Theorem fsetsniunop 47710

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 487	. . . . . . 7 ⊢ (((𝑔‘𝑆) ∈ 𝐵 ∧ 𝑔 = {⟨𝑆, (𝑔‘𝑆)⟩}) → (𝑔‘𝑆) ∈ 𝐵)
3		opeq2 4841	. . . . . . . . . 10 ⊢ (𝑏 = (𝑔‘𝑆) → ⟨𝑆, 𝑏⟩ = ⟨𝑆, (𝑔‘𝑆)⟩)
4	3	sneqd 4604	. . . . . . . . 9 ⊢ (𝑏 = (𝑔‘𝑆) → {⟨𝑆, 𝑏⟩} = {⟨𝑆, (𝑔‘𝑆)⟩})
5	4	eqeq2d 2780	. . . . . . . 8 ⊢ (𝑏 = (𝑔‘𝑆) → (𝑔 = {⟨𝑆, 𝑏⟩} ↔ 𝑔 = {⟨𝑆, (𝑔‘𝑆)⟩}))
6	5	adantl 486	. . . . . . 7 ⊢ ((((𝑔‘𝑆) ∈ 𝐵 ∧ 𝑔 = {⟨𝑆, (𝑔‘𝑆)⟩}) ∧ 𝑏 = (𝑔‘𝑆)) → (𝑔 = {⟨𝑆, 𝑏⟩} ↔ 𝑔 = {⟨𝑆, (𝑔‘𝑆)⟩}))
7		simpr 489	. . . . . . 7 ⊢ (((𝑔‘𝑆) ∈ 𝐵 ∧ 𝑔 = {⟨𝑆, (𝑔‘𝑆)⟩}) → 𝑔 = {⟨𝑆, (𝑔‘𝑆)⟩})
8	2, 6, 7	rspcedvd 3590	. . . . . 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 487	. . . . . . . . . 10 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑏 ∈ 𝐵) → 𝑆 ∈ 𝑉)
11		simpr 489	. . . . . . . . . 10 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑏 ∈ 𝐵) → 𝑏 ∈ 𝐵)
12	10, 11	fsnd 6866	. . . . . . . . 9 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑏 ∈ 𝐵) → {⟨𝑆, 𝑏⟩}:{𝑆}⟶𝐵)
13	12	adantr 485	. . . . . . . 8 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑏 ∈ 𝐵) ∧ 𝑔 = {⟨𝑆, 𝑏⟩}) → {⟨𝑆, 𝑏⟩}:{𝑆}⟶𝐵)
14		simpr 489	. . . . . . . . 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 417	. . . . . 6 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑏 ∈ 𝐵) → (𝑔 = {⟨𝑆, 𝑏⟩} → 𝑔:{𝑆}⟶𝐵))
18	17	rexlimdva 3172	. . . . 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 4608	. . . . . 6 ⊢ (𝑔 ∈ {{⟨𝑆, 𝑏⟩}} ↔ 𝑔 = {⟨𝑆, 𝑏⟩})
21	20	bicomi 227	. . . . 5 ⊢ (𝑔 = {⟨𝑆, 𝑏⟩} ↔ 𝑔 ∈ {{⟨𝑆, 𝑏⟩}})
22	21	rexbii 3118	. . . 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 3465	. . . 4 ⊢ 𝑔 ∈ V
25		feq1 6684	. . . 4 ⊢ (𝑓 = 𝑔 → (𝑓:{𝑆}⟶𝐵 ↔ 𝑔:{𝑆}⟶𝐵))
26	24, 25	elab 3645	. . 3 ⊢ (𝑔 ∈ {𝑓 ∣ 𝑓:{𝑆}⟶𝐵} ↔ 𝑔:{𝑆}⟶𝐵)
27		eliun 4962	. . 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 2767	1 ⊢ (𝑆 ∈ 𝑉 → {𝑓 ∣ 𝑓:{𝑆}⟶𝐵} = ∪ 𝑏 ∈ 𝐵 {{⟨𝑆, 𝑏⟩}})

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567   ∈ wcel 2149  {cab 2747  ∃wrex 3095  {csn 4592  ⟨cop 4598  ∪ ciun 4958  ⟶wf 6533  ‘cfv 6537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-12 2219  ax-ext 2741  ax-sep 5259  ax-nul 5271  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3376  df-rab 3423  df-v 3463  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4491  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  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 referenced by:  fsetabsnop  47711