Theorem joindmss 17360

Description: Subset property of domain of join. (Contributed by NM, 12-Sep-2018.)

Hypotheses

Ref	Expression
joindmss.b	⊢ 𝐵 = (Base‘𝐾)
joindmss.j	⊢ ∨ = (join‘𝐾)
joindmss.k	⊢ (𝜑 → 𝐾 ∈ 𝑉)

Assertion

Ref	Expression
joindmss	⊢ (𝜑 → dom ∨ ⊆ (𝐵 × 𝐵))

Proof of Theorem joindmss

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		relopab 5480	. . 3 ⊢ Rel {⟨𝑥, 𝑦⟩ ∣ {𝑥, 𝑦} ∈ dom (lub‘𝐾)}
2		joindmss.k	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ 𝑉)
3		eqid 2825	. . . . . 6 ⊢ (lub‘𝐾) = (lub‘𝐾)
4		joindmss.j	. . . . . 6 ⊢ ∨ = (join‘𝐾)
5	3, 4	joindm 17356	. . . . 5 ⊢ (𝐾 ∈ 𝑉 → dom ∨ = {⟨𝑥, 𝑦⟩ ∣ {𝑥, 𝑦} ∈ dom (lub‘𝐾)})
6	2, 5	syl 17	. . . 4 ⊢ (𝜑 → dom ∨ = {⟨𝑥, 𝑦⟩ ∣ {𝑥, 𝑦} ∈ dom (lub‘𝐾)})
7	6	releqd 5438	. . 3 ⊢ (𝜑 → (Rel dom ∨ ↔ Rel {⟨𝑥, 𝑦⟩ ∣ {𝑥, 𝑦} ∈ dom (lub‘𝐾)}))
8	1, 7	mpbiri 250	. 2 ⊢ (𝜑 → Rel dom ∨ )
9		vex 3417	. . . . 5 ⊢ 𝑥 ∈ V
10	9	a1i 11	. . . 4 ⊢ (𝜑 → 𝑥 ∈ V)
11		vex 3417	. . . . 5 ⊢ 𝑦 ∈ V
12	11	a1i 11	. . . 4 ⊢ (𝜑 → 𝑦 ∈ V)
13	3, 4, 2, 10, 12	joindef 17357	. . 3 ⊢ (𝜑 → (⟨𝑥, 𝑦⟩ ∈ dom ∨ ↔ {𝑥, 𝑦} ∈ dom (lub‘𝐾)))
14		joindmss.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐾)
15		eqid 2825	. . . . . 6 ⊢ (le‘𝐾) = (le‘𝐾)
16	2	adantr 474	. . . . . 6 ⊢ ((𝜑 ∧ {𝑥, 𝑦} ∈ dom (lub‘𝐾)) → 𝐾 ∈ 𝑉)
17		simpr 479	. . . . . 6 ⊢ ((𝜑 ∧ {𝑥, 𝑦} ∈ dom (lub‘𝐾)) → {𝑥, 𝑦} ∈ dom (lub‘𝐾))
18	14, 15, 3, 16, 17	lubelss 17335	. . . . 5 ⊢ ((𝜑 ∧ {𝑥, 𝑦} ∈ dom (lub‘𝐾)) → {𝑥, 𝑦} ⊆ 𝐵)
19	18	ex 403	. . . 4 ⊢ (𝜑 → ({𝑥, 𝑦} ∈ dom (lub‘𝐾) → {𝑥, 𝑦} ⊆ 𝐵))
20	9, 11	prss 4569	. . . . 5 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ↔ {𝑥, 𝑦} ⊆ 𝐵)
21		opelxpi 5379	. . . . 5 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → ⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵))
22	20, 21	sylbir 227	. . . 4 ⊢ ({𝑥, 𝑦} ⊆ 𝐵 → ⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵))
23	19, 22	syl6 35	. . 3 ⊢ (𝜑 → ({𝑥, 𝑦} ∈ dom (lub‘𝐾) → ⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵)))
24	13, 23	sylbid 232	. 2 ⊢ (𝜑 → (⟨𝑥, 𝑦⟩ ∈ dom ∨ → ⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵)))
25	8, 24	relssdv 5446	1 ⊢ (𝜑 → dom ∨ ⊆ (𝐵 × 𝐵))

Syntax hints:   → wi 4   ∧ wa 386   = wceq 1656   ∈ wcel 2164  Vcvv 3414   ⊆ wss 3798  {cpr 4399  ⟨cop 4403  {copab 4935   × cxp 5340  dom cdm 5342  Rel wrel 5347  ‘cfv 6123  Basecbs 16222  lecple 16312  lubclub 17295  joincjn 17297

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4994  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-riota 6866  df-oprab 6909  df-lub 17327  df-join 17329

This theorem is referenced by:  clatl  17469