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Theorem mdbr3 30066
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 30063 . 2 ((𝐴C𝐵C ) → (𝐴 𝑀 𝐵 ↔ ∀𝑦C (𝑦𝐵 → ((𝑦 𝐴) ∩ 𝐵) = (𝑦 (𝐴𝐵)))))
2 chincl 29268 . . . . . . . 8 ((𝑥C𝐵C ) → (𝑥𝐵) ∈ C )
3 inss2 4204 . . . . . . . . 9 (𝑥𝐵) ⊆ 𝐵
4 sseq1 3990 . . . . . . . . . . 11 (𝑦 = (𝑥𝐵) → (𝑦𝐵 ↔ (𝑥𝐵) ⊆ 𝐵))
5 oveq1 7155 . . . . . . . . . . . . 13 (𝑦 = (𝑥𝐵) → (𝑦 𝐴) = ((𝑥𝐵) ∨ 𝐴))
65ineq1d 4186 . . . . . . . . . . . 12 (𝑦 = (𝑥𝐵) → ((𝑦 𝐴) ∩ 𝐵) = (((𝑥𝐵) ∨ 𝐴) ∩ 𝐵))
7 oveq1 7155 . . . . . . . . . . . 12 (𝑦 = (𝑥𝐵) → (𝑦 (𝐴𝐵)) = ((𝑥𝐵) ∨ (𝐴𝐵)))
86, 7eqeq12d 2835 . . . . . . . . . . 11 (𝑦 = (𝑥𝐵) → (((𝑦 𝐴) ∩ 𝐵) = (𝑦 (𝐴𝐵)) ↔ (((𝑥𝐵) ∨ 𝐴) ∩ 𝐵) = ((𝑥𝐵) ∨ (𝐴𝐵))))
94, 8imbi12d 347 . . . . . . . . . 10 (𝑦 = (𝑥𝐵) → ((𝑦𝐵 → ((𝑦 𝐴) ∩ 𝐵) = (𝑦 (𝐴𝐵))) ↔ ((𝑥𝐵) ⊆ 𝐵 → (((𝑥𝐵) ∨ 𝐴) ∩ 𝐵) = ((𝑥𝐵) ∨ (𝐴𝐵)))))
109rspcv 3616 . . . . . . . . 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 415 . . . . . 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 3186 . . . 4 (𝐵C → (∀𝑦C (𝑦𝐵 → ((𝑦 𝐴) ∩ 𝐵) = (𝑦 (𝐴𝐵))) → ∀𝑥C (((𝑥𝐵) ∨ 𝐴) ∩ 𝐵) = ((𝑥𝐵) ∨ (𝐴𝐵))))
16 dfss 3951 . . . . . . . . . . 11 (𝑥𝐵𝑥 = (𝑥𝐵))
1716biimpi 218 . . . . . . . . . 10 (𝑥𝐵𝑥 = (𝑥𝐵))
1817oveq1d 7163 . . . . . . . . 9 (𝑥𝐵 → (𝑥 𝐴) = ((𝑥𝐵) ∨ 𝐴))
1918ineq1d 4186 . . . . . . . 8 (𝑥𝐵 → ((𝑥 𝐴) ∩ 𝐵) = (((𝑥𝐵) ∨ 𝐴) ∩ 𝐵))
2017oveq1d 7163 . . . . . . . 8 (𝑥𝐵 → (𝑥 (𝐴𝐵)) = ((𝑥𝐵) ∨ (𝐴𝐵)))
2119, 20eqeq12d 2835 . . . . . . 7 (𝑥𝐵 → (((𝑥 𝐴) ∩ 𝐵) = (𝑥 (𝐴𝐵)) ↔ (((𝑥𝐵) ∨ 𝐴) ∩ 𝐵) = ((𝑥𝐵) ∨ (𝐴𝐵))))
2221biimprcd 252 . . . . . 6 ((((𝑥𝐵) ∨ 𝐴) ∩ 𝐵) = ((𝑥𝐵) ∨ (𝐴𝐵)) → (𝑥𝐵 → ((𝑥 𝐴) ∩ 𝐵) = (𝑥 (𝐴𝐵))))
2322ralimi 3158 . . . . 5 (∀𝑥C (((𝑥𝐵) ∨ 𝐴) ∩ 𝐵) = ((𝑥𝐵) ∨ (𝐴𝐵)) → ∀𝑥C (𝑥𝐵 → ((𝑥 𝐴) ∩ 𝐵) = (𝑥 (𝐴𝐵))))
24 sseq1 3990 . . . . . . 7 (𝑥 = 𝑦 → (𝑥𝐵𝑦𝐵))
25 oveq1 7155 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑥 𝐴) = (𝑦 𝐴))
2625ineq1d 4186 . . . . . . . 8 (𝑥 = 𝑦 → ((𝑥 𝐴) ∩ 𝐵) = ((𝑦 𝐴) ∩ 𝐵))
27 oveq1 7155 . . . . . . . 8 (𝑥 = 𝑦 → (𝑥 (𝐴𝐵)) = (𝑦 (𝐴𝐵)))
2826, 27eqeq12d 2835 . . . . . . 7 (𝑥 = 𝑦 → (((𝑥 𝐴) ∩ 𝐵) = (𝑥 (𝐴𝐵)) ↔ ((𝑦 𝐴) ∩ 𝐵) = (𝑦 (𝐴𝐵))))
2924, 28imbi12d 347 . . . . . 6 (𝑥 = 𝑦 → ((𝑥𝐵 → ((𝑥 𝐴) ∩ 𝐵) = (𝑥 (𝐴𝐵))) ↔ (𝑦𝐵 → ((𝑦 𝐴) ∩ 𝐵) = (𝑦 (𝐴𝐵)))))
3029cbvralvw 3448 . . . . 5 (∀𝑥C (𝑥𝐵 → ((𝑥 𝐴) ∩ 𝐵) = (𝑥 (𝐴𝐵))) ↔ ∀𝑦C (𝑦𝐵 → ((𝑦 𝐴) ∩ 𝐵) = (𝑦 (𝐴𝐵))))
3123, 30sylib 220 . . . 4 (∀𝑥C (((𝑥𝐵) ∨ 𝐴) ∩ 𝐵) = ((𝑥𝐵) ∨ (𝐴𝐵)) → ∀𝑦C (𝑦𝐵 → ((𝑦 𝐴) ∩ 𝐵) = (𝑦 (𝐴𝐵))))
3215, 31impbid1 227 . . 3 (𝐵C → (∀𝑦C (𝑦𝐵 → ((𝑦 𝐴) ∩ 𝐵) = (𝑦 (𝐴𝐵))) ↔ ∀𝑥C (((𝑥𝐵) ∨ 𝐴) ∩ 𝐵) = ((𝑥𝐵) ∨ (𝐴𝐵))))
3332adantl 484 . 2 ((𝐴C𝐵C ) → (∀𝑦C (𝑦𝐵 → ((𝑦 𝐴) ∩ 𝐵) = (𝑦 (𝐴𝐵))) ↔ ∀𝑥C (((𝑥𝐵) ∨ 𝐴) ∩ 𝐵) = ((𝑥𝐵) ∨ (𝐴𝐵))))
341, 33bitrd 281 1 ((𝐴C𝐵C ) → (𝐴 𝑀 𝐵 ↔ ∀𝑥C (((𝑥𝐵) ∨ 𝐴) ∩ 𝐵) = ((𝑥𝐵) ∨ (𝐴𝐵))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398   = wceq 1530  wcel 2107  wral 3136  cin 3933  wss 3934   class class class wbr 5057  (class class class)co 7148   C cch 28698   chj 28702   𝑀 cmd 28735
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453  ax-cnex 10585  ax-1cn 10587  ax-addcl 10589  ax-hilex 28768  ax-hfvadd 28769  ax-hv0cl 28772  ax-hfvmul 28774
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-reu 3143  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7573  df-wrecs 7939  df-recs 8000  df-rdg 8038  df-map 8400  df-nn 11631  df-hlim 28741  df-sh 28976  df-ch 28990  df-md 30049
This theorem is referenced by: (None)
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