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Theorem clsun 35208
Description: A pairwise union of closures is the closure of the union. (Contributed by Jeff Hankins, 31-Aug-2009.)
Hypothesis
Ref Expression
clsun.1 𝑋 = βˆͺ 𝐽
Assertion
Ref Expression
clsun ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋 ∧ 𝐡 βŠ† 𝑋) β†’ ((clsβ€˜π½)β€˜(𝐴 βˆͺ 𝐡)) = (((clsβ€˜π½)β€˜π΄) βˆͺ ((clsβ€˜π½)β€˜π΅)))
Proof of Theorem clsun
StepHypRef Expression
1 difundi 4279 . . . . . 6 (𝑋 βˆ– (𝐴 βˆͺ 𝐡)) = ((𝑋 βˆ– 𝐴) ∩ (𝑋 βˆ– 𝐡))
21fveq2i 6894 . . . . 5 ((intβ€˜π½)β€˜(𝑋 βˆ– (𝐴 βˆͺ 𝐡))) = ((intβ€˜π½)β€˜((𝑋 βˆ– 𝐴) ∩ (𝑋 βˆ– 𝐡)))
3 difss 4131 . . . . . . 7 (𝑋 βˆ– 𝐴) βŠ† 𝑋
4 difss 4131 . . . . . . 7 (𝑋 βˆ– 𝐡) βŠ† 𝑋
5 clsun.1 . . . . . . . 8 𝑋 = βˆͺ 𝐽
65ntrin 22564 . . . . . . 7 ((𝐽 ∈ Top ∧ (𝑋 βˆ– 𝐴) βŠ† 𝑋 ∧ (𝑋 βˆ– 𝐡) βŠ† 𝑋) β†’ ((intβ€˜π½)β€˜((𝑋 βˆ– 𝐴) ∩ (𝑋 βˆ– 𝐡))) = (((intβ€˜π½)β€˜(𝑋 βˆ– 𝐴)) ∩ ((intβ€˜π½)β€˜(𝑋 βˆ– 𝐡))))
73, 4, 6mp3an23 1453 . . . . . 6 (𝐽 ∈ Top β†’ ((intβ€˜π½)β€˜((𝑋 βˆ– 𝐴) ∩ (𝑋 βˆ– 𝐡))) = (((intβ€˜π½)β€˜(𝑋 βˆ– 𝐴)) ∩ ((intβ€˜π½)β€˜(𝑋 βˆ– 𝐡))))
873ad2ant1 1133 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋 ∧ 𝐡 βŠ† 𝑋) β†’ ((intβ€˜π½)β€˜((𝑋 βˆ– 𝐴) ∩ (𝑋 βˆ– 𝐡))) = (((intβ€˜π½)β€˜(𝑋 βˆ– 𝐴)) ∩ ((intβ€˜π½)β€˜(𝑋 βˆ– 𝐡))))
92, 8eqtrid 2784 . . . 4 ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋 ∧ 𝐡 βŠ† 𝑋) β†’ ((intβ€˜π½)β€˜(𝑋 βˆ– (𝐴 βˆͺ 𝐡))) = (((intβ€˜π½)β€˜(𝑋 βˆ– 𝐴)) ∩ ((intβ€˜π½)β€˜(𝑋 βˆ– 𝐡))))
10 simp1 1136 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋 ∧ 𝐡 βŠ† 𝑋) β†’ 𝐽 ∈ Top)
11 unss 4184 . . . . . . 7 ((𝐴 βŠ† 𝑋 ∧ 𝐡 βŠ† 𝑋) ↔ (𝐴 βˆͺ 𝐡) βŠ† 𝑋)
1211biimpi 215 . . . . . 6 ((𝐴 βŠ† 𝑋 ∧ 𝐡 βŠ† 𝑋) β†’ (𝐴 βˆͺ 𝐡) βŠ† 𝑋)
13123adant1 1130 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋 ∧ 𝐡 βŠ† 𝑋) β†’ (𝐴 βˆͺ 𝐡) βŠ† 𝑋)
145ntrdif 22555 . . . . 5 ((𝐽 ∈ Top ∧ (𝐴 βˆͺ 𝐡) βŠ† 𝑋) β†’ ((intβ€˜π½)β€˜(𝑋 βˆ– (𝐴 βˆͺ 𝐡))) = (𝑋 βˆ– ((clsβ€˜π½)β€˜(𝐴 βˆͺ 𝐡))))
1510, 13, 14syl2anc 584 . . . 4 ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋 ∧ 𝐡 βŠ† 𝑋) β†’ ((intβ€˜π½)β€˜(𝑋 βˆ– (𝐴 βˆͺ 𝐡))) = (𝑋 βˆ– ((clsβ€˜π½)β€˜(𝐴 βˆͺ 𝐡))))
165ntrdif 22555 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋) β†’ ((intβ€˜π½)β€˜(𝑋 βˆ– 𝐴)) = (𝑋 βˆ– ((clsβ€˜π½)β€˜π΄)))
17163adant3 1132 . . . . . 6 ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋 ∧ 𝐡 βŠ† 𝑋) β†’ ((intβ€˜π½)β€˜(𝑋 βˆ– 𝐴)) = (𝑋 βˆ– ((clsβ€˜π½)β€˜π΄)))
185ntrdif 22555 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝐡 βŠ† 𝑋) β†’ ((intβ€˜π½)β€˜(𝑋 βˆ– 𝐡)) = (𝑋 βˆ– ((clsβ€˜π½)β€˜π΅)))
19183adant2 1131 . . . . . 6 ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋 ∧ 𝐡 βŠ† 𝑋) β†’ ((intβ€˜π½)β€˜(𝑋 βˆ– 𝐡)) = (𝑋 βˆ– ((clsβ€˜π½)β€˜π΅)))
2017, 19ineq12d 4213 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋 ∧ 𝐡 βŠ† 𝑋) β†’ (((intβ€˜π½)β€˜(𝑋 βˆ– 𝐴)) ∩ ((intβ€˜π½)β€˜(𝑋 βˆ– 𝐡))) = ((𝑋 βˆ– ((clsβ€˜π½)β€˜π΄)) ∩ (𝑋 βˆ– ((clsβ€˜π½)β€˜π΅))))
21 difundi 4279 . . . . 5 (𝑋 βˆ– (((clsβ€˜π½)β€˜π΄) βˆͺ ((clsβ€˜π½)β€˜π΅))) = ((𝑋 βˆ– ((clsβ€˜π½)β€˜π΄)) ∩ (𝑋 βˆ– ((clsβ€˜π½)β€˜π΅)))
2220, 21eqtr4di 2790 . . . 4 ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋 ∧ 𝐡 βŠ† 𝑋) β†’ (((intβ€˜π½)β€˜(𝑋 βˆ– 𝐴)) ∩ ((intβ€˜π½)β€˜(𝑋 βˆ– 𝐡))) = (𝑋 βˆ– (((clsβ€˜π½)β€˜π΄) βˆͺ ((clsβ€˜π½)β€˜π΅))))
239, 15, 223eqtr3d 2780 . . 3 ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋 ∧ 𝐡 βŠ† 𝑋) β†’ (𝑋 βˆ– ((clsβ€˜π½)β€˜(𝐴 βˆͺ 𝐡))) = (𝑋 βˆ– (((clsβ€˜π½)β€˜π΄) βˆͺ ((clsβ€˜π½)β€˜π΅))))
2423difeq2d 4122 . 2 ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋 ∧ 𝐡 βŠ† 𝑋) β†’ (𝑋 βˆ– (𝑋 βˆ– ((clsβ€˜π½)β€˜(𝐴 βˆͺ 𝐡)))) = (𝑋 βˆ– (𝑋 βˆ– (((clsβ€˜π½)β€˜π΄) βˆͺ ((clsβ€˜π½)β€˜π΅)))))
255clscld 22550 . . . . 5 ((𝐽 ∈ Top ∧ (𝐴 βˆͺ 𝐡) βŠ† 𝑋) β†’ ((clsβ€˜π½)β€˜(𝐴 βˆͺ 𝐡)) ∈ (Clsdβ€˜π½))
2610, 13, 25syl2anc 584 . . . 4 ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋 ∧ 𝐡 βŠ† 𝑋) β†’ ((clsβ€˜π½)β€˜(𝐴 βˆͺ 𝐡)) ∈ (Clsdβ€˜π½))
275cldss 22532 . . . 4 (((clsβ€˜π½)β€˜(𝐴 βˆͺ 𝐡)) ∈ (Clsdβ€˜π½) β†’ ((clsβ€˜π½)β€˜(𝐴 βˆͺ 𝐡)) βŠ† 𝑋)
2826, 27syl 17 . . 3 ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋 ∧ 𝐡 βŠ† 𝑋) β†’ ((clsβ€˜π½)β€˜(𝐴 βˆͺ 𝐡)) βŠ† 𝑋)
29 dfss4 4258 . . 3 (((clsβ€˜π½)β€˜(𝐴 βˆͺ 𝐡)) βŠ† 𝑋 ↔ (𝑋 βˆ– (𝑋 βˆ– ((clsβ€˜π½)β€˜(𝐴 βˆͺ 𝐡)))) = ((clsβ€˜π½)β€˜(𝐴 βˆͺ 𝐡)))
3028, 29sylib 217 . 2 ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋 ∧ 𝐡 βŠ† 𝑋) β†’ (𝑋 βˆ– (𝑋 βˆ– ((clsβ€˜π½)β€˜(𝐴 βˆͺ 𝐡)))) = ((clsβ€˜π½)β€˜(𝐴 βˆͺ 𝐡)))
315clsss3 22562 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋) β†’ ((clsβ€˜π½)β€˜π΄) βŠ† 𝑋)
32313adant3 1132 . . . 4 ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋 ∧ 𝐡 βŠ† 𝑋) β†’ ((clsβ€˜π½)β€˜π΄) βŠ† 𝑋)
335clsss3 22562 . . . . 5 ((𝐽 ∈ Top ∧ 𝐡 βŠ† 𝑋) β†’ ((clsβ€˜π½)β€˜π΅) βŠ† 𝑋)
34333adant2 1131 . . . 4 ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋 ∧ 𝐡 βŠ† 𝑋) β†’ ((clsβ€˜π½)β€˜π΅) βŠ† 𝑋)
3532, 34jca 512 . . 3 ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋 ∧ 𝐡 βŠ† 𝑋) β†’ (((clsβ€˜π½)β€˜π΄) βŠ† 𝑋 ∧ ((clsβ€˜π½)β€˜π΅) βŠ† 𝑋))
36 unss 4184 . . . 4 ((((clsβ€˜π½)β€˜π΄) βŠ† 𝑋 ∧ ((clsβ€˜π½)β€˜π΅) βŠ† 𝑋) ↔ (((clsβ€˜π½)β€˜π΄) βˆͺ ((clsβ€˜π½)β€˜π΅)) βŠ† 𝑋)
37 dfss4 4258 . . . 4 ((((clsβ€˜π½)β€˜π΄) βˆͺ ((clsβ€˜π½)β€˜π΅)) βŠ† 𝑋 ↔ (𝑋 βˆ– (𝑋 βˆ– (((clsβ€˜π½)β€˜π΄) βˆͺ ((clsβ€˜π½)β€˜π΅)))) = (((clsβ€˜π½)β€˜π΄) βˆͺ ((clsβ€˜π½)β€˜π΅)))
3836, 37bitri 274 . . 3 ((((clsβ€˜π½)β€˜π΄) βŠ† 𝑋 ∧ ((clsβ€˜π½)β€˜π΅) βŠ† 𝑋) ↔ (𝑋 βˆ– (𝑋 βˆ– (((clsβ€˜π½)β€˜π΄) βˆͺ ((clsβ€˜π½)β€˜π΅)))) = (((clsβ€˜π½)β€˜π΄) βˆͺ ((clsβ€˜π½)β€˜π΅)))
3935, 38sylib 217 . 2 ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋 ∧ 𝐡 βŠ† 𝑋) β†’ (𝑋 βˆ– (𝑋 βˆ– (((clsβ€˜π½)β€˜π΄) βˆͺ ((clsβ€˜π½)β€˜π΅)))) = (((clsβ€˜π½)β€˜π΄) βˆͺ ((clsβ€˜π½)β€˜π΅)))
4024, 30, 393eqtr3d 2780 1 ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋 ∧ 𝐡 βŠ† 𝑋) β†’ ((clsβ€˜π½)β€˜(𝐴 βˆͺ 𝐡)) = (((clsβ€˜π½)β€˜π΄) βˆͺ ((clsβ€˜π½)β€˜π΅)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   βˆ– cdif 3945   βˆͺ cun 3946   ∩ cin 3947   βŠ† wss 3948  βˆͺ cuni 4908  β€˜cfv 6543  Topctop 22394  Clsdccld 22519  intcnt 22520  clsccl 22521
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 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  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 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-top 22395  df-cld 22522  df-ntr 22523  df-cls 22524
This theorem is referenced by: (None)
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