵))
16	13, 15	mpbird 260	. . . . . . 7 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑏 ∈ 𝐵) ∧ 𝑔 = {⟨𝑆, 𝑏⟩}) → 𝑔:{𝑆}⟶𝐵)
17	16	ex 418	. . . . . 6 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑏 ∈ 𝐵) → (𝑔 = {⟨𝑆, 𝑏⟩} → 𝑔:{𝑆}⟶𝐵))
18	17	rexlimdva 3165	. . . . 5 ⊢ (𝑆 ∈ 𝑉 → (∃𝑏 ∈ 𝐵 𝑔 = {⟨𝑆, 𝑏⟩} → 𝑔:{𝑆}⟶𝐵))
19	9, 18	impbid 215	. . . 4 ⊢ (𝑆 ∈ 𝑉 → (𝑔:{𝑆}⟶𝐵 ↔ ∃𝑏 ∈ 𝐵 𝑔 = {⟨𝑆, 𝑏⟩}))
20		velsn 4603	. . . . . 6 ⊢ (𝑔 ∈ {{⟨𝑆, 𝑏⟩}} ↔ 𝑔 = {⟨𝑆, 𝑏⟩})
21	20	bicomi 227	. . . . 5 ⊢ (𝑔 = {⟨𝑆, 𝑏⟩} ↔ 𝑔 ∈ {{⟨𝑆, 𝑏⟩}})
22	21	rexbii 3111	. . . 4 ⊢ (∃𝑏 ∈ 𝐵 𝑔 = {⟨𝑆, 𝑏⟩} ↔ ∃𝑏 ∈ 𝐵 𝑔 ∈ {{⟨𝑆, 𝑏⟩}})
23	19, 22	bitrdi 290	. . 3 ⊢ (𝑆 ∈ 𝑉 → (𝑔:{𝑆}⟶𝐵 ↔ ∃𝑏 ∈ 𝐵 𝑔 ∈ {{⟨𝑆, 𝑏⟩}}))
24		vex 3457	. . . 4 ⊢ 𝑔 ∈ V
25		feq1 6684	. . . 4 ⊢ (𝑓 = 𝑔 → (𝑓:{𝑆}⟶𝐵 ↔ 𝑔:{𝑆}⟶𝐵))
26	24, 25	elab 3636	. . 3 ⊢ (𝑔 ∈ {𝑓 ∣ 𝑓:{𝑆}⟶𝐵} ↔ 𝑔:{𝑆}⟶𝐵)
27		eliun 4958	. . 3 ⊢ (𝑔 ∈ ∪ 𝑏 ∈ 𝐵 {{⟨𝑆, 𝑏⟩}} ↔ ∃𝑏 ∈ 𝐵 𝑔 ∈ {{⟨𝑆, 𝑏⟩}})
28	23, 26, 27	3bitr4g 317	. 2 ⊢ (𝑆 ∈ 𝑉 → (𝑔 ∈ {𝑓 ∣ 𝑓:{𝑆}⟶𝐵} ↔ 𝑔 ∈ ∪ 𝑏 ∈ 𝐵 {{⟨𝑆, 𝑏⟩}}))
29	28	eqrdv 2760	1 ⊢ (𝑆 ∈ 𝑉 → {𝑓 ∣ 𝑓:{𝑆}⟶𝐵} = ∪ 𝑏 ∈ 𝐵 {{⟨𝑆, 𝑏⟩}})

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  {cab 2740  ∃wrex 3088  {csn 4587  ⟨cop 4593  ∪ ciun 4954  ⟶wf 6533  ‘cfv 6537

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-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  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-ne 2958  df-ral 3079  df-rex 3089  df-reu 3368  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-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  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-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545

This theorem is used by:  fsetabsnop  47925