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Theorem ntrin 22557
Description: A pairwise intersection of interiors is the interior of the intersection. This does not always hold for arbitrary intersections. (Contributed by Jeff Hankins, 31-Aug-2009.)
Hypothesis
Ref Expression
clscld.1 𝑋 = βˆͺ 𝐽
Assertion
Ref Expression
ntrin ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋 ∧ 𝐡 βŠ† 𝑋) β†’ ((intβ€˜π½)β€˜(𝐴 ∩ 𝐡)) = (((intβ€˜π½)β€˜π΄) ∩ ((intβ€˜π½)β€˜π΅)))
Proof of Theorem ntrin
StepHypRef Expression
1 inss1 4228 . . . . 5 (𝐴 ∩ 𝐡) βŠ† 𝐴
2 clscld.1 . . . . . 6 𝑋 = βˆͺ 𝐽
32ntrss 22551 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋 ∧ (𝐴 ∩ 𝐡) βŠ† 𝐴) β†’ ((intβ€˜π½)β€˜(𝐴 ∩ 𝐡)) βŠ† ((intβ€˜π½)β€˜π΄))
41, 3mp3an3 1451 . . . 4 ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋) β†’ ((intβ€˜π½)β€˜(𝐴 ∩ 𝐡)) βŠ† ((intβ€˜π½)β€˜π΄))
543adant3 1133 . . 3 ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋 ∧ 𝐡 βŠ† 𝑋) β†’ ((intβ€˜π½)β€˜(𝐴 ∩ 𝐡)) βŠ† ((intβ€˜π½)β€˜π΄))
6 inss2 4229 . . . . 5 (𝐴 ∩ 𝐡) βŠ† 𝐡
72ntrss 22551 . . . . 5 ((𝐽 ∈ Top ∧ 𝐡 βŠ† 𝑋 ∧ (𝐴 ∩ 𝐡) βŠ† 𝐡) β†’ ((intβ€˜π½)β€˜(𝐴 ∩ 𝐡)) βŠ† ((intβ€˜π½)β€˜π΅))
86, 7mp3an3 1451 . . . 4 ((𝐽 ∈ Top ∧ 𝐡 βŠ† 𝑋) β†’ ((intβ€˜π½)β€˜(𝐴 ∩ 𝐡)) βŠ† ((intβ€˜π½)β€˜π΅))
983adant2 1132 . . 3 ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋 ∧ 𝐡 βŠ† 𝑋) β†’ ((intβ€˜π½)β€˜(𝐴 ∩ 𝐡)) βŠ† ((intβ€˜π½)β€˜π΅))
105, 9ssind 4232 . 2 ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋 ∧ 𝐡 βŠ† 𝑋) β†’ ((intβ€˜π½)β€˜(𝐴 ∩ 𝐡)) βŠ† (((intβ€˜π½)β€˜π΄) ∩ ((intβ€˜π½)β€˜π΅)))
11 simp1 1137 . . 3 ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋 ∧ 𝐡 βŠ† 𝑋) β†’ 𝐽 ∈ Top)
12 ssinss1 4237 . . . 4 (𝐴 βŠ† 𝑋 β†’ (𝐴 ∩ 𝐡) βŠ† 𝑋)
13123ad2ant2 1135 . . 3 ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋 ∧ 𝐡 βŠ† 𝑋) β†’ (𝐴 ∩ 𝐡) βŠ† 𝑋)
142ntropn 22545 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋) β†’ ((intβ€˜π½)β€˜π΄) ∈ 𝐽)
15143adant3 1133 . . . 4 ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋 ∧ 𝐡 βŠ† 𝑋) β†’ ((intβ€˜π½)β€˜π΄) ∈ 𝐽)
162ntropn 22545 . . . . 5 ((𝐽 ∈ Top ∧ 𝐡 βŠ† 𝑋) β†’ ((intβ€˜π½)β€˜π΅) ∈ 𝐽)
17163adant2 1132 . . . 4 ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋 ∧ 𝐡 βŠ† 𝑋) β†’ ((intβ€˜π½)β€˜π΅) ∈ 𝐽)
18 inopn 22393 . . . 4 ((𝐽 ∈ Top ∧ ((intβ€˜π½)β€˜π΄) ∈ 𝐽 ∧ ((intβ€˜π½)β€˜π΅) ∈ 𝐽) β†’ (((intβ€˜π½)β€˜π΄) ∩ ((intβ€˜π½)β€˜π΅)) ∈ 𝐽)
1911, 15, 17, 18syl3anc 1372 . . 3 ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋 ∧ 𝐡 βŠ† 𝑋) β†’ (((intβ€˜π½)β€˜π΄) ∩ ((intβ€˜π½)β€˜π΅)) ∈ 𝐽)
20 inss1 4228 . . . . 5 (((intβ€˜π½)β€˜π΄) ∩ ((intβ€˜π½)β€˜π΅)) βŠ† ((intβ€˜π½)β€˜π΄)
212ntrss2 22553 . . . . . 6 ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋) β†’ ((intβ€˜π½)β€˜π΄) βŠ† 𝐴)
22213adant3 1133 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋 ∧ 𝐡 βŠ† 𝑋) β†’ ((intβ€˜π½)β€˜π΄) βŠ† 𝐴)
2320, 22sstrid 3993 . . . 4 ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋 ∧ 𝐡 βŠ† 𝑋) β†’ (((intβ€˜π½)β€˜π΄) ∩ ((intβ€˜π½)β€˜π΅)) βŠ† 𝐴)
24 inss2 4229 . . . . 5 (((intβ€˜π½)β€˜π΄) ∩ ((intβ€˜π½)β€˜π΅)) βŠ† ((intβ€˜π½)β€˜π΅)
252ntrss2 22553 . . . . . 6 ((𝐽 ∈ Top ∧ 𝐡 βŠ† 𝑋) β†’ ((intβ€˜π½)β€˜π΅) βŠ† 𝐡)
26253adant2 1132 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋 ∧ 𝐡 βŠ† 𝑋) β†’ ((intβ€˜π½)β€˜π΅) βŠ† 𝐡)
2724, 26sstrid 3993 . . . 4 ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋 ∧ 𝐡 βŠ† 𝑋) β†’ (((intβ€˜π½)β€˜π΄) ∩ ((intβ€˜π½)β€˜π΅)) βŠ† 𝐡)
2823, 27ssind 4232 . . 3 ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋 ∧ 𝐡 βŠ† 𝑋) β†’ (((intβ€˜π½)β€˜π΄) ∩ ((intβ€˜π½)β€˜π΅)) βŠ† (𝐴 ∩ 𝐡))
292ssntr 22554 . . 3 (((𝐽 ∈ Top ∧ (𝐴 ∩ 𝐡) βŠ† 𝑋) ∧ ((((intβ€˜π½)β€˜π΄) ∩ ((intβ€˜π½)β€˜π΅)) ∈ 𝐽 ∧ (((intβ€˜π½)β€˜π΄) ∩ ((intβ€˜π½)β€˜π΅)) βŠ† (𝐴 ∩ 𝐡))) β†’ (((intβ€˜π½)β€˜π΄) ∩ ((intβ€˜π½)β€˜π΅)) βŠ† ((intβ€˜π½)β€˜(𝐴 ∩ 𝐡)))
3011, 13, 19, 28, 29syl22anc 838 . 2 ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋 ∧ 𝐡 βŠ† 𝑋) β†’ (((intβ€˜π½)β€˜π΄) ∩ ((intβ€˜π½)β€˜π΅)) βŠ† ((intβ€˜π½)β€˜(𝐴 ∩ 𝐡)))
3110, 30eqssd 3999 1 ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋 ∧ 𝐡 βŠ† 𝑋) β†’ ((intβ€˜π½)β€˜(𝐴 ∩ 𝐡)) = (((intβ€˜π½)β€˜π΄) ∩ ((intβ€˜π½)β€˜π΅)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   ∩ cin 3947   βŠ† wss 3948  βˆͺ cuni 4908  β€˜cfv 6541  Topctop 22387  intcnt 22513
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-iin 5000  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-top 22388  df-cld 22515  df-ntr 22516  df-cls 22517
This theorem is referenced by:  dvreslem  25418  dvaddbr  25447  dvmulbr  25448  gg-dvmulbr  35164  clsun  35202  neiin  35206
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