Theorem mdbr 5726

Description: Binary relation expressing ⟨A, B⟩ is a modular pair. Definition 1.1 of [MaedaMaeda] p. 1.

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 group(s):   x,A   x,B

Proof of Theorem mdbr

Step	Hyp	Ref	Expression
1		eleq1 1149	. . . . 5 ⊢ (y = A → (y ∈ Cℋ ↔ A ∈ Cℋ ))
2	1	anbi1d 469	. . . 4 ⊢ (y = A → ((y ∈ Cℋ ∧ z ∈ Cℋ ) ↔ (A ∈ Cℋ ∧ z ∈ Cℋ )))
3		opreq2 3007	. . . . . . . 8 ⊢ (y = A → (x ∨ℋ y) = (x ∨ℋ A))
4	3	ineq1d 1644	. . . . . . 7 ⊢ (y = A → ((x ∨ℋ y) ∩ z) = ((x ∨ℋ A) ∩ z))
5		ineq1 1638	. . . . . . . 8 ⊢ (y = A → (y ∩ z) = (A ∩ z))
6	5	opreq2d 3013	. . . . . . 7 ⊢ (y = A → (x ∨ℋ (y ∩ z)) = (x ∨ℋ (A ∩ z)))
7	4, 6	cleq12d 1115	. . . . . 6 ⊢ (y = A → (((x ∨ℋ y) ∩ z) = (x ∨ℋ (y ∩ z)) ↔ ((x ∨ℋ A) ∩ z) = (x ∨ℋ (A ∩ z))))
8	7	imbi2d 464	. . . . 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	biraldv 1219	. . . 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 476	. . 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 1149	. . . . 5 ⊢ (z = B → (z ∈ Cℋ ↔ B ∈ Cℋ ))
12	11	anbi2d 468	. . . 4 ⊢ (z = B → ((A ∈ Cℋ ∧ z ∈ Cℋ ) ↔ (A ∈ Cℋ ∧ B ∈ Cℋ )))
13		sseq2 1522	. . . . . 6 ⊢ (z = B → (x ⊆ z ↔ x ⊆ B))
14		ineq2 1639	. . . . . . 7 ⊢ (z = B → ((x ∨ℋ A) ∩ z) = ((x ∨ℋ A) ∩ B))
15		ineq2 1639	. . . . . . . 8 ⊢ (z = B → (A ∩ z) = (A ∩ B))
16	15	opreq2d 3013	. . . . . . 7 ⊢ (z = B → (x ∨ℋ (A ∩ z)) = (x ∨ℋ (A ∩ B)))
17	14, 16	cleq12d 1115	. . . . . 6 ⊢ (z = B → (((x ∨ℋ A) ∩ z) = (x ∨ℋ (A ∩ z)) ↔ ((x ∨ℋ A) ∩ B) = (x ∨ℋ (A ∩ B))))
18	13, 17	imbi12d 474	. . . . 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	biraldv 1219	. . . 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 476	. . 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 5713	. . 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 2116	. 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		ibar 487	. 2 ⊢ ((A ∈ Cℋ ∧ B ∈ Cℋ ) → (∀x ∈ Cℋ (x ⊆ B → ((x ∨ℋ A) ∩ B) = (x ∨ℋ (A ∩ B))) ↔ ((A ∈ Cℋ ∧ B ∈ Cℋ ) ∧ ∀x ∈ Cℋ (x ⊆ B → ((x ∨ℋ A) ∩ B) = (x ∨ℋ (A ∩ B))))))
24	22, 23	bitr4d 409	1 ⊢ ((A ∈ Cℋ ∧ B ∈ Cℋ ) → (A Mℋ B ↔ ∀x ∈ Cℋ (x ⊆ B → ((x ∨ℋ A) ∩ B) = (x ∨ℋ (A ∩ B)))))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196   = wceq 1091   ∈ wcel 1092  ∀wral 1201   ∩ cin 1486   ⊆ wss 1487   class class class wbr 2054  (class class class)co 3001   C_ℋ cch 4968   ∨_ℋ chj 4972   M_ℋ cmd 4982

This theorem is referenced by:  mdi 5727  mdbr2 5728  mdbr3 5729  dmdbr 5731

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-xp 2424  df-cnv 2426  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fv 2438  df-opr 3003  df-md 5713

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