Theorem lubeldm2d 49082

Description: Member of the domain of the least upper bound function of a poset. (Contributed by Zhi Wang, 28-Sep-2024.)

Hypotheses

Ref	Expression
lubeldm2d.b	⊢ (𝜑 → 𝐵 = (Base‘𝐾))
lubeldm2d.l	⊢ (𝜑 → ≤ = (le‘𝐾))
lubeldm2d.u	⊢ (𝜑 → 𝑈 = (lub‘𝐾))
lubeldm2d.p	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝜓 ↔ (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧))))
lubeldm2d.k	⊢ (𝜑 → 𝐾 ∈ Poset)

Assertion

Ref	Expression
lubeldm2d	⊢ (𝜑 → (𝑆 ∈ dom 𝑈 ↔ (𝑆 ⊆ 𝐵 ∧ ∃𝑥 ∈ 𝐵 𝜓)))

Distinct variable groups:   𝑥,𝐾,𝑦,𝑧   𝑥,𝑆,𝑦,𝑧   𝜑,𝑥,𝑦,𝑧

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

Proof of Theorem lubeldm2d

Step	Hyp	Ref	Expression
1		eqid 2733	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		eqid 2733	. . 3 ⊢ (le‘𝐾) = (le‘𝐾)
3		eqid 2733	. . 3 ⊢ (lub‘𝐾) = (lub‘𝐾)
4		biid 261	. . 3 ⊢ ((∀𝑦 ∈ 𝑆 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑆 𝑦(le‘𝐾)𝑧 → 𝑥(le‘𝐾)𝑧)) ↔ (∀𝑦 ∈ 𝑆 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑆 𝑦(le‘𝐾)𝑧 → 𝑥(le‘𝐾)𝑧)))
5		lubeldm2d.k	. . 3 ⊢ (𝜑 → 𝐾 ∈ Poset)
6	1, 2, 3, 4, 5	lubeldm2 49080	. 2 ⊢ (𝜑 → (𝑆 ∈ dom (lub‘𝐾) ↔ (𝑆 ⊆ (Base‘𝐾) ∧ ∃𝑥 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑆 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑆 𝑦(le‘𝐾)𝑧 → 𝑥(le‘𝐾)𝑧)))))
7		lubeldm2d.u	. . . 4 ⊢ (𝜑 → 𝑈 = (lub‘𝐾))
8	7	dmeqd 5849	. . 3 ⊢ (𝜑 → dom 𝑈 = dom (lub‘𝐾))
9	8	eleq2d 2819	. 2 ⊢ (𝜑 → (𝑆 ∈ dom 𝑈 ↔ 𝑆 ∈ dom (lub‘𝐾)))
10		lubeldm2d.b	. . . 4 ⊢ (𝜑 → 𝐵 = (Base‘𝐾))
11	10	sseq2d 3963	. . 3 ⊢ (𝜑 → (𝑆 ⊆ 𝐵 ↔ 𝑆 ⊆ (Base‘𝐾)))
12		lubeldm2d.p	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝜓 ↔ (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧))))
13		lubeldm2d.l	. . . . . . . . . . 11 ⊢ (𝜑 → ≤ = (le‘𝐾))
14	13	breqd 5104	. . . . . . . . . 10 ⊢ (𝜑 → (𝑦 ≤ 𝑥 ↔ 𝑦(le‘𝐾)𝑥))
15	14	ralbidv 3156	. . . . . . . . 9 ⊢ (𝜑 → (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥 ↔ ∀𝑦 ∈ 𝑆 𝑦(le‘𝐾)𝑥))
16	13	breqd 5104	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑦 ≤ 𝑧 ↔ 𝑦(le‘𝐾)𝑧))
17	16	ralbidv 3156	. . . . . . . . . . 11 ⊢ (𝜑 → (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 ↔ ∀𝑦 ∈ 𝑆 𝑦(le‘𝐾)𝑧))
18	13	breqd 5104	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥 ≤ 𝑧 ↔ 𝑥(le‘𝐾)𝑧))
19	17, 18	imbi12d 344	. . . . . . . . . 10 ⊢ (𝜑 → ((∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧) ↔ (∀𝑦 ∈ 𝑆 𝑦(le‘𝐾)𝑧 → 𝑥(le‘𝐾)𝑧)))
20	10, 19	raleqbidv 3313	. . . . . . . . 9 ⊢ (𝜑 → (∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧) ↔ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑆 𝑦(le‘𝐾)𝑧 → 𝑥(le‘𝐾)𝑧)))
21	15, 20	anbi12d 632	. . . . . . . 8 ⊢ (𝜑 → ((∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧)) ↔ (∀𝑦 ∈ 𝑆 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑆 𝑦(le‘𝐾)𝑧 → 𝑥(le‘𝐾)𝑧))))
22	21	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧)) ↔ (∀𝑦 ∈ 𝑆 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑆 𝑦(le‘𝐾)𝑧 → 𝑥(le‘𝐾)𝑧))))
23	12, 22	bitrd 279	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝜓 ↔ (∀𝑦 ∈ 𝑆 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑆 𝑦(le‘𝐾)𝑧 → 𝑥(le‘𝐾)𝑧))))
24	23	pm5.32da 579	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ 𝐵 ∧ 𝜓) ↔ (𝑥 ∈ 𝐵 ∧ (∀𝑦 ∈ 𝑆 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑆 𝑦(le‘𝐾)𝑧 → 𝑥(le‘𝐾)𝑧)))))
25	10	eleq2d 2819	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↔ 𝑥 ∈ (Base‘𝐾)))
26	25	anbi1d 631	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ 𝐵 ∧ (∀𝑦 ∈ 𝑆 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑆 𝑦(le‘𝐾)𝑧 → 𝑥(le‘𝐾)𝑧))) ↔ (𝑥 ∈ (Base‘𝐾) ∧ (∀𝑦 ∈ 𝑆 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑆 𝑦(le‘𝐾)𝑧 → 𝑥(le‘𝐾)𝑧)))))
27	24, 26	bitrd 279	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝐵 ∧ 𝜓) ↔ (𝑥 ∈ (Base‘𝐾) ∧ (∀𝑦 ∈ 𝑆 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑆 𝑦(le‘𝐾)𝑧 → 𝑥(le‘𝐾)𝑧)))))
28	27	rexbidv2 3153	. . 3 ⊢ (𝜑 → (∃𝑥 ∈ 𝐵 𝜓 ↔ ∃𝑥 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑆 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑆 𝑦(le‘𝐾)𝑧 → 𝑥(le‘𝐾)𝑧))))
29	11, 28	anbi12d 632	. 2 ⊢ (𝜑 → ((𝑆 ⊆ 𝐵 ∧ ∃𝑥 ∈ 𝐵 𝜓) ↔ (𝑆 ⊆ (Base‘𝐾) ∧ ∃𝑥 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑆 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑆 𝑦(le‘𝐾)𝑧 → 𝑥(le‘𝐾)𝑧)))))
30	6, 9, 29	3bitr4d 311	1 ⊢ (𝜑 → (𝑆 ∈ dom 𝑈 ↔ (𝑆 ⊆ 𝐵 ∧ ∃𝑥 ∈ 𝐵 𝜓)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∀wral 3048  ∃wrex 3057   ⊆ wss 3898   class class class wbr 5093  dom cdm 5619  ‘cfv 6486  Basecbs 17122  lecple 17170  Posetcpo 18215  lubclub 18217

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 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372

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 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rmo 3347  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-riota 7309  df-proset 18202  df-poset 18221  df-lub 18252

This theorem is referenced by:  ipolubdm  49111