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Theorem clsun 34853
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 4243 . . . . . 6 (𝑋 βˆ– (𝐴 βˆͺ 𝐡)) = ((𝑋 βˆ– 𝐴) ∩ (𝑋 βˆ– 𝐡))
21fveq2i 6849 . . . . 5 ((intβ€˜π½)β€˜(𝑋 βˆ– (𝐴 βˆͺ 𝐡))) = ((intβ€˜π½)β€˜((𝑋 βˆ– 𝐴) ∩ (𝑋 βˆ– 𝐡)))
3 difss 4095 . . . . . . 7 (𝑋 βˆ– 𝐴) βŠ† 𝑋
4 difss 4095 . . . . . . 7 (𝑋 βˆ– 𝐡) βŠ† 𝑋
5 clsun.1 . . . . . . . 8 𝑋 = βˆͺ 𝐽
65ntrin 22435 . . . . . . 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 1454 . . . . . 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 1134 . . . . 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 2785 . . . 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 1137 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋 ∧ 𝐡 βŠ† 𝑋) β†’ 𝐽 ∈ Top)
11 unss 4148 . . . . . . 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 1131 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋 ∧ 𝐡 βŠ† 𝑋) β†’ (𝐴 βˆͺ 𝐡) βŠ† 𝑋)
145ntrdif 22426 . . . . 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 585 . . . 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 22426 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋) β†’ ((intβ€˜π½)β€˜(𝑋 βˆ– 𝐴)) = (𝑋 βˆ– ((clsβ€˜π½)β€˜π΄)))
17163adant3 1133 . . . . . 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 22426 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝐡 βŠ† 𝑋) β†’ ((intβ€˜π½)β€˜(𝑋 βˆ– 𝐡)) = (𝑋 βˆ– ((clsβ€˜π½)β€˜π΅)))
19183adant2 1132 . . . . . 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 4177 . . . . 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 4243 . . . . 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 2791 . . . 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 2781 . . 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 4086 . 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 22421 . . . . 5 ((𝐽 ∈ Top ∧ (𝐴 βˆͺ 𝐡) βŠ† 𝑋) β†’ ((clsβ€˜π½)β€˜(𝐴 βˆͺ 𝐡)) ∈ (Clsdβ€˜π½))
2610, 13, 25syl2anc 585 . . . 4 ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋 ∧ 𝐡 βŠ† 𝑋) β†’ ((clsβ€˜π½)β€˜(𝐴 βˆͺ 𝐡)) ∈ (Clsdβ€˜π½))
275cldss 22403 . . . 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 4222 . . 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 22433 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋) β†’ ((clsβ€˜π½)β€˜π΄) βŠ† 𝑋)
32313adant3 1133 . . . 4 ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋 ∧ 𝐡 βŠ† 𝑋) β†’ ((clsβ€˜π½)β€˜π΄) βŠ† 𝑋)
335clsss3 22433 . . . . 5 ((𝐽 ∈ Top ∧ 𝐡 βŠ† 𝑋) β†’ ((clsβ€˜π½)β€˜π΅) βŠ† 𝑋)
34333adant2 1132 . . . 4 ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋 ∧ 𝐡 βŠ† 𝑋) β†’ ((clsβ€˜π½)β€˜π΅) βŠ† 𝑋)
3532, 34jca 513 . . 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 4148 . . . 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 4222 . . . 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 2781 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 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   βˆ– cdif 3911   βˆͺ cun 3912   ∩ cin 3913   βŠ† wss 3914  βˆͺ cuni 4869  β€˜cfv 6500  Topctop 22265  Clsdccld 22390  intcnt 22391  clsccl 22392
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 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
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 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-int 4912  df-iun 4960  df-iin 4961  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-top 22266  df-cld 22393  df-ntr 22394  df-cls 22395
This theorem is referenced by: (None)
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