Theorem lubid 18422

Description: The LUB of elements less than or equal to a fixed value equals that value. (Contributed by NM, 19-Oct-2011.) (Revised by NM, 7-Sep-2018.)

Hypotheses

Ref	Expression
lubid.b	⊢ 𝐵 = (Base‘𝐾)
lubid.l	⊢ ≤ = (le‘𝐾)
lubid.u	⊢ 𝑈 = (lub‘𝐾)
lubid.k	⊢ (𝜑 → 𝐾 ∈ Poset)
lubid.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)

Assertion

Ref	Expression
lubid	⊢ (𝜑 → (𝑈‘{𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑋}) = 𝑋)

Distinct variable groups:   𝑦, ≤   𝑦,𝐵   𝑦,𝑋

Allowed substitution hints:   𝜑(𝑦)   𝑈(𝑦)   𝐾(𝑦)

Proof of Theorem lubid

Dummy variables 𝑥 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		lubid.b	. . 3 ⊢ 𝐵 = (Base‘𝐾)
2		lubid.l	. . 3 ⊢ ≤ = (le‘𝐾)
3		lubid.u	. . 3 ⊢ 𝑈 = (lub‘𝐾)
4		biid 264	. . 3 ⊢ ((∀𝑧 ∈ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑋}𝑧 ≤ 𝑥 ∧ ∀𝑤 ∈ 𝐵 (∀𝑧 ∈ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑋}𝑧 ≤ 𝑤 → 𝑥 ≤ 𝑤)) ↔ (∀𝑧 ∈ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑋}𝑧 ≤ 𝑥 ∧ ∀𝑤 ∈ 𝐵 (∀𝑧 ∈ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑋}𝑧 ≤ 𝑤 → 𝑥 ≤ 𝑤)))
5		lubid.k	. . 3 ⊢ (𝜑 → 𝐾 ∈ Poset)
6		ssrab2 4033	. . . 4 ⊢ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑋} ⊆ 𝐵
7	6	a1i 11	. . 3 ⊢ (𝜑 → {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑋} ⊆ 𝐵)
8	1, 2, 3, 4, 5, 7	lubval 18416	. 2 ⊢ (𝜑 → (𝑈‘{𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑋}) = (℩𝑥 ∈ 𝐵 (∀𝑧 ∈ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑋}𝑧 ≤ 𝑥 ∧ ∀𝑤 ∈ 𝐵 (∀𝑧 ∈ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑋}𝑧 ≤ 𝑤 → 𝑥 ≤ 𝑤))))
9		lubid.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
10	1, 2, 3, 5, 9	lublecllem 18420	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((∀𝑧 ∈ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑋}𝑧 ≤ 𝑥 ∧ ∀𝑤 ∈ 𝐵 (∀𝑧 ∈ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑋}𝑧 ≤ 𝑤 → 𝑥 ≤ 𝑤)) ↔ 𝑥 = 𝑋))
11	9, 10	riota5 7398	. 2 ⊢ (𝜑 → (℩𝑥 ∈ 𝐵 (∀𝑧 ∈ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑋}𝑧 ≤ 𝑥 ∧ ∀𝑤 ∈ 𝐵 (∀𝑧 ∈ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑋}𝑧 ≤ 𝑤 → 𝑥 ≤ 𝑤))) = 𝑋)
12	8, 11	eqtrd 2797	1 ⊢ (𝜑 → (𝑈‘{𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑋}) = 𝑋)

This proof depends on syntax axioms:   → wi 4   ∧ wa 400   = wceq 1569   ∈ wcel 2142  ∀wral 3078  {crab 3415   ⊆ wss 3904   class class class wbr 5108  ‘cfv 6536  ℩crio 7368  Basecbs 17275  lecple 17323  Posetcpo 18369  lubclub 18371

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5335  ax-pr 5403

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rmo 3368  df-reu 3369  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5555  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7369  df-proset 18356  df-poset 18375  df-lub 18406

This theorem is used by:  atlatmstc  40121  lubprlem  49768