Theorem mdbr 32719

Description: Binary relation expressing ⟨𝐴, 𝐵⟩ is a modular pair. Definition 1.1 of [MaedaMaeda] p. 1. (Contributed by NM, 14-Jun-2004.) (New usage is discouraged.)

Assertion

Ref	Expression
mdbr	⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 𝑀ℋ 𝐵 ↔ ∀𝑥 ∈ Cℋ (𝑥 ⊆ 𝐵 → ((𝑥 ∨ℋ 𝐴) ∩ 𝐵) = (𝑥 ∨ℋ (𝐴 ∩ 𝐵)))))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem mdbr

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

Step	Hyp	Ref	Expression
1		eleq1 2853	. . . . 5 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ Cℋ ↔ 𝐴 ∈ Cℋ ))
2	1	anbi1d 643	. . . 4 ⊢ (𝑦 = 𝐴 → ((𝑦 ∈ Cℋ ∧ 𝑧 ∈ Cℋ ) ↔ (𝐴 ∈ Cℋ ∧ 𝑧 ∈ Cℋ )))
3		oveq2 7427	. . . . . . . 8 ⊢ (𝑦 = 𝐴 → (𝑥 ∨ℋ 𝑦) = (𝑥 ∨ℋ 𝐴))
4	3	ineq1d 4172	. . . . . . 7 ⊢ (𝑦 = 𝐴 → ((𝑥 ∨ℋ 𝑦) ∩ 𝑧) = ((𝑥 ∨ℋ 𝐴) ∩ 𝑧))
5		ineq1 4166	. . . . . . . 8 ⊢ (𝑦 = 𝐴 → (𝑦 ∩ 𝑧) = (𝐴 ∩ 𝑧))
6	5	oveq2d 7435	. . . . . . 7 ⊢ (𝑦 = 𝐴 → (𝑥 ∨ℋ (𝑦 ∩ 𝑧)) = (𝑥 ∨ℋ (𝐴 ∩ 𝑧)))
7	4, 6	eqeq12d 2781	. . . . . 6 ⊢ (𝑦 = 𝐴 → (((𝑥 ∨ℋ 𝑦) ∩ 𝑧) = (𝑥 ∨ℋ (𝑦 ∩ 𝑧)) ↔ ((𝑥 ∨ℋ 𝐴) ∩ 𝑧) = (𝑥 ∨ℋ (𝐴 ∩ 𝑧))))
8	7	imbi2d 343	. . . . 5 ⊢ (𝑦 = 𝐴 → ((𝑥 ⊆ 𝑧 → ((𝑥 ∨ℋ 𝑦) ∩ 𝑧) = (𝑥 ∨ℋ (𝑦 ∩ 𝑧))) ↔ (𝑥 ⊆ 𝑧 → ((𝑥 ∨ℋ 𝐴) ∩ 𝑧) = (𝑥 ∨ℋ (𝐴 ∩ 𝑧)))))
9	8	ralbidv 3190	. . . 4 ⊢ (𝑦 = 𝐴 → (∀𝑥 ∈ Cℋ (𝑥 ⊆ 𝑧 → ((𝑥 ∨ℋ 𝑦) ∩ 𝑧) = (𝑥 ∨ℋ (𝑦 ∩ 𝑧))) ↔ ∀𝑥 ∈ Cℋ (𝑥 ⊆ 𝑧 → ((𝑥 ∨ℋ 𝐴) ∩ 𝑧) = (𝑥 ∨ℋ (𝐴 ∩ 𝑧)))))
10	2, 9	anbi12d 644	. . 3 ⊢ (𝑦 = 𝐴 → (((𝑦 ∈ Cℋ ∧ 𝑧 ∈ Cℋ ) ∧ ∀𝑥 ∈ Cℋ (𝑥 ⊆ 𝑧 → ((𝑥 ∨ℋ 𝑦) ∩ 𝑧) = (𝑥 ∨ℋ (𝑦 ∩ 𝑧)))) ↔ ((𝐴 ∈ Cℋ ∧ 𝑧 ∈ Cℋ ) ∧ ∀𝑥 ∈ Cℋ (𝑥 ⊆ 𝑧 → ((𝑥 ∨ℋ 𝐴) ∩ 𝑧) = (𝑥 ∨ℋ (𝐴 ∩ 𝑧))))))
11		eleq1 2853	. . . . 5 ⊢ (𝑧 = 𝐵 → (𝑧 ∈ Cℋ ↔ 𝐵 ∈ Cℋ ))
12	11	anbi2d 642	. . . 4 ⊢ (𝑧 = 𝐵 → ((𝐴 ∈ Cℋ ∧ 𝑧 ∈ Cℋ ) ↔ (𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ )))
13		sseq2 3964	. . . . . 6 ⊢ (𝑧 = 𝐵 → (𝑥 ⊆ 𝑧 ↔ 𝑥 ⊆ 𝐵))
14		ineq2 4167	. . . . . . 7 ⊢ (𝑧 = 𝐵 → ((𝑥 ∨ℋ 𝐴) ∩ 𝑧) = ((𝑥 ∨ℋ 𝐴) ∩ 𝐵))
15		ineq2 4167	. . . . . . . 8 ⊢ (𝑧 = 𝐵 → (𝐴 ∩ 𝑧) = (𝐴 ∩ 𝐵))
16	15	oveq2d 7435	. . . . . . 7 ⊢ (𝑧 = 𝐵 → (𝑥 ∨ℋ (𝐴 ∩ 𝑧)) = (𝑥 ∨ℋ (𝐴 ∩ 𝐵)))
17	14, 16	eqeq12d 2781	. . . . . 6 ⊢ (𝑧 = 𝐵 → (((𝑥 ∨ℋ 𝐴) ∩ 𝑧) = (𝑥 ∨ℋ (𝐴 ∩ 𝑧)) ↔ ((𝑥 ∨ℋ 𝐴) ∩ 𝐵) = (𝑥 ∨ℋ (𝐴 ∩ 𝐵))))
18	13, 17	imbi12d 347	. . . . 5 ⊢ (𝑧 = 𝐵 → ((𝑥 ⊆ 𝑧 → ((𝑥 ∨ℋ 𝐴) ∩ 𝑧) = (𝑥 ∨ℋ (𝐴 ∩ 𝑧))) ↔ (𝑥 ⊆ 𝐵 → ((𝑥 ∨ℋ 𝐴) ∩ 𝐵) = (𝑥 ∨ℋ (𝐴 ∩ 𝐵)))))
19	18	ralbidv 3190	. . . 4 ⊢ (𝑧 = 𝐵 → (∀𝑥 ∈ Cℋ (𝑥 ⊆ 𝑧 → ((𝑥 ∨ℋ 𝐴) ∩ 𝑧) = (𝑥 ∨ℋ (𝐴 ∩ 𝑧))) ↔ ∀𝑥 ∈ Cℋ (𝑥 ⊆ 𝐵 → ((𝑥 ∨ℋ 𝐴) ∩ 𝐵) = (𝑥 ∨ℋ (𝐴 ∩ 𝐵)))))
20	12, 19	anbi12d 644	. . 3 ⊢ (𝑧 = 𝐵 → (((𝐴 ∈ Cℋ ∧ 𝑧 ∈ Cℋ ) ∧ ∀𝑥 ∈ Cℋ (𝑥 ⊆ 𝑧 → ((𝑥 ∨ℋ 𝐴) ∩ 𝑧) = (𝑥 ∨ℋ (𝐴 ∩ 𝑧)))) ↔ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) ∧ ∀𝑥 ∈ Cℋ (𝑥 ⊆ 𝐵 → ((𝑥 ∨ℋ 𝐴) ∩ 𝐵) = (𝑥 ∨ℋ (𝐴 ∩ 𝐵))))))
21		df-md 32705	. . 3 ⊢ 𝑀ℋ = {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ Cℋ ∧ 𝑧 ∈ Cℋ ) ∧ ∀𝑥 ∈ Cℋ (𝑥 ⊆ 𝑧 → ((𝑥 ∨ℋ 𝑦) ∩ 𝑧) = (𝑥 ∨ℋ (𝑦 ∩ 𝑧))))}
22	10, 20, 21	brabg 5526	. 2 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 𝑀ℋ 𝐵 ↔ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) ∧ ∀𝑥 ∈ Cℋ (𝑥 ⊆ 𝐵 → ((𝑥 ∨ℋ 𝐴) ∩ 𝐵) = (𝑥 ∨ℋ (𝐴 ∩ 𝐵))))))
23	22	bianabs 551	1 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 𝑀ℋ 𝐵 ↔ ∀𝑥 ∈ Cℋ (𝑥 ⊆ 𝐵 → ((𝑥 ∨ℋ 𝐴) ∩ 𝐵) = (𝑥 ∨ℋ (𝐴 ∩ 𝐵)))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2146  ∀wral 3081   ∩ cin 3905   ⊆ wss 3906   class class class wbr 5111  (class class class)co 7419   C_ℋ cch 31354   ∨_ℋ chj 31358   𝑀_ℋ cmd 31391

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737  ax-sep 5259  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3082  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-iota 6496  df-fv 6548  df-ov 7422  df-md 32705

This theorem is used by:  mdi  32720  mdbr2  32721  mdbr3  32722  dmdmd  32725  mddmd2  32734  mdsl1i  32746