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Theorem mdbr3 31084
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 ((𝐴C𝐵C ) → (𝐴 𝑀 𝐵 ↔ ∀𝑥C (((𝑥𝐵) ∨ 𝐴) ∩ 𝐵) = ((𝑥𝐵) ∨ (𝐴𝐵))))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem mdbr3
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 mdbr 31081 . 2 ((𝐴C𝐵C ) → (𝐴 𝑀 𝐵 ↔ ∀𝑦C (𝑦𝐵 → ((𝑦 𝐴) ∩ 𝐵) = (𝑦 (𝐴𝐵)))))
2 chincl 30286 . . . . . . . 8 ((𝑥C𝐵C ) → (𝑥𝐵) ∈ C )
3 inss2 4187 . . . . . . . . 9 (𝑥𝐵) ⊆ 𝐵
4 sseq1 3967 . . . . . . . . . . 11 (𝑦 = (𝑥𝐵) → (𝑦𝐵 ↔ (𝑥𝐵) ⊆ 𝐵))
5 oveq1 7358 . . . . . . . . . . . . 13 (𝑦 = (𝑥𝐵) → (𝑦 𝐴) = ((𝑥𝐵) ∨ 𝐴))
65ineq1d 4169 . . . . . . . . . . . 12 (𝑦 = (𝑥𝐵) → ((𝑦 𝐴) ∩ 𝐵) = (((𝑥𝐵) ∨ 𝐴) ∩ 𝐵))
7 oveq1 7358 . . . . . . . . . . . 12 (𝑦 = (𝑥𝐵) → (𝑦 (𝐴𝐵)) = ((𝑥𝐵) ∨ (𝐴𝐵)))
86, 7eqeq12d 2752 . . . . . . . . . . 11 (𝑦 = (𝑥𝐵) → (((𝑦 𝐴) ∩ 𝐵) = (𝑦 (𝐴𝐵)) ↔ (((𝑥𝐵) ∨ 𝐴) ∩ 𝐵) = ((𝑥𝐵) ∨ (𝐴𝐵))))
94, 8imbi12d 344 . . . . . . . . . 10 (𝑦 = (𝑥𝐵) → ((𝑦𝐵 → ((𝑦 𝐴) ∩ 𝐵) = (𝑦 (𝐴𝐵))) ↔ ((𝑥𝐵) ⊆ 𝐵 → (((𝑥𝐵) ∨ 𝐴) ∩ 𝐵) = ((𝑥𝐵) ∨ (𝐴𝐵)))))
109rspcv 3575 . . . . . . . . 9 ((𝑥𝐵) ∈ C → (∀𝑦C (𝑦𝐵 → ((𝑦 𝐴) ∩ 𝐵) = (𝑦 (𝐴𝐵))) → ((𝑥𝐵) ⊆ 𝐵 → (((𝑥𝐵) ∨ 𝐴) ∩ 𝐵) = ((𝑥𝐵) ∨ (𝐴𝐵)))))
113, 10mpii 46 . . . . . . . 8 ((𝑥𝐵) ∈ C → (∀𝑦C (𝑦𝐵 → ((𝑦 𝐴) ∩ 𝐵) = (𝑦 (𝐴𝐵))) → (((𝑥𝐵) ∨ 𝐴) ∩ 𝐵) = ((𝑥𝐵) ∨ (𝐴𝐵))))
122, 11syl 17 . . . . . . 7 ((𝑥C𝐵C ) → (∀𝑦C (𝑦𝐵 → ((𝑦 𝐴) ∩ 𝐵) = (𝑦 (𝐴𝐵))) → (((𝑥𝐵) ∨ 𝐴) ∩ 𝐵) = ((𝑥𝐵) ∨ (𝐴𝐵))))
1312ex 413 . . . . . 6 (𝑥C → (𝐵C → (∀𝑦C (𝑦𝐵 → ((𝑦 𝐴) ∩ 𝐵) = (𝑦 (𝐴𝐵))) → (((𝑥𝐵) ∨ 𝐴) ∩ 𝐵) = ((𝑥𝐵) ∨ (𝐴𝐵)))))
1413com3l 89 . . . . 5 (𝐵C → (∀𝑦C (𝑦𝐵 → ((𝑦 𝐴) ∩ 𝐵) = (𝑦 (𝐴𝐵))) → (𝑥C → (((𝑥𝐵) ∨ 𝐴) ∩ 𝐵) = ((𝑥𝐵) ∨ (𝐴𝐵)))))
1514ralrimdv 3147 . . . 4 (𝐵C → (∀𝑦C (𝑦𝐵 → ((𝑦 𝐴) ∩ 𝐵) = (𝑦 (𝐴𝐵))) → ∀𝑥C (((𝑥𝐵) ∨ 𝐴) ∩ 𝐵) = ((𝑥𝐵) ∨ (𝐴𝐵))))
16 dfss 3926 . . . . . . . . . . 11 (𝑥𝐵𝑥 = (𝑥𝐵))
1716biimpi 215 . . . . . . . . . 10 (𝑥𝐵𝑥 = (𝑥𝐵))
1817oveq1d 7366 . . . . . . . . 9 (𝑥𝐵 → (𝑥 𝐴) = ((𝑥𝐵) ∨ 𝐴))
1918ineq1d 4169 . . . . . . . 8 (𝑥𝐵 → ((𝑥 𝐴) ∩ 𝐵) = (((𝑥𝐵) ∨ 𝐴) ∩ 𝐵))
2017oveq1d 7366 . . . . . . . 8 (𝑥𝐵 → (𝑥 (𝐴𝐵)) = ((𝑥𝐵) ∨ (𝐴𝐵)))
2119, 20eqeq12d 2752 . . . . . . 7 (𝑥𝐵 → (((𝑥 𝐴) ∩ 𝐵) = (𝑥 (𝐴𝐵)) ↔ (((𝑥𝐵) ∨ 𝐴) ∩ 𝐵) = ((𝑥𝐵) ∨ (𝐴𝐵))))
2221biimprcd 249 . . . . . 6 ((((𝑥𝐵) ∨ 𝐴) ∩ 𝐵) = ((𝑥𝐵) ∨ (𝐴𝐵)) → (𝑥𝐵 → ((𝑥 𝐴) ∩ 𝐵) = (𝑥 (𝐴𝐵))))
2322ralimi 3084 . . . . 5 (∀𝑥C (((𝑥𝐵) ∨ 𝐴) ∩ 𝐵) = ((𝑥𝐵) ∨ (𝐴𝐵)) → ∀𝑥C (𝑥𝐵 → ((𝑥 𝐴) ∩ 𝐵) = (𝑥 (𝐴𝐵))))
24 sseq1 3967 . . . . . . 7 (𝑥 = 𝑦 → (𝑥𝐵𝑦𝐵))
25 oveq1 7358 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑥 𝐴) = (𝑦 𝐴))
2625ineq1d 4169 . . . . . . . 8 (𝑥 = 𝑦 → ((𝑥 𝐴) ∩ 𝐵) = ((𝑦 𝐴) ∩ 𝐵))
27 oveq1 7358 . . . . . . . 8 (𝑥 = 𝑦 → (𝑥 (𝐴𝐵)) = (𝑦 (𝐴𝐵)))
2826, 27eqeq12d 2752 . . . . . . 7 (𝑥 = 𝑦 → (((𝑥 𝐴) ∩ 𝐵) = (𝑥 (𝐴𝐵)) ↔ ((𝑦 𝐴) ∩ 𝐵) = (𝑦 (𝐴𝐵))))
2924, 28imbi12d 344 . . . . . 6 (𝑥 = 𝑦 → ((𝑥𝐵 → ((𝑥 𝐴) ∩ 𝐵) = (𝑥 (𝐴𝐵))) ↔ (𝑦𝐵 → ((𝑦 𝐴) ∩ 𝐵) = (𝑦 (𝐴𝐵)))))
3029cbvralvw 3223 . . . . 5 (∀𝑥C (𝑥𝐵 → ((𝑥 𝐴) ∩ 𝐵) = (𝑥 (𝐴𝐵))) ↔ ∀𝑦C (𝑦𝐵 → ((𝑦 𝐴) ∩ 𝐵) = (𝑦 (𝐴𝐵))))
3123, 30sylib 217 . . . 4 (∀𝑥C (((𝑥𝐵) ∨ 𝐴) ∩ 𝐵) = ((𝑥𝐵) ∨ (𝐴𝐵)) → ∀𝑦C (𝑦𝐵 → ((𝑦 𝐴) ∩ 𝐵) = (𝑦 (𝐴𝐵))))
3215, 31impbid1 224 . . 3 (𝐵C → (∀𝑦C (𝑦𝐵 → ((𝑦 𝐴) ∩ 𝐵) = (𝑦 (𝐴𝐵))) ↔ ∀𝑥C (((𝑥𝐵) ∨ 𝐴) ∩ 𝐵) = ((𝑥𝐵) ∨ (𝐴𝐵))))
3332adantl 482 . 2 ((𝐴C𝐵C ) → (∀𝑦C (𝑦𝐵 → ((𝑦 𝐴) ∩ 𝐵) = (𝑦 (𝐴𝐵))) ↔ ∀𝑥C (((𝑥𝐵) ∨ 𝐴) ∩ 𝐵) = ((𝑥𝐵) ∨ (𝐴𝐵))))
341, 33bitrd 278 1 ((𝐴C𝐵C ) → (𝐴 𝑀 𝐵 ↔ ∀𝑥C (((𝑥𝐵) ∨ 𝐴) ∩ 𝐵) = ((𝑥𝐵) ∨ (𝐴𝐵))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  wral 3062  cin 3907  wss 3908   class class class wbr 5103  (class class class)co 7351   C cch 29716   chj 29720   𝑀 cmd 29753
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5240  ax-sep 5254  ax-nul 5261  ax-pow 5318  ax-pr 5382  ax-un 7664  ax-cnex 11065  ax-1cn 11067  ax-addcl 11069  ax-hilex 29786  ax-hfvadd 29787  ax-hv0cl 29790  ax-hfvmul 29792
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3352  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-int 4906  df-iun 4954  df-br 5104  df-opab 5166  df-mpt 5187  df-tr 5221  df-id 5529  df-eprel 5535  df-po 5543  df-so 5544  df-fr 5586  df-we 5588  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6251  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-ov 7354  df-oprab 7355  df-mpo 7356  df-om 7795  df-2nd 7914  df-frecs 8204  df-wrecs 8235  df-recs 8309  df-rdg 8348  df-map 8725  df-nn 12112  df-hlim 29759  df-sh 29994  df-ch 30008  df-md 31067
This theorem is referenced by: (None)
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