Theorem joindmss 18095

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		relopabv 5730	. . 3 ⊢ Rel {⟨𝑥, 𝑦⟩ ∣ {𝑥, 𝑦} ∈ dom (lub‘𝐾)}
2		joindmss.k	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ 𝑉)
3		eqid 2740	. . . . . 6 ⊢ (lub‘𝐾) = (lub‘𝐾)
4		joindmss.j	. . . . . 6 ⊢ ∨ = (join‘𝐾)
5	3, 4	joindm 18091	. . . . 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 5689	. . 3 ⊢ (𝜑 → (Rel dom ∨ ↔ Rel {⟨𝑥, 𝑦⟩ ∣ {𝑥, 𝑦} ∈ dom (lub‘𝐾)}))
8	1, 7	mpbiri 257	. 2 ⊢ (𝜑 → Rel dom ∨ )
9		vex 3435	. . . . 5 ⊢ 𝑥 ∈ V
10	9	a1i 11	. . . 4 ⊢ (𝜑 → 𝑥 ∈ V)
11		vex 3435	. . . . 5 ⊢ 𝑦 ∈ V
12	11	a1i 11	. . . 4 ⊢ (𝜑 → 𝑦 ∈ V)
13	3, 4, 2, 10, 12	joindef 18092	. . 3 ⊢ (𝜑 → (⟨𝑥, 𝑦⟩ ∈ dom ∨ ↔ {𝑥, 𝑦} ∈ dom (lub‘𝐾)))
14		joindmss.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐾)
15		eqid 2740	. . . . . 6 ⊢ (le‘𝐾) = (le‘𝐾)
16	2	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ {𝑥, 𝑦} ∈ dom (lub‘𝐾)) → 𝐾 ∈ 𝑉)
17		simpr 485	. . . . . 6 ⊢ ((𝜑 ∧ {𝑥, 𝑦} ∈ dom (lub‘𝐾)) → {𝑥, 𝑦} ∈ dom (lub‘𝐾))
18	14, 15, 3, 16, 17	lubelss 18070	. . . . 5 ⊢ ((𝜑 ∧ {𝑥, 𝑦} ∈ dom (lub‘𝐾)) → {𝑥, 𝑦} ⊆ 𝐵)
19	18	ex 413	. . . 4 ⊢ (𝜑 → ({𝑥, 𝑦} ∈ dom (lub‘𝐾) → {𝑥, 𝑦} ⊆ 𝐵))
20	9, 11	prss 4759	. . . . 5 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ↔ {𝑥, 𝑦} ⊆ 𝐵)
21		opelxpi 5627	. . . . 5 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → ⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵))
22	20, 21	sylbir 234	. . . 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 239	. 2 ⊢ (𝜑 → (⟨𝑥, 𝑦⟩ ∈ dom ∨ → ⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵)))
25	8, 24	relssdv 5697	1 ⊢ (𝜑 → dom ∨ ⊆ (𝐵 × 𝐵))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1542   ∈ wcel 2110  Vcvv 3431   ⊆ wss 3892  {cpr 4569  ⟨cop 4573  {copab 5141   × cxp 5588  dom cdm 5590  Rel wrel 5595  ‘cfv 6432  Basecbs 16910  lecple 16967  lubclub 18025  joincjn 18027

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-rep 5214  ax-sep 5227  ax-nul 5234  ax-pow 5292  ax-pr 5356  ax-un 7582

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-ral 3071  df-rex 3072  df-reu 3073  df-rab 3075  df-v 3433  df-sbc 3721  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-iun 4932  df-br 5080  df-opab 5142  df-mpt 5163  df-id 5490  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-rn 5601  df-res 5602  df-ima 5603  df-iota 6390  df-fun 6434  df-fn 6435  df-f 6436  df-f1 6437  df-fo 6438  df-f1o 6439  df-fv 6440  df-riota 7228  df-oprab 7275  df-lub 18062  df-join 18064

This theorem is referenced by:  clatl  18224  joindm2  46231