Theorem mdbr 30065

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 2900	. . . . 5 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ Cℋ ↔ 𝐴 ∈ Cℋ ))
2	1	anbi1d 631	. . . 4 ⊢ (𝑦 = 𝐴 → ((𝑦 ∈ Cℋ ∧ 𝑧 ∈ Cℋ ) ↔ (𝐴 ∈ Cℋ ∧ 𝑧 ∈ Cℋ )))
3		oveq2 7158	. . . . . . . 8 ⊢ (𝑦 = 𝐴 → (𝑥 ∨ℋ 𝑦) = (𝑥 ∨ℋ 𝐴))
4	3	ineq1d 4187	. . . . . . 7 ⊢ (𝑦 = 𝐴 → ((𝑥 ∨ℋ 𝑦) ∩ 𝑧) = ((𝑥 ∨ℋ 𝐴) ∩ 𝑧))
5		ineq1 4180	. . . . . . . 8 ⊢ (𝑦 = 𝐴 → (𝑦 ∩ 𝑧) = (𝐴 ∩ 𝑧))
6	5	oveq2d 7166	. . . . . . 7 ⊢ (𝑦 = 𝐴 → (𝑥 ∨ℋ (𝑦 ∩ 𝑧)) = (𝑥 ∨ℋ (𝐴 ∩ 𝑧)))
7	4, 6	eqeq12d 2837	. . . . . 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 3197	. . . 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 632	. . 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 2900	. . . . 5 ⊢ (𝑧 = 𝐵 → (𝑧 ∈ Cℋ ↔ 𝐵 ∈ Cℋ ))
12	11	anbi2d 630	. . . 4 ⊢ (𝑧 = 𝐵 → ((𝐴 ∈ Cℋ ∧ 𝑧 ∈ Cℋ ) ↔ (𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ )))
13		sseq2 3992	. . . . . 6 ⊢ (𝑧 = 𝐵 → (𝑥 ⊆ 𝑧 ↔ 𝑥 ⊆ 𝐵))
14		ineq2 4182	. . . . . . 7 ⊢ (𝑧 = 𝐵 → ((𝑥 ∨ℋ 𝐴) ∩ 𝑧) = ((𝑥 ∨ℋ 𝐴) ∩ 𝐵))
15		ineq2 4182	. . . . . . . 8 ⊢ (𝑧 = 𝐵 → (𝐴 ∩ 𝑧) = (𝐴 ∩ 𝐵))
16	15	oveq2d 7166	. . . . . . 7 ⊢ (𝑧 = 𝐵 → (𝑥 ∨ℋ (𝐴 ∩ 𝑧)) = (𝑥 ∨ℋ (𝐴 ∩ 𝐵)))
17	14, 16	eqeq12d 2837	. . . . . 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 3197	. . . 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 632	. . 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 30051	. . 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 5418	. 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 544	1 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 𝑀ℋ 𝐵 ↔ ∀𝑥 ∈ Cℋ (𝑥 ⊆ 𝐵 → ((𝑥 ∨ℋ 𝐴) ∩ 𝐵) = (𝑥 ∨ℋ (𝐴 ∩ 𝐵)))))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1533   ∈ wcel 2110  ∀wral 3138   ∩ cin 3934   ⊆ wss 3935   class class class wbr 5058  (class class class)co 7150   C_ℋ cch 28700   ∨_ℋ chj 28704   𝑀_ℋ cmd 28737

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pr 5321

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-br 5059  df-opab 5121  df-iota 6308  df-fv 6357  df-ov 7153  df-md 30051

This theorem is referenced by:  mdi  30066  mdbr2  30067  mdbr3  30068  dmdmd  30071  mddmd2  30080  mdsl1i  30092