Theorem lubfval 18428

Description: Value of the least upper bound function of a poset. (Contributed by NM, 12-Sep-2011.) (Revised by NM, 6-Sep-2018.)

Hypotheses

Ref	Expression
lubfval.b	⊢ 𝐵 = (Base‘𝐾)
lubfval.l	⊢ ≤ = (le‘𝐾)
lubfval.u	⊢ 𝑈 = (lub‘𝐾)
lubfval.p	⊢ (𝜓 ↔ (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧)))
lubfval.k	⊢ (𝜑 → 𝐾 ∈ 𝑉)

Assertion

Ref	Expression
lubfval	⊢ (𝜑 → 𝑈 = ((𝑠 ∈ 𝒫 𝐵 ↦ (℩𝑥 ∈ 𝐵 𝜓)) ↾ {𝑠 ∣ ∃!𝑥 ∈ 𝐵 𝜓}))

Distinct variable groups:   𝑥,𝑠,𝑧,𝐵   𝑦,𝑠,𝐾,𝑥,𝑧

Allowed substitution hints:   𝜑(𝑥, 𝑦, 𝑧, 𝑠)   𝜓(𝑥, 𝑦, 𝑧, 𝑠)   𝐵(𝑦)   𝑈(𝑥, 𝑦, 𝑧, 𝑠)   ≤ (𝑥, 𝑦, 𝑧, 𝑠)   𝑉(𝑥, 𝑦, 𝑧, 𝑠)

Proof of Theorem lubfval

Dummy variable 𝑝 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		lubfval.k	. 2 ⊢ (𝜑 → 𝐾 ∈ 𝑉)
2		elex 3478	. 2 ⊢ (𝐾 ∈ 𝑉 → 𝐾 ∈ V)
3		fveq2 6885	. . . . . . . 8 ⊢ (𝑝 = 𝐾 → (Base‘𝑝) = (Base‘𝐾))
4		lubfval.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝐾)
5	3, 4	eqtr4di 2818	. . . . . . 7 ⊢ (𝑝 = 𝐾 → (Base‘𝑝) = 𝐵)
6	5	pweqd 4581	. . . . . 6 ⊢ (𝑝 = 𝐾 → 𝒫 (Base‘𝑝) = 𝒫 𝐵)
7		fveq2 6885	. . . . . . . . . . 11 ⊢ (𝑝 = 𝐾 → (le‘𝑝) = (le‘𝐾))
8		lubfval.l	. . . . . . . . . . 11 ⊢ ≤ = (le‘𝐾)
9	7, 8	eqtr4di 2818	. . . . . . . . . 10 ⊢ (𝑝 = 𝐾 → (le‘𝑝) = ≤ )
10	9	breqd 5122	. . . . . . . . 9 ⊢ (𝑝 = 𝐾 → (𝑦(le‘𝑝)𝑥 ↔ 𝑦 ≤ 𝑥))
11	10	ralbidv 3190	. . . . . . . 8 ⊢ (𝑝 = 𝐾 → (∀𝑦 ∈ 𝑠 𝑦(le‘𝑝)𝑥 ↔ ∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑥))
12	9	breqd 5122	. . . . . . . . . . 11 ⊢ (𝑝 = 𝐾 → (𝑦(le‘𝑝)𝑧 ↔ 𝑦 ≤ 𝑧))
13	12	ralbidv 3190	. . . . . . . . . 10 ⊢ (𝑝 = 𝐾 → (∀𝑦 ∈ 𝑠 𝑦(le‘𝑝)𝑧 ↔ ∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑧))
14	9	breqd 5122	. . . . . . . . . 10 ⊢ (𝑝 = 𝐾 → (𝑥(le‘𝑝)𝑧 ↔ 𝑥 ≤ 𝑧))
15	13, 14	imbi12d 347	. . . . . . . . 9 ⊢ (𝑝 = 𝐾 → ((∀𝑦 ∈ 𝑠 𝑦(le‘𝑝)𝑧 → 𝑥(le‘𝑝)𝑧) ↔ (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧)))
16	5, 15	raleqbidv 3340	. . . . . . . 8 ⊢ (𝑝 = 𝐾 → (∀𝑧 ∈ (Base‘𝑝)(∀𝑦 ∈ 𝑠 𝑦(le‘𝑝)𝑧 → 𝑥(le‘𝑝)𝑧) ↔ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧)))
17	11, 16	anbi12d 644	. . . . . . 7 ⊢ (𝑝 = 𝐾 → ((∀𝑦 ∈ 𝑠 𝑦(le‘𝑝)𝑥 ∧ ∀𝑧 ∈ (Base‘𝑝)(∀𝑦 ∈ 𝑠 𝑦(le‘𝑝)𝑧 → 𝑥(le‘𝑝)𝑧)) ↔ (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧))))
18	5, 17	riotaeqbidv 7379	. . . . . 6 ⊢ (𝑝 = 𝐾 → (℩𝑥 ∈ (Base‘𝑝)(∀𝑦 ∈ 𝑠 𝑦(le‘𝑝)𝑥 ∧ ∀𝑧 ∈ (Base‘𝑝)(∀𝑦 ∈ 𝑠 𝑦(le‘𝑝)𝑧 → 𝑥(le‘𝑝)𝑧))) = (℩𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧))))
19	6, 18	mpteq12dv 5200	. . . . 5 ⊢ (𝑝 = 𝐾 → (𝑠 ∈ 𝒫 (Base‘𝑝) ↦ (℩𝑥 ∈ (Base‘𝑝)(∀𝑦 ∈ 𝑠 𝑦(le‘𝑝)𝑥 ∧ ∀𝑧 ∈ (Base‘𝑝)(∀𝑦 ∈ 𝑠 𝑦(le‘𝑝)𝑧 → 𝑥(le‘𝑝)𝑧)))) = (𝑠 ∈ 𝒫 𝐵 ↦ (℩𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧)))))
20	17	reubidv 3387	. . . . . . 7 ⊢ (𝑝 = 𝐾 → (∃!𝑥 ∈ (Base‘𝑝)(∀𝑦 ∈ 𝑠 𝑦(le‘𝑝)𝑥 ∧ ∀𝑧 ∈ (Base‘𝑝)(∀𝑦 ∈ 𝑠 𝑦(le‘𝑝)𝑧 → 𝑥(le‘𝑝)𝑧)) ↔ ∃!𝑥 ∈ (Base‘𝑝)(∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧))))
21		reueq1 3403	. . . . . . . 8 ⊢ ((Base‘𝑝) = 𝐵 → (∃!𝑥 ∈ (Base‘𝑝)(∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧)) ↔ ∃!𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧))))
22	5, 21	syl 18	. . . . . . 7 ⊢ (𝑝 = 𝐾 → (∃!𝑥 ∈ (Base‘𝑝)(∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧)) ↔ ∃!𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧))))
23	20, 22	bitrd 282	. . . . . 6 ⊢ (𝑝 = 𝐾 → (∃!𝑥 ∈ (Base‘𝑝)(∀𝑦 ∈ 𝑠 𝑦(le‘𝑝)𝑥 ∧ ∀𝑧 ∈ (Base‘𝑝)(∀𝑦 ∈ 𝑠 𝑦(le‘𝑝)𝑧 → 𝑥(le‘𝑝)𝑧)) ↔ ∃!𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧))))
24	23	abbidv 2831	. . . . 5 ⊢ (𝑝 = 𝐾 → {𝑠 ∣ ∃!𝑥 ∈ (Base‘𝑝)(∀𝑦 ∈ 𝑠 𝑦(le‘𝑝)𝑥 ∧ ∀𝑧 ∈ (Base‘𝑝)(∀𝑦 ∈ 𝑠 𝑦(le‘𝑝)𝑧 → 𝑥(le‘𝑝)𝑧))} = {𝑠 ∣ ∃!𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧))})
25	19, 24	reseq12d 5981	. . . 4 ⊢ (𝑝 = 𝐾 → ((𝑠 ∈ 𝒫 (Base‘𝑝) ↦ (℩𝑥 ∈ (Base‘𝑝)(∀𝑦 ∈ 𝑠 𝑦(le‘𝑝)𝑥 ∧ ∀𝑧 ∈ (Base‘𝑝)(∀𝑦 ∈ 𝑠 𝑦(le‘𝑝)𝑧 → 𝑥(le‘𝑝)𝑧)))) ↾ {𝑠 ∣ ∃!𝑥 ∈ (Base‘𝑝)(∀𝑦 ∈ 𝑠 𝑦(le‘𝑝)𝑥 ∧ ∀𝑧 ∈ (Base‘𝑝)(∀𝑦 ∈ 𝑠 𝑦(le‘𝑝)𝑧 → 𝑥(le‘𝑝)𝑧))}) = ((𝑠 ∈ 𝒫 𝐵 ↦ (℩𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧)))) ↾ {𝑠 ∣ ∃!𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧))}))
26		df-lub 18424	. . . 4 ⊢ lub = (𝑝 ∈ V ↦ ((𝑠 ∈ 𝒫 (Base‘𝑝) ↦ (℩𝑥 ∈ (Base‘𝑝)(∀𝑦 ∈ 𝑠 𝑦(le‘𝑝)𝑥 ∧ ∀𝑧 ∈ (Base‘𝑝)(∀𝑦 ∈ 𝑠 𝑦(le‘𝑝)𝑧 → 𝑥(le‘𝑝)𝑧)))) ↾ {𝑠 ∣ ∃!𝑥 ∈ (Base‘𝑝)(∀𝑦 ∈ 𝑠 𝑦(le‘𝑝)𝑥 ∧ ∀𝑧 ∈ (Base‘𝑝)(∀𝑦 ∈ 𝑠 𝑦(le‘𝑝)𝑧 → 𝑥(le‘𝑝)𝑧))}))
27	4	fvexi 6899	. . . . . . 7 ⊢ 𝐵 ∈ V
28	27	pwex 5353	. . . . . 6 ⊢ 𝒫 𝐵 ∈ V
29	28	mptex 7228	. . . . 5 ⊢ (𝑠 ∈ 𝒫 𝐵 ↦ (℩𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧)))) ∈ V
30	29	resex 6030	. . . 4 ⊢ ((𝑠 ∈ 𝒫 𝐵 ↦ (℩𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧)))) ↾ {𝑠 ∣ ∃!𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧))}) ∈ V
31	25, 26, 30	fvmpt 6993	. . 3 ⊢ (𝐾 ∈ V → (lub‘𝐾) = ((𝑠 ∈ 𝒫 𝐵 ↦ (℩𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧)))) ↾ {𝑠 ∣ ∃!𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧))}))
32		lubfval.u	. . 3 ⊢ 𝑈 = (lub‘𝐾)
33		lubfval.p	. . . . . . 7 ⊢ (𝜓 ↔ (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧)))
34	33	a1i 11	. . . . . 6 ⊢ (𝑥 ∈ 𝐵 → (𝜓 ↔ (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧))))
35	34	riotabiia 7396	. . . . 5 ⊢ (℩𝑥 ∈ 𝐵 𝜓) = (℩𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧)))
36	35	mpteq2i 5209	. . . 4 ⊢ (𝑠 ∈ 𝒫 𝐵 ↦ (℩𝑥 ∈ 𝐵 𝜓)) = (𝑠 ∈ 𝒫 𝐵 ↦ (℩𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧))))
37	33	reubii 3380	. . . . 5 ⊢ (∃!𝑥 ∈ 𝐵 𝜓 ↔ ∃!𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧)))
38	37	abbii 2832	. . . 4 ⊢ {𝑠 ∣ ∃!𝑥 ∈ 𝐵 𝜓} = {𝑠 ∣ ∃!𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧))}
39	36, 38	reseq12i 5978	. . 3 ⊢ ((𝑠 ∈ 𝒫 𝐵 ↦ (℩𝑥 ∈ 𝐵 𝜓)) ↾ {𝑠 ∣ ∃!𝑥 ∈ 𝐵 𝜓}) = ((𝑠 ∈ 𝒫 𝐵 ↦ (℩𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧)))) ↾ {𝑠 ∣ ∃!𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧))})
40	31, 32, 39	3eqtr4g 2825	. 2 ⊢ (𝐾 ∈ V → 𝑈 = ((𝑠 ∈ 𝒫 𝐵 ↦ (℩𝑥 ∈ 𝐵 𝜓)) ↾ {𝑠 ∣ ∃!𝑥 ∈ 𝐵 𝜓}))
41	1, 2, 40	3syl 19	1 ⊢ (𝜑 → 𝑈 = ((𝑠 ∈ 𝒫 𝐵 ↦ (℩𝑥 ∈ 𝐵 𝜓)) ↾ {𝑠 ∣ ∃!𝑥 ∈ 𝐵 𝜓}))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2146  {cab 2743  ∀wral 3081  ∃!wreu 3369  Vcvv 3457  𝒫 cpw 4564   class class class wbr 5111   ↦ cmpt 5194   ↾ cres 5665  ‘cfv 6540  ℩crio 7375  Basecbs 17293  lecple 17341  lubclub 18389

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406

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 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7376  df-lub 18424

This theorem is used by:  lubdm  18429  lubfun  18430  lubval  18434  join0  18483  odulub  18485  oduglb  18487