Theorem lubeldm2d 47680

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 2731	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		eqid 2731	. . 3 ⊢ (le‘𝐾) = (le‘𝐾)
3		eqid 2731	. . 3 ⊢ (lub‘𝐾) = (lub‘𝐾)
4		biid 260	. . 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 47678	. 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 5906	. . 3 ⊢ (𝜑 → dom 𝑈 = dom (lub‘𝐾))
9	8	eleq2d 2818	. 2 ⊢ (𝜑 → (𝑆 ∈ dom 𝑈 ↔ 𝑆 ∈ dom (lub‘𝐾)))
10		lubeldm2d.b	. . . 4 ⊢ (𝜑 → 𝐵 = (Base‘𝐾))
11	10	sseq2d 4015	. . 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 5160	. . . . . . . . . 10 ⊢ (𝜑 → (𝑦 ≤ 𝑥 ↔ 𝑦(le‘𝐾)𝑥))
15	14	ralbidv 3176	. . . . . . . . 9 ⊢ (𝜑 → (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥 ↔ ∀𝑦 ∈ 𝑆 𝑦(le‘𝐾)𝑥))
16	13	breqd 5160	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑦 ≤ 𝑧 ↔ 𝑦(le‘𝐾)𝑧))
17	16	ralbidv 3176	. . . . . . . . . . 11 ⊢ (𝜑 → (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 ↔ ∀𝑦 ∈ 𝑆 𝑦(le‘𝐾)𝑧))
18	13	breqd 5160	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥 ≤ 𝑧 ↔ 𝑥(le‘𝐾)𝑧))
19	17, 18	imbi12d 343	. . . . . . . . . 10 ⊢ (𝜑 → ((∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧) ↔ (∀𝑦 ∈ 𝑆 𝑦(le‘𝐾)𝑧 → 𝑥(le‘𝐾)𝑧)))
20	10, 19	raleqbidv 3341	. . . . . . . . 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 630	. . . . . . . 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 278	. . . . . 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 578	. . . . 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 2818	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↔ 𝑥 ∈ (Base‘𝐾)))
26	25	anbi1d 629	. . . . 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 278	. . . 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 3173	. . 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 630	. 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 310	1 ⊢ (𝜑 → (𝑆 ∈ dom 𝑈 ↔ (𝑆 ⊆ 𝐵 ∧ ∃𝑥 ∈ 𝐵 𝜓)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1540   ∈ wcel 2105  ∀wral 3060  ∃wrex 3069   ⊆ wss 3949   class class class wbr 5149  dom cdm 5677  ‘cfv 6544  Basecbs 17149  lecple 17209  Posetcpo 18265  lubclub 18267

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7368  df-proset 18253  df-poset 18271  df-lub 18304

This theorem is referenced by:  ipolubdm  47701