Theorem mdbr3 31815

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 31812	. 2 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 𝑀ℋ 𝐵 ↔ ∀𝑦 ∈ Cℋ (𝑦 ⊆ 𝐵 → ((𝑦 ∨ℋ 𝐴) ∩ 𝐵) = (𝑦 ∨ℋ (𝐴 ∩ 𝐵)))))
2		chincl 31017	. . . . . . . 8 ⊢ ((𝑥 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝑥 ∩ 𝐵) ∈ Cℋ )
3		inss2 4230	. . . . . . . . 9 ⊢ (𝑥 ∩ 𝐵) ⊆ 𝐵
4		sseq1 4008	. . . . . . . . . . 11 ⊢ (𝑦 = (𝑥 ∩ 𝐵) → (𝑦 ⊆ 𝐵 ↔ (𝑥 ∩ 𝐵) ⊆ 𝐵))
5		oveq1 7420	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝑥 ∩ 𝐵) → (𝑦 ∨ℋ 𝐴) = ((𝑥 ∩ 𝐵) ∨ℋ 𝐴))
6	5	ineq1d 4212	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝑥 ∩ 𝐵) → ((𝑦 ∨ℋ 𝐴) ∩ 𝐵) = (((𝑥 ∩ 𝐵) ∨ℋ 𝐴) ∩ 𝐵))
7		oveq1 7420	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝑥 ∩ 𝐵) → (𝑦 ∨ℋ (𝐴 ∩ 𝐵)) = ((𝑥 ∩ 𝐵) ∨ℋ (𝐴 ∩ 𝐵)))
8	6, 7	eqeq12d 2746	. . . . . . . . . . 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 343	. . . . . . . . . 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 3609	. . . . . . . . 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 411	. . . . . 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 3150	. . . 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 3967	. . . . . . . . . . 11 ⊢ (𝑥 ⊆ 𝐵 ↔ 𝑥 = (𝑥 ∩ 𝐵))
17	16	biimpi 215	. . . . . . . . . 10 ⊢ (𝑥 ⊆ 𝐵 → 𝑥 = (𝑥 ∩ 𝐵))
18	17	oveq1d 7428	. . . . . . . . 9 ⊢ (𝑥 ⊆ 𝐵 → (𝑥 ∨ℋ 𝐴) = ((𝑥 ∩ 𝐵) ∨ℋ 𝐴))
19	18	ineq1d 4212	. . . . . . . 8 ⊢ (𝑥 ⊆ 𝐵 → ((𝑥 ∨ℋ 𝐴) ∩ 𝐵) = (((𝑥 ∩ 𝐵) ∨ℋ 𝐴) ∩ 𝐵))
20	17	oveq1d 7428	. . . . . . . 8 ⊢ (𝑥 ⊆ 𝐵 → (𝑥 ∨ℋ (𝐴 ∩ 𝐵)) = ((𝑥 ∩ 𝐵) ∨ℋ (𝐴 ∩ 𝐵)))
21	19, 20	eqeq12d 2746	. . . . . . 7 ⊢ (𝑥 ⊆ 𝐵 → (((𝑥 ∨ℋ 𝐴) ∩ 𝐵) = (𝑥 ∨ℋ (𝐴 ∩ 𝐵)) ↔ (((𝑥 ∩ 𝐵) ∨ℋ 𝐴) ∩ 𝐵) = ((𝑥 ∩ 𝐵) ∨ℋ (𝐴 ∩ 𝐵))))
22	21	biimprcd 249	. . . . . 6 ⊢ ((((𝑥 ∩ 𝐵) ∨ℋ 𝐴) ∩ 𝐵) = ((𝑥 ∩ 𝐵) ∨ℋ (𝐴 ∩ 𝐵)) → (𝑥 ⊆ 𝐵 → ((𝑥 ∨ℋ 𝐴) ∩ 𝐵) = (𝑥 ∨ℋ (𝐴 ∩ 𝐵))))
23	22	ralimi 3081	. . . . 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 4008	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑥 ⊆ 𝐵 ↔ 𝑦 ⊆ 𝐵))
25		oveq1 7420	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑥 ∨ℋ 𝐴) = (𝑦 ∨ℋ 𝐴))
26	25	ineq1d 4212	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ((𝑥 ∨ℋ 𝐴) ∩ 𝐵) = ((𝑦 ∨ℋ 𝐴) ∩ 𝐵))
27		oveq1 7420	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥 ∨ℋ (𝐴 ∩ 𝐵)) = (𝑦 ∨ℋ (𝐴 ∩ 𝐵)))
28	26, 27	eqeq12d 2746	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (((𝑥 ∨ℋ 𝐴) ∩ 𝐵) = (𝑥 ∨ℋ (𝐴 ∩ 𝐵)) ↔ ((𝑦 ∨ℋ 𝐴) ∩ 𝐵) = (𝑦 ∨ℋ (𝐴 ∩ 𝐵))))
29	24, 28	imbi12d 343	. . . . . 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 3232	. . . . 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 217	. . . 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 224	. . 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 480	. 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 278	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 205   ∧ wa 394   = wceq 1539   ∈ wcel 2104  ∀wral 3059   ∩ cin 3948   ⊆ wss 3949   class class class wbr 5149  (class class class)co 7413   C_ℋ cch 30447   ∨_ℋ chj 30451   𝑀_ℋ cmd 30484

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7729  ax-cnex 11170  ax-1cn 11172  ax-addcl 11174  ax-hilex 30517  ax-hfvadd 30518  ax-hv0cl 30521  ax-hfvmul 30523

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7860  df-2nd 7980  df-frecs 8270  df-wrecs 8301  df-recs 8375  df-rdg 8414  df-map 8826  df-nn 12219  df-hlim 30490  df-sh 30725  df-ch 30739  df-md 31798

This theorem is referenced by: (None)