Theorem mdbr 30557

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 2826	. . . . 5 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ Cℋ ↔ 𝐴 ∈ Cℋ ))
2	1	anbi1d 629	. . . 4 ⊢ (𝑦 = 𝐴 → ((𝑦 ∈ Cℋ ∧ 𝑧 ∈ Cℋ ) ↔ (𝐴 ∈ Cℋ ∧ 𝑧 ∈ Cℋ )))
3		oveq2 7263	. . . . . . . 8 ⊢ (𝑦 = 𝐴 → (𝑥 ∨ℋ 𝑦) = (𝑥 ∨ℋ 𝐴))
4	3	ineq1d 4142	. . . . . . 7 ⊢ (𝑦 = 𝐴 → ((𝑥 ∨ℋ 𝑦) ∩ 𝑧) = ((𝑥 ∨ℋ 𝐴) ∩ 𝑧))
5		ineq1 4136	. . . . . . . 8 ⊢ (𝑦 = 𝐴 → (𝑦 ∩ 𝑧) = (𝐴 ∩ 𝑧))
6	5	oveq2d 7271	. . . . . . 7 ⊢ (𝑦 = 𝐴 → (𝑥 ∨ℋ (𝑦 ∩ 𝑧)) = (𝑥 ∨ℋ (𝐴 ∩ 𝑧)))
7	4, 6	eqeq12d 2754	. . . . . 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 3120	. . . 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 630	. . 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 2826	. . . . 5 ⊢ (𝑧 = 𝐵 → (𝑧 ∈ Cℋ ↔ 𝐵 ∈ Cℋ ))
12	11	anbi2d 628	. . . 4 ⊢ (𝑧 = 𝐵 → ((𝐴 ∈ Cℋ ∧ 𝑧 ∈ Cℋ ) ↔ (𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ )))
13		sseq2 3943	. . . . . 6 ⊢ (𝑧 = 𝐵 → (𝑥 ⊆ 𝑧 ↔ 𝑥 ⊆ 𝐵))
14		ineq2 4137	. . . . . . 7 ⊢ (𝑧 = 𝐵 → ((𝑥 ∨ℋ 𝐴) ∩ 𝑧) = ((𝑥 ∨ℋ 𝐴) ∩ 𝐵))
15		ineq2 4137	. . . . . . . 8 ⊢ (𝑧 = 𝐵 → (𝐴 ∩ 𝑧) = (𝐴 ∩ 𝐵))
16	15	oveq2d 7271	. . . . . . 7 ⊢ (𝑧 = 𝐵 → (𝑥 ∨ℋ (𝐴 ∩ 𝑧)) = (𝑥 ∨ℋ (𝐴 ∩ 𝐵)))
17	14, 16	eqeq12d 2754	. . . . . 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 3120	. . . 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 630	. . 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 30543	. . 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 5445	. 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 1539   ∈ wcel 2108  ∀wral 3063   ∩ cin 3882   ⊆ wss 3883   class class class wbr 5070  (class class class)co 7255   C_ℋ cch 29192   ∨_ℋ chj 29196   𝑀_ℋ cmd 29229

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-iota 6376  df-fv 6426  df-ov 7258  df-md 30543

This theorem is referenced by:  mdi  30558  mdbr2  30559  mdbr3  30560  dmdmd  30563  mddmd2  30572  mdsl1i  30584