Theorem lubid 18266

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 261	. . 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 4027	. . . 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 18260	. 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 18264	. . 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 7332	. 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 2766	1 ⊢ (𝜑 → (𝑈‘{𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑋}) = 𝑋)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111  ∀wral 3047  {crab 3395   ⊆ wss 3897   class class class wbr 5089  ‘cfv 6481  ℩crio 7302  Basecbs 17120  lecple 17168  Posetcpo 18213  lubclub 18215

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-riota 7303  df-proset 18200  df-poset 18219  df-lub 18250

This theorem is referenced by:  atlatmstc  39428  lubprlem  49072