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Theorem clsun 35713
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 4272 . . . . . 6 (𝑋 βˆ– (𝐴 βˆͺ 𝐡)) = ((𝑋 βˆ– 𝐴) ∩ (𝑋 βˆ– 𝐡))
21fveq2i 6885 . . . . 5 ((intβ€˜π½)β€˜(𝑋 βˆ– (𝐴 βˆͺ 𝐡))) = ((intβ€˜π½)β€˜((𝑋 βˆ– 𝐴) ∩ (𝑋 βˆ– 𝐡)))
3 difss 4124 . . . . . . 7 (𝑋 βˆ– 𝐴) βŠ† 𝑋
4 difss 4124 . . . . . . 7 (𝑋 βˆ– 𝐡) βŠ† 𝑋
5 clsun.1 . . . . . . . 8 𝑋 = βˆͺ 𝐽
65ntrin 22909 . . . . . . 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 1449 . . . . . 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 1130 . . . . 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 2776 . . . 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 1133 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋 ∧ 𝐡 βŠ† 𝑋) β†’ 𝐽 ∈ Top)
11 unss 4177 . . . . . . 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 1127 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋 ∧ 𝐡 βŠ† 𝑋) β†’ (𝐴 βˆͺ 𝐡) βŠ† 𝑋)
145ntrdif 22900 . . . . 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 583 . . . 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 22900 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋) β†’ ((intβ€˜π½)β€˜(𝑋 βˆ– 𝐴)) = (𝑋 βˆ– ((clsβ€˜π½)β€˜π΄)))
17163adant3 1129 . . . . . 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 22900 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝐡 βŠ† 𝑋) β†’ ((intβ€˜π½)β€˜(𝑋 βˆ– 𝐡)) = (𝑋 βˆ– ((clsβ€˜π½)β€˜π΅)))
19183adant2 1128 . . . . . 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 4206 . . . . 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 4272 . . . . 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 2782 . . . 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 2772 . . 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 4115 . 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 22895 . . . . 5 ((𝐽 ∈ Top ∧ (𝐴 βˆͺ 𝐡) βŠ† 𝑋) β†’ ((clsβ€˜π½)β€˜(𝐴 βˆͺ 𝐡)) ∈ (Clsdβ€˜π½))
2610, 13, 25syl2anc 583 . . . 4 ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋 ∧ 𝐡 βŠ† 𝑋) β†’ ((clsβ€˜π½)β€˜(𝐴 βˆͺ 𝐡)) ∈ (Clsdβ€˜π½))
275cldss 22877 . . . 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 4251 . . 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 22907 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋) β†’ ((clsβ€˜π½)β€˜π΄) βŠ† 𝑋)
32313adant3 1129 . . . 4 ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋 ∧ 𝐡 βŠ† 𝑋) β†’ ((clsβ€˜π½)β€˜π΄) βŠ† 𝑋)
335clsss3 22907 . . . . 5 ((𝐽 ∈ Top ∧ 𝐡 βŠ† 𝑋) β†’ ((clsβ€˜π½)β€˜π΅) βŠ† 𝑋)
34333adant2 1128 . . . 4 ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋 ∧ 𝐡 βŠ† 𝑋) β†’ ((clsβ€˜π½)β€˜π΅) βŠ† 𝑋)
3532, 34jca 511 . . 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 4177 . . . 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 4251 . . . 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 275 . . 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 2772 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 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   βˆ– cdif 3938   βˆͺ cun 3939   ∩ cin 3940   βŠ† wss 3941  βˆͺ cuni 4900  β€˜cfv 6534  Topctop 22739  Clsdccld 22864  intcnt 22865  clsccl 22866
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5276  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-int 4942  df-iun 4990  df-iin 4991  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-top 22740  df-cld 22867  df-ntr 22868  df-cls 22869
This theorem is referenced by: (None)
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