Theorem mdbr3 30232

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 30229	. 2 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 𝑀ℋ 𝐵 ↔ ∀𝑦 ∈ Cℋ (𝑦 ⊆ 𝐵 → ((𝑦 ∨ℋ 𝐴) ∩ 𝐵) = (𝑦 ∨ℋ (𝐴 ∩ 𝐵)))))
2		chincl 29434	. . . . . . . 8 ⊢ ((𝑥 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝑥 ∩ 𝐵) ∈ Cℋ )
3		inss2 4121	. . . . . . . . 9 ⊢ (𝑥 ∩ 𝐵) ⊆ 𝐵
4		sseq1 3903	. . . . . . . . . . 11 ⊢ (𝑦 = (𝑥 ∩ 𝐵) → (𝑦 ⊆ 𝐵 ↔ (𝑥 ∩ 𝐵) ⊆ 𝐵))
5		oveq1 7178	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝑥 ∩ 𝐵) → (𝑦 ∨ℋ 𝐴) = ((𝑥 ∩ 𝐵) ∨ℋ 𝐴))
6	5	ineq1d 4103	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝑥 ∩ 𝐵) → ((𝑦 ∨ℋ 𝐴) ∩ 𝐵) = (((𝑥 ∩ 𝐵) ∨ℋ 𝐴) ∩ 𝐵))
7		oveq1 7178	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝑥 ∩ 𝐵) → (𝑦 ∨ℋ (𝐴 ∩ 𝐵)) = ((𝑥 ∩ 𝐵) ∨ℋ (𝐴 ∩ 𝐵)))
8	6, 7	eqeq12d 2754	. . . . . . . . . . 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 348	. . . . . . . . . 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 3522	. . . . . . . . 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 416	. . . . . 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 3100	. . . 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 3862	. . . . . . . . . . 11 ⊢ (𝑥 ⊆ 𝐵 ↔ 𝑥 = (𝑥 ∩ 𝐵))
17	16	biimpi 219	. . . . . . . . . 10 ⊢ (𝑥 ⊆ 𝐵 → 𝑥 = (𝑥 ∩ 𝐵))
18	17	oveq1d 7186	. . . . . . . . 9 ⊢ (𝑥 ⊆ 𝐵 → (𝑥 ∨ℋ 𝐴) = ((𝑥 ∩ 𝐵) ∨ℋ 𝐴))
19	18	ineq1d 4103	. . . . . . . 8 ⊢ (𝑥 ⊆ 𝐵 → ((𝑥 ∨ℋ 𝐴) ∩ 𝐵) = (((𝑥 ∩ 𝐵) ∨ℋ 𝐴) ∩ 𝐵))
20	17	oveq1d 7186	. . . . . . . 8 ⊢ (𝑥 ⊆ 𝐵 → (𝑥 ∨ℋ (𝐴 ∩ 𝐵)) = ((𝑥 ∩ 𝐵) ∨ℋ (𝐴 ∩ 𝐵)))
21	19, 20	eqeq12d 2754	. . . . . . 7 ⊢ (𝑥 ⊆ 𝐵 → (((𝑥 ∨ℋ 𝐴) ∩ 𝐵) = (𝑥 ∨ℋ (𝐴 ∩ 𝐵)) ↔ (((𝑥 ∩ 𝐵) ∨ℋ 𝐴) ∩ 𝐵) = ((𝑥 ∩ 𝐵) ∨ℋ (𝐴 ∩ 𝐵))))
22	21	biimprcd 253	. . . . . 6 ⊢ ((((𝑥 ∩ 𝐵) ∨ℋ 𝐴) ∩ 𝐵) = ((𝑥 ∩ 𝐵) ∨ℋ (𝐴 ∩ 𝐵)) → (𝑥 ⊆ 𝐵 → ((𝑥 ∨ℋ 𝐴) ∩ 𝐵) = (𝑥 ∨ℋ (𝐴 ∩ 𝐵))))
23	22	ralimi 3075	. . . . 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 3903	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑥 ⊆ 𝐵 ↔ 𝑦 ⊆ 𝐵))
25		oveq1 7178	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑥 ∨ℋ 𝐴) = (𝑦 ∨ℋ 𝐴))
26	25	ineq1d 4103	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ((𝑥 ∨ℋ 𝐴) ∩ 𝐵) = ((𝑦 ∨ℋ 𝐴) ∩ 𝐵))
27		oveq1 7178	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥 ∨ℋ (𝐴 ∩ 𝐵)) = (𝑦 ∨ℋ (𝐴 ∩ 𝐵)))
28	26, 27	eqeq12d 2754	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (((𝑥 ∨ℋ 𝐴) ∩ 𝐵) = (𝑥 ∨ℋ (𝐴 ∩ 𝐵)) ↔ ((𝑦 ∨ℋ 𝐴) ∩ 𝐵) = (𝑦 ∨ℋ (𝐴 ∩ 𝐵))))
29	24, 28	imbi12d 348	. . . . . 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 3349	. . . . 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 221	. . . 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 228	. . 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 485	. 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 282	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 209   ∧ wa 399   = wceq 1542   ∈ wcel 2113  ∀wral 3053   ∩ cin 3843   ⊆ wss 3844   class class class wbr 5031  (class class class)co 7171   C_ℋ cch 28864   ∨_ℋ chj 28868   𝑀_ℋ cmd 28901

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-rep 5155  ax-sep 5168  ax-nul 5175  ax-pow 5233  ax-pr 5297  ax-un 7480  ax-cnex 10672  ax-1cn 10674  ax-addcl 10676  ax-hilex 28934  ax-hfvadd 28935  ax-hv0cl 28938  ax-hfvmul 28940

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rab 3062  df-v 3400  df-sbc 3683  df-csb 3792  df-dif 3847  df-un 3849  df-in 3851  df-ss 3861  df-pss 3863  df-nul 4213  df-if 4416  df-pw 4491  df-sn 4518  df-pr 4520  df-tp 4522  df-op 4524  df-uni 4798  df-int 4838  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-tr 5138  df-id 5430  df-eprel 5435  df-po 5443  df-so 5444  df-fr 5484  df-we 5486  df-xp 5532  df-rel 5533  df-cnv 5534  df-co 5535  df-dm 5536  df-rn 5537  df-res 5538  df-ima 5539  df-pred 6130  df-ord 6176  df-on 6177  df-lim 6178  df-suc 6179  df-iota 6298  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-ov 7174  df-oprab 7175  df-mpo 7176  df-om 7601  df-wrecs 7977  df-recs 8038  df-rdg 8076  df-map 8440  df-nn 11718  df-hlim 28907  df-sh 29142  df-ch 29156  df-md 30215

This theorem is referenced by: (None)