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Theorem ntrin 13663
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 3357 . . . . 5 (𝐴 ∩ 𝐡) βŠ† 𝐴
2 clscld.1 . . . . . 6 𝑋 = βˆͺ 𝐽
32ntrss 13658 . . . . 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 1326 . . . 4 ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋) β†’ ((intβ€˜π½)β€˜(𝐴 ∩ 𝐡)) βŠ† ((intβ€˜π½)β€˜π΄))
543adant3 1017 . . 3 ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋 ∧ 𝐡 βŠ† 𝑋) β†’ ((intβ€˜π½)β€˜(𝐴 ∩ 𝐡)) βŠ† ((intβ€˜π½)β€˜π΄))
6 inss2 3358 . . . . 5 (𝐴 ∩ 𝐡) βŠ† 𝐡
72ntrss 13658 . . . . 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 1326 . . . 4 ((𝐽 ∈ Top ∧ 𝐡 βŠ† 𝑋) β†’ ((intβ€˜π½)β€˜(𝐴 ∩ 𝐡)) βŠ† ((intβ€˜π½)β€˜π΅))
983adant2 1016 . . 3 ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋 ∧ 𝐡 βŠ† 𝑋) β†’ ((intβ€˜π½)β€˜(𝐴 ∩ 𝐡)) βŠ† ((intβ€˜π½)β€˜π΅))
105, 9ssind 3361 . 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 997 . . 3 ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋 ∧ 𝐡 βŠ† 𝑋) β†’ 𝐽 ∈ Top)
12 ssinss1 3366 . . . 4 (𝐴 βŠ† 𝑋 β†’ (𝐴 ∩ 𝐡) βŠ† 𝑋)
13123ad2ant2 1019 . . 3 ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋 ∧ 𝐡 βŠ† 𝑋) β†’ (𝐴 ∩ 𝐡) βŠ† 𝑋)
142ntropn 13656 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋) β†’ ((intβ€˜π½)β€˜π΄) ∈ 𝐽)
15143adant3 1017 . . . 4 ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋 ∧ 𝐡 βŠ† 𝑋) β†’ ((intβ€˜π½)β€˜π΄) ∈ 𝐽)
162ntropn 13656 . . . . 5 ((𝐽 ∈ Top ∧ 𝐡 βŠ† 𝑋) β†’ ((intβ€˜π½)β€˜π΅) ∈ 𝐽)
17163adant2 1016 . . . 4 ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋 ∧ 𝐡 βŠ† 𝑋) β†’ ((intβ€˜π½)β€˜π΅) ∈ 𝐽)
18 inopn 13542 . . . 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 1238 . . 3 ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋 ∧ 𝐡 βŠ† 𝑋) β†’ (((intβ€˜π½)β€˜π΄) ∩ ((intβ€˜π½)β€˜π΅)) ∈ 𝐽)
20 inss1 3357 . . . . 5 (((intβ€˜π½)β€˜π΄) ∩ ((intβ€˜π½)β€˜π΅)) βŠ† ((intβ€˜π½)β€˜π΄)
212ntrss2 13660 . . . . . 6 ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋) β†’ ((intβ€˜π½)β€˜π΄) βŠ† 𝐴)
22213adant3 1017 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋 ∧ 𝐡 βŠ† 𝑋) β†’ ((intβ€˜π½)β€˜π΄) βŠ† 𝐴)
2320, 22sstrid 3168 . . . 4 ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋 ∧ 𝐡 βŠ† 𝑋) β†’ (((intβ€˜π½)β€˜π΄) ∩ ((intβ€˜π½)β€˜π΅)) βŠ† 𝐴)
24 inss2 3358 . . . . 5 (((intβ€˜π½)β€˜π΄) ∩ ((intβ€˜π½)β€˜π΅)) βŠ† ((intβ€˜π½)β€˜π΅)
252ntrss2 13660 . . . . . 6 ((𝐽 ∈ Top ∧ 𝐡 βŠ† 𝑋) β†’ ((intβ€˜π½)β€˜π΅) βŠ† 𝐡)
26253adant2 1016 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋 ∧ 𝐡 βŠ† 𝑋) β†’ ((intβ€˜π½)β€˜π΅) βŠ† 𝐡)
2724, 26sstrid 3168 . . . 4 ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋 ∧ 𝐡 βŠ† 𝑋) β†’ (((intβ€˜π½)β€˜π΄) ∩ ((intβ€˜π½)β€˜π΅)) βŠ† 𝐡)
2823, 27ssind 3361 . . 3 ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋 ∧ 𝐡 βŠ† 𝑋) β†’ (((intβ€˜π½)β€˜π΄) ∩ ((intβ€˜π½)β€˜π΅)) βŠ† (𝐴 ∩ 𝐡))
292ssntr 13661 . . 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 1239 . 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 3174 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 set class
Syntax hints:   β†’ wi 4   ∧ w3a 978   = wceq 1353   ∈ wcel 2148   ∩ cin 3130   βŠ† wss 3131  βˆͺ cuni 3811  β€˜cfv 5218  Topctop 13536  intcnt 13632
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-top 13537  df-ntr 13635
This theorem is referenced by: (None)
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