Theorem fsetsniunop 46354

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 7141	. . . . . 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 4870	. . . . . . . . . 10 ⊢ (𝑏 = (𝑔‘𝑆) → ⟨𝑆, 𝑏⟩ = ⟨𝑆, (𝑔‘𝑆)⟩)
4	3	sneqd 4636	. . . . . . . . 9 ⊢ (𝑏 = (𝑔‘𝑆) → {⟨𝑆, 𝑏⟩} = {⟨𝑆, (𝑔‘𝑆)⟩})
5	4	eqeq2d 2738	. . . . . . . 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 3609	. . . . . 6 ⊢ (((𝑔‘𝑆) ∈ 𝐵 ∧ 𝑔 = {⟨𝑆, (𝑔‘𝑆)⟩}) → ∃𝑏 ∈ 𝐵 𝑔 = {⟨𝑆, 𝑏⟩})
9	1, 8	syl6bi 253	. . . . 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 6876	. . . . . . . . 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 6701	. . . . . . . 8 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑏 ∈ 𝐵) ∧ 𝑔 = {⟨𝑆, 𝑏⟩}) → (𝑔:{𝑆}⟶𝐵 ↔ {⟨𝑆, 𝑏⟩}:{𝑆}⟶𝐵))
16	13, 15	mpbird 257	. . . . . . 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 3150	. . . . 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 4640	. . . . . 6 ⊢ (𝑔 ∈ {{⟨𝑆, 𝑏⟩}} ↔ 𝑔 = {⟨𝑆, 𝑏⟩})
21	20	bicomi 223	. . . . 5 ⊢ (𝑔 = {⟨𝑆, 𝑏⟩} ↔ 𝑔 ∈ {{⟨𝑆, 𝑏⟩}})
22	21	rexbii 3089	. . . 4 ⊢ (∃𝑏 ∈ 𝐵 𝑔 = {⟨𝑆, 𝑏⟩} ↔ ∃𝑏 ∈ 𝐵 𝑔 ∈ {{⟨𝑆, 𝑏⟩}})
23	19, 22	bitrdi 287	. . 3 ⊢ (𝑆 ∈ 𝑉 → (𝑔:{𝑆}⟶𝐵 ↔ ∃𝑏 ∈ 𝐵 𝑔 ∈ {{⟨𝑆, 𝑏⟩}}))
24		vex 3473	. . . 4 ⊢ 𝑔 ∈ V
25		feq1 6697	. . . 4 ⊢ (𝑓 = 𝑔 → (𝑓:{𝑆}⟶𝐵 ↔ 𝑔:{𝑆}⟶𝐵))
26	24, 25	elab 3665	. . 3 ⊢ (𝑔 ∈ {𝑓 ∣ 𝑓:{𝑆}⟶𝐵} ↔ 𝑔:{𝑆}⟶𝐵)
27		eliun 4995	. . 3 ⊢ (𝑔 ∈ ∪ 𝑏 ∈ 𝐵 {{⟨𝑆, 𝑏⟩}} ↔ ∃𝑏 ∈ 𝐵 𝑔 ∈ {{⟨𝑆, 𝑏⟩}})
28	23, 26, 27	3bitr4g 314	. 2 ⊢ (𝑆 ∈ 𝑉 → (𝑔 ∈ {𝑓 ∣ 𝑓:{𝑆}⟶𝐵} ↔ 𝑔 ∈ ∪ 𝑏 ∈ 𝐵 {{⟨𝑆, 𝑏⟩}}))
29	28	eqrdv 2725	1 ⊢ (𝑆 ∈ 𝑉 → {𝑓 ∣ 𝑓:{𝑆}⟶𝐵} = ∪ 𝑏 ∈ 𝐵 {{⟨𝑆, 𝑏⟩}})

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1534   ∈ wcel 2099  {cab 2704  ∃wrex 3065  {csn 4624  ⟨cop 4630  ∪ ciun 4991  ⟶wf 6538  ‘cfv 6542

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pr 5423

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-ne 2936  df-ral 3057  df-rex 3066  df-reu 3372  df-rab 3428  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550

This theorem is referenced by:  fsetabsnop  46355