Theorem mdbr3 32329

Description: Binary relation expressing the modular pair property. This version quantifies an equality instead of an inference. (Contributed by NM, 6-Jul-2004.) (New usage is discouraged.)

Assertion

Ref	Expression
mdbr3	⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 𝑀ℋ 𝐵 ↔ ∀𝑥 ∈ Cℋ (((𝑥 ∩ 𝐵) ∨ℋ 𝐴) ∩ 𝐵) = ((𝑥 ∩ 𝐵) ∨ℋ (𝐴 ∩ 𝐵))))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem mdbr3

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mdbr 32326	. 2 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 𝑀ℋ 𝐵 ↔ ∀𝑦 ∈ Cℋ (𝑦 ⊆ 𝐵 → ((𝑦 ∨ℋ 𝐴) ∩ 𝐵) = (𝑦 ∨ℋ (𝐴 ∩ 𝐵)))))
2		chincl 31531	. . . . . . . 8 ⊢ ((𝑥 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝑥 ∩ 𝐵) ∈ Cℋ )
3		inss2 4259	. . . . . . . . 9 ⊢ (𝑥 ∩ 𝐵) ⊆ 𝐵
4		sseq1 4034	. . . . . . . . . . 11 ⊢ (𝑦 = (𝑥 ∩ 𝐵) → (𝑦 ⊆ 𝐵 ↔ (𝑥 ∩ 𝐵) ⊆ 𝐵))
5		oveq1 7455	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝑥 ∩ 𝐵) → (𝑦 ∨ℋ 𝐴) = ((𝑥 ∩ 𝐵) ∨ℋ 𝐴))
6	5	ineq1d 4240	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝑥 ∩ 𝐵) → ((𝑦 ∨ℋ 𝐴) ∩ 𝐵) = (((𝑥 ∩ 𝐵) ∨ℋ 𝐴) ∩ 𝐵))
7		oveq1 7455	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝑥 ∩ 𝐵) → (𝑦 ∨ℋ (𝐴 ∩ 𝐵)) = ((𝑥 ∩ 𝐵) ∨ℋ (𝐴 ∩ 𝐵)))
8	6, 7	eqeq12d 2756	. . . . . . . . . . 11 ⊢ (𝑦 = (𝑥 ∩ 𝐵) → (((𝑦 ∨ℋ 𝐴) ∩ 𝐵) = (𝑦 ∨ℋ (𝐴 ∩ 𝐵)) ↔ (((𝑥 ∩ 𝐵) ∨ℋ 𝐴) ∩ 𝐵) = ((𝑥 ∩ 𝐵) ∨ℋ (𝐴 ∩ 𝐵))))
9	4, 8	imbi12d 344	. . . . . . . . . 10 ⊢ (𝑦 = (𝑥 ∩ 𝐵) → ((𝑦 ⊆ 𝐵 → ((𝑦 ∨ℋ 𝐴) ∩ 𝐵) = (𝑦 ∨ℋ (𝐴 ∩ 𝐵))) ↔ ((𝑥 ∩ 𝐵) ⊆ 𝐵 → (((𝑥 ∩ 𝐵) ∨ℋ 𝐴) ∩ 𝐵) = ((𝑥 ∩ 𝐵) ∨ℋ (𝐴 ∩ 𝐵)))))
10	9	rspcv 3631	. . . . . . . . 9 ⊢ ((𝑥 ∩ 𝐵) ∈ Cℋ → (∀𝑦 ∈ Cℋ (𝑦 ⊆ 𝐵 → ((𝑦 ∨ℋ 𝐴) ∩ 𝐵) = (𝑦 ∨ℋ (𝐴 ∩ 𝐵))) → ((𝑥 ∩ 𝐵) ⊆ 𝐵 → (((𝑥 ∩ 𝐵) ∨ℋ 𝐴) ∩ 𝐵) = ((𝑥 ∩ 𝐵) ∨ℋ (𝐴 ∩ 𝐵)))))
11	3, 10	mpii 46	. . . . . . . 8 ⊢ ((𝑥 ∩ 𝐵) ∈ Cℋ → (∀𝑦 ∈ Cℋ (𝑦 ⊆ 𝐵 → ((𝑦 ∨ℋ 𝐴) ∩ 𝐵) = (𝑦 ∨ℋ (𝐴 ∩ 𝐵))) → (((𝑥 ∩ 𝐵) ∨ℋ 𝐴) ∩ 𝐵) = ((𝑥 ∩ 𝐵) ∨ℋ (𝐴 ∩ 𝐵))))
12	2, 11	syl 17	. . . . . . 7 ⊢ ((𝑥 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (∀𝑦 ∈ Cℋ (𝑦 ⊆ 𝐵 → ((𝑦 ∨ℋ 𝐴) ∩ 𝐵) = (𝑦 ∨ℋ (𝐴 ∩ 𝐵))) → (((𝑥 ∩ 𝐵) ∨ℋ 𝐴) ∩ 𝐵) = ((𝑥 ∩ 𝐵) ∨ℋ (𝐴 ∩ 𝐵))))
13	12	ex 412	. . . . . 6 ⊢ (𝑥 ∈ Cℋ → (𝐵 ∈ Cℋ → (∀𝑦 ∈ Cℋ (𝑦 ⊆ 𝐵 → ((𝑦 ∨ℋ 𝐴) ∩ 𝐵) = (𝑦 ∨ℋ (𝐴 ∩ 𝐵))) → (((𝑥 ∩ 𝐵) ∨ℋ 𝐴) ∩ 𝐵) = ((𝑥 ∩ 𝐵) ∨ℋ (𝐴 ∩ 𝐵)))))
14	13	com3l 89	. . . . 5 ⊢ (𝐵 ∈ Cℋ → (∀𝑦 ∈ Cℋ (𝑦 ⊆ 𝐵 → ((𝑦 ∨ℋ 𝐴) ∩ 𝐵) = (𝑦 ∨ℋ (𝐴 ∩ 𝐵))) → (𝑥 ∈ Cℋ → (((𝑥 ∩ 𝐵) ∨ℋ 𝐴) ∩ 𝐵) = ((𝑥 ∩ 𝐵) ∨ℋ (𝐴 ∩ 𝐵)))))
15	14	ralrimdv 3158	. . . 4 ⊢ (𝐵 ∈ Cℋ → (∀𝑦 ∈ Cℋ (𝑦 ⊆ 𝐵 → ((𝑦 ∨ℋ 𝐴) ∩ 𝐵) = (𝑦 ∨ℋ (𝐴 ∩ 𝐵))) → ∀𝑥 ∈ Cℋ (((𝑥 ∩ 𝐵) ∨ℋ 𝐴) ∩ 𝐵) = ((𝑥 ∩ 𝐵) ∨ℋ (𝐴 ∩ 𝐵))))
16		dfss 3995	. . . . . . . . . . 11 ⊢ (𝑥 ⊆ 𝐵 ↔ 𝑥 = (𝑥 ∩ 𝐵))
17	16	biimpi 216	. . . . . . . . . 10 ⊢ (𝑥 ⊆ 𝐵 → 𝑥 = (𝑥 ∩ 𝐵))
18	17	oveq1d 7463	. . . . . . . . 9 ⊢ (𝑥 ⊆ 𝐵 → (𝑥 ∨ℋ 𝐴) = ((𝑥 ∩ 𝐵) ∨ℋ 𝐴))
19	18	ineq1d 4240	. . . . . . . 8 ⊢ (𝑥 ⊆ 𝐵 → ((𝑥 ∨ℋ 𝐴) ∩ 𝐵) = (((𝑥 ∩ 𝐵) ∨ℋ 𝐴) ∩ 𝐵))
20	17	oveq1d 7463	. . . . . . . 8 ⊢ (𝑥 ⊆ 𝐵 → (𝑥 ∨ℋ (𝐴 ∩ 𝐵)) = ((𝑥 ∩ 𝐵) ∨ℋ (𝐴 ∩ 𝐵)))
21	19, 20	eqeq12d 2756	. . . . . . 7 ⊢ (𝑥 ⊆ 𝐵 → (((𝑥 ∨ℋ 𝐴) ∩ 𝐵) = (𝑥 ∨ℋ (𝐴 ∩ 𝐵)) ↔ (((𝑥 ∩ 𝐵) ∨ℋ 𝐴) ∩ 𝐵) = ((𝑥 ∩ 𝐵) ∨ℋ (𝐴 ∩ 𝐵))))
22	21	biimprcd 250	. . . . . 6 ⊢ ((((𝑥 ∩ 𝐵) ∨ℋ 𝐴) ∩ 𝐵) = ((𝑥 ∩ 𝐵) ∨ℋ (𝐴 ∩ 𝐵)) → (𝑥 ⊆ 𝐵 → ((𝑥 ∨ℋ 𝐴) ∩ 𝐵) = (𝑥 ∨ℋ (𝐴 ∩ 𝐵))))
23	22	ralimi 3089	. . . . 5 ⊢ (∀𝑥 ∈ Cℋ (((𝑥 ∩ 𝐵) ∨ℋ 𝐴) ∩ 𝐵) = ((𝑥 ∩ 𝐵) ∨ℋ (𝐴 ∩ 𝐵)) → ∀𝑥 ∈ Cℋ (𝑥 ⊆ 𝐵 → ((𝑥 ∨ℋ 𝐴) ∩ 𝐵) = (𝑥 ∨ℋ (𝐴 ∩ 𝐵))))
24		sseq1 4034	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑥 ⊆ 𝐵 ↔ 𝑦 ⊆ 𝐵))
25		oveq1 7455	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑥 ∨ℋ 𝐴) = (𝑦 ∨ℋ 𝐴))
26	25	ineq1d 4240	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ((𝑥 ∨ℋ 𝐴) ∩ 𝐵) = ((𝑦 ∨ℋ 𝐴) ∩ 𝐵))
27		oveq1 7455	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥 ∨ℋ (𝐴 ∩ 𝐵)) = (𝑦 ∨ℋ (𝐴 ∩ 𝐵)))
28	26, 27	eqeq12d 2756	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (((𝑥 ∨ℋ 𝐴) ∩ 𝐵) = (𝑥 ∨ℋ (𝐴 ∩ 𝐵)) ↔ ((𝑦 ∨ℋ 𝐴) ∩ 𝐵) = (𝑦 ∨ℋ (𝐴 ∩ 𝐵))))
29	24, 28	imbi12d 344	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝑥 ⊆ 𝐵 → ((𝑥 ∨ℋ 𝐴) ∩ 𝐵) = (𝑥 ∨ℋ (𝐴 ∩ 𝐵))) ↔ (𝑦 ⊆ 𝐵 → ((𝑦 ∨ℋ 𝐴) ∩ 𝐵) = (𝑦 ∨ℋ (𝐴 ∩ 𝐵)))))
30	29	cbvralvw 3243	. . . . 5 ⊢ (∀𝑥 ∈ Cℋ (𝑥 ⊆ 𝐵 → ((𝑥 ∨ℋ 𝐴) ∩ 𝐵) = (𝑥 ∨ℋ (𝐴 ∩ 𝐵))) ↔ ∀𝑦 ∈ Cℋ (𝑦 ⊆ 𝐵 → ((𝑦 ∨ℋ 𝐴) ∩ 𝐵) = (𝑦 ∨ℋ (𝐴 ∩ 𝐵))))
31	23, 30	sylib 218	. . . 4 ⊢ (∀𝑥 ∈ Cℋ (((𝑥 ∩ 𝐵) ∨ℋ 𝐴) ∩ 𝐵) = ((𝑥 ∩ 𝐵) ∨ℋ (𝐴 ∩ 𝐵)) → ∀𝑦 ∈ Cℋ (𝑦 ⊆ 𝐵 → ((𝑦 ∨ℋ 𝐴) ∩ 𝐵) = (𝑦 ∨ℋ (𝐴 ∩ 𝐵))))
32	15, 31	impbid1 225	. . 3 ⊢ (𝐵 ∈ Cℋ → (∀𝑦 ∈ Cℋ (𝑦 ⊆ 𝐵 → ((𝑦 ∨ℋ 𝐴) ∩ 𝐵) = (𝑦 ∨ℋ (𝐴 ∩ 𝐵))) ↔ ∀𝑥 ∈ Cℋ (((𝑥 ∩ 𝐵) ∨ℋ 𝐴) ∩ 𝐵) = ((𝑥 ∩ 𝐵) ∨ℋ (𝐴 ∩ 𝐵))))
33	32	adantl 481	. 2 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (∀𝑦 ∈ Cℋ (𝑦 ⊆ 𝐵 → ((𝑦 ∨ℋ 𝐴) ∩ 𝐵) = (𝑦 ∨ℋ (𝐴 ∩ 𝐵))) ↔ ∀𝑥 ∈ Cℋ (((𝑥 ∩ 𝐵) ∨ℋ 𝐴) ∩ 𝐵) = ((𝑥 ∩ 𝐵) ∨ℋ (𝐴 ∩ 𝐵))))
34	1, 33	bitrd 279	1 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 𝑀ℋ 𝐵 ↔ ∀𝑥 ∈ Cℋ (((𝑥 ∩ 𝐵) ∨ℋ 𝐴) ∩ 𝐵) = ((𝑥 ∩ 𝐵) ∨ℋ (𝐴 ∩ 𝐵))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2108  ∀wral 3067   ∩ cin 3975   ⊆ wss 3976   class class class wbr 5166  (class class class)co 7448   C_ℋ cch 30961   ∨_ℋ chj 30965   𝑀_ℋ cmd 30998

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-1cn 11242  ax-addcl 11244  ax-hilex 31031  ax-hfvadd 31032  ax-hv0cl 31035  ax-hfvmul 31037

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-map 8886  df-nn 12294  df-hlim 31004  df-sh 31239  df-ch 31253  df-md 32312

This theorem is referenced by: (None)