Theorem mdbr 32097

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 2817	. . . . 5 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ Cℋ ↔ 𝐴 ∈ Cℋ ))
2	1	anbi1d 630	. . . 4 ⊢ (𝑦 = 𝐴 → ((𝑦 ∈ Cℋ ∧ 𝑧 ∈ Cℋ ) ↔ (𝐴 ∈ Cℋ ∧ 𝑧 ∈ Cℋ )))
3		oveq2 7422	. . . . . . . 8 ⊢ (𝑦 = 𝐴 → (𝑥 ∨ℋ 𝑦) = (𝑥 ∨ℋ 𝐴))
4	3	ineq1d 4207	. . . . . . 7 ⊢ (𝑦 = 𝐴 → ((𝑥 ∨ℋ 𝑦) ∩ 𝑧) = ((𝑥 ∨ℋ 𝐴) ∩ 𝑧))
5		ineq1 4201	. . . . . . . 8 ⊢ (𝑦 = 𝐴 → (𝑦 ∩ 𝑧) = (𝐴 ∩ 𝑧))
6	5	oveq2d 7430	. . . . . . 7 ⊢ (𝑦 = 𝐴 → (𝑥 ∨ℋ (𝑦 ∩ 𝑧)) = (𝑥 ∨ℋ (𝐴 ∩ 𝑧)))
7	4, 6	eqeq12d 2744	. . . . . 6 ⊢ (𝑦 = 𝐴 → (((𝑥 ∨ℋ 𝑦) ∩ 𝑧) = (𝑥 ∨ℋ (𝑦 ∩ 𝑧)) ↔ ((𝑥 ∨ℋ 𝐴) ∩ 𝑧) = (𝑥 ∨ℋ (𝐴 ∩ 𝑧))))
8	7	imbi2d 340	. . . . 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 3173	. . . 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 631	. . 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 2817	. . . . 5 ⊢ (𝑧 = 𝐵 → (𝑧 ∈ Cℋ ↔ 𝐵 ∈ Cℋ ))
12	11	anbi2d 629	. . . 4 ⊢ (𝑧 = 𝐵 → ((𝐴 ∈ Cℋ ∧ 𝑧 ∈ Cℋ ) ↔ (𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ )))
13		sseq2 4004	. . . . . 6 ⊢ (𝑧 = 𝐵 → (𝑥 ⊆ 𝑧 ↔ 𝑥 ⊆ 𝐵))
14		ineq2 4202	. . . . . . 7 ⊢ (𝑧 = 𝐵 → ((𝑥 ∨ℋ 𝐴) ∩ 𝑧) = ((𝑥 ∨ℋ 𝐴) ∩ 𝐵))
15		ineq2 4202	. . . . . . . 8 ⊢ (𝑧 = 𝐵 → (𝐴 ∩ 𝑧) = (𝐴 ∩ 𝐵))
16	15	oveq2d 7430	. . . . . . 7 ⊢ (𝑧 = 𝐵 → (𝑥 ∨ℋ (𝐴 ∩ 𝑧)) = (𝑥 ∨ℋ (𝐴 ∩ 𝐵)))
17	14, 16	eqeq12d 2744	. . . . . 6 ⊢ (𝑧 = 𝐵 → (((𝑥 ∨ℋ 𝐴) ∩ 𝑧) = (𝑥 ∨ℋ (𝐴 ∩ 𝑧)) ↔ ((𝑥 ∨ℋ 𝐴) ∩ 𝐵) = (𝑥 ∨ℋ (𝐴 ∩ 𝐵))))
18	13, 17	imbi12d 344	. . . . 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 3173	. . . 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 631	. . 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 32083	. . 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 5535	. 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 541	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 205   ∧ wa 395   = wceq 1534   ∈ wcel 2099  ∀wral 3057   ∩ cin 3944   ⊆ wss 3945   class class class wbr 5142  (class class class)co 7414   C_ℋ cch 30732   ∨_ℋ chj 30736   𝑀_ℋ cmd 30769

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699  ax-sep 5293  ax-nul 5300  ax-pr 5423

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3058  df-rab 3429  df-v 3472  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-iota 6494  df-fv 6550  df-ov 7417  df-md 32083

This theorem is referenced by:  mdi  32098  mdbr2  32099  mdbr3  32100  dmdmd  32103  mddmd2  32112  mdsl1i  32124