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Theorem clsun 33573
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 = 𝐽
Assertion
Ref Expression
clsun ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → ((cls‘𝐽)‘(𝐴𝐵)) = (((cls‘𝐽)‘𝐴) ∪ ((cls‘𝐽)‘𝐵)))

Proof of Theorem clsun
StepHypRef Expression
1 difundi 4253 . . . . . 6 (𝑋 ∖ (𝐴𝐵)) = ((𝑋𝐴) ∩ (𝑋𝐵))
21fveq2i 6666 . . . . 5 ((int‘𝐽)‘(𝑋 ∖ (𝐴𝐵))) = ((int‘𝐽)‘((𝑋𝐴) ∩ (𝑋𝐵)))
3 difss 4105 . . . . . . 7 (𝑋𝐴) ⊆ 𝑋
4 difss 4105 . . . . . . 7 (𝑋𝐵) ⊆ 𝑋
5 clsun.1 . . . . . . . 8 𝑋 = 𝐽
65ntrin 21597 . . . . . . 7 ((𝐽 ∈ Top ∧ (𝑋𝐴) ⊆ 𝑋 ∧ (𝑋𝐵) ⊆ 𝑋) → ((int‘𝐽)‘((𝑋𝐴) ∩ (𝑋𝐵))) = (((int‘𝐽)‘(𝑋𝐴)) ∩ ((int‘𝐽)‘(𝑋𝐵))))
73, 4, 6mp3an23 1444 . . . . . 6 (𝐽 ∈ Top → ((int‘𝐽)‘((𝑋𝐴) ∩ (𝑋𝐵))) = (((int‘𝐽)‘(𝑋𝐴)) ∩ ((int‘𝐽)‘(𝑋𝐵))))
873ad2ant1 1125 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → ((int‘𝐽)‘((𝑋𝐴) ∩ (𝑋𝐵))) = (((int‘𝐽)‘(𝑋𝐴)) ∩ ((int‘𝐽)‘(𝑋𝐵))))
92, 8syl5eq 2865 . . . 4 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → ((int‘𝐽)‘(𝑋 ∖ (𝐴𝐵))) = (((int‘𝐽)‘(𝑋𝐴)) ∩ ((int‘𝐽)‘(𝑋𝐵))))
10 simp1 1128 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → 𝐽 ∈ Top)
11 unss 4157 . . . . . . 7 ((𝐴𝑋𝐵𝑋) ↔ (𝐴𝐵) ⊆ 𝑋)
1211biimpi 217 . . . . . 6 ((𝐴𝑋𝐵𝑋) → (𝐴𝐵) ⊆ 𝑋)
13123adant1 1122 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐵) ⊆ 𝑋)
145ntrdif 21588 . . . . 5 ((𝐽 ∈ Top ∧ (𝐴𝐵) ⊆ 𝑋) → ((int‘𝐽)‘(𝑋 ∖ (𝐴𝐵))) = (𝑋 ∖ ((cls‘𝐽)‘(𝐴𝐵))))
1510, 13, 14syl2anc 584 . . . 4 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → ((int‘𝐽)‘(𝑋 ∖ (𝐴𝐵))) = (𝑋 ∖ ((cls‘𝐽)‘(𝐴𝐵))))
165ntrdif 21588 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝐴𝑋) → ((int‘𝐽)‘(𝑋𝐴)) = (𝑋 ∖ ((cls‘𝐽)‘𝐴)))
17163adant3 1124 . . . . . 6 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → ((int‘𝐽)‘(𝑋𝐴)) = (𝑋 ∖ ((cls‘𝐽)‘𝐴)))
185ntrdif 21588 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝐵𝑋) → ((int‘𝐽)‘(𝑋𝐵)) = (𝑋 ∖ ((cls‘𝐽)‘𝐵)))
19183adant2 1123 . . . . . 6 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → ((int‘𝐽)‘(𝑋𝐵)) = (𝑋 ∖ ((cls‘𝐽)‘𝐵)))
2017, 19ineq12d 4187 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → (((int‘𝐽)‘(𝑋𝐴)) ∩ ((int‘𝐽)‘(𝑋𝐵))) = ((𝑋 ∖ ((cls‘𝐽)‘𝐴)) ∩ (𝑋 ∖ ((cls‘𝐽)‘𝐵))))
21 difundi 4253 . . . . 5 (𝑋 ∖ (((cls‘𝐽)‘𝐴) ∪ ((cls‘𝐽)‘𝐵))) = ((𝑋 ∖ ((cls‘𝐽)‘𝐴)) ∩ (𝑋 ∖ ((cls‘𝐽)‘𝐵)))
2220, 21syl6eqr 2871 . . . 4 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → (((int‘𝐽)‘(𝑋𝐴)) ∩ ((int‘𝐽)‘(𝑋𝐵))) = (𝑋 ∖ (((cls‘𝐽)‘𝐴) ∪ ((cls‘𝐽)‘𝐵))))
239, 15, 223eqtr3d 2861 . . 3 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → (𝑋 ∖ ((cls‘𝐽)‘(𝐴𝐵))) = (𝑋 ∖ (((cls‘𝐽)‘𝐴) ∪ ((cls‘𝐽)‘𝐵))))
2423difeq2d 4096 . 2 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → (𝑋 ∖ (𝑋 ∖ ((cls‘𝐽)‘(𝐴𝐵)))) = (𝑋 ∖ (𝑋 ∖ (((cls‘𝐽)‘𝐴) ∪ ((cls‘𝐽)‘𝐵)))))
255clscld 21583 . . . . 5 ((𝐽 ∈ Top ∧ (𝐴𝐵) ⊆ 𝑋) → ((cls‘𝐽)‘(𝐴𝐵)) ∈ (Clsd‘𝐽))
2610, 13, 25syl2anc 584 . . . 4 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → ((cls‘𝐽)‘(𝐴𝐵)) ∈ (Clsd‘𝐽))
275cldss 21565 . . . 4 (((cls‘𝐽)‘(𝐴𝐵)) ∈ (Clsd‘𝐽) → ((cls‘𝐽)‘(𝐴𝐵)) ⊆ 𝑋)
2826, 27syl 17 . . 3 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → ((cls‘𝐽)‘(𝐴𝐵)) ⊆ 𝑋)
29 dfss4 4232 . . 3 (((cls‘𝐽)‘(𝐴𝐵)) ⊆ 𝑋 ↔ (𝑋 ∖ (𝑋 ∖ ((cls‘𝐽)‘(𝐴𝐵)))) = ((cls‘𝐽)‘(𝐴𝐵)))
3028, 29sylib 219 . 2 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → (𝑋 ∖ (𝑋 ∖ ((cls‘𝐽)‘(𝐴𝐵)))) = ((cls‘𝐽)‘(𝐴𝐵)))
315clsss3 21595 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴𝑋) → ((cls‘𝐽)‘𝐴) ⊆ 𝑋)
32313adant3 1124 . . . 4 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → ((cls‘𝐽)‘𝐴) ⊆ 𝑋)
335clsss3 21595 . . . . 5 ((𝐽 ∈ Top ∧ 𝐵𝑋) → ((cls‘𝐽)‘𝐵) ⊆ 𝑋)
34333adant2 1123 . . . 4 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → ((cls‘𝐽)‘𝐵) ⊆ 𝑋)
3532, 34jca 512 . . 3 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → (((cls‘𝐽)‘𝐴) ⊆ 𝑋 ∧ ((cls‘𝐽)‘𝐵) ⊆ 𝑋))
36 unss 4157 . . . 4 ((((cls‘𝐽)‘𝐴) ⊆ 𝑋 ∧ ((cls‘𝐽)‘𝐵) ⊆ 𝑋) ↔ (((cls‘𝐽)‘𝐴) ∪ ((cls‘𝐽)‘𝐵)) ⊆ 𝑋)
37 dfss4 4232 . . . 4 ((((cls‘𝐽)‘𝐴) ∪ ((cls‘𝐽)‘𝐵)) ⊆ 𝑋 ↔ (𝑋 ∖ (𝑋 ∖ (((cls‘𝐽)‘𝐴) ∪ ((cls‘𝐽)‘𝐵)))) = (((cls‘𝐽)‘𝐴) ∪ ((cls‘𝐽)‘𝐵)))
3836, 37bitri 276 . . 3 ((((cls‘𝐽)‘𝐴) ⊆ 𝑋 ∧ ((cls‘𝐽)‘𝐵) ⊆ 𝑋) ↔ (𝑋 ∖ (𝑋 ∖ (((cls‘𝐽)‘𝐴) ∪ ((cls‘𝐽)‘𝐵)))) = (((cls‘𝐽)‘𝐴) ∪ ((cls‘𝐽)‘𝐵)))
3935, 38sylib 219 . 2 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → (𝑋 ∖ (𝑋 ∖ (((cls‘𝐽)‘𝐴) ∪ ((cls‘𝐽)‘𝐵)))) = (((cls‘𝐽)‘𝐴) ∪ ((cls‘𝐽)‘𝐵)))
4024, 30, 393eqtr3d 2861 1 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → ((cls‘𝐽)‘(𝐴𝐵)) = (((cls‘𝐽)‘𝐴) ∪ ((cls‘𝐽)‘𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1079   = wceq 1528  wcel 2105  cdif 3930  cun 3931  cin 3932  wss 3933   cuni 4830  cfv 6348  Topctop 21429  Clsdccld 21552  intcnt 21553  clsccl 21554
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-iin 4913  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-top 21430  df-cld 21555  df-ntr 21556  df-cls 21557
This theorem is referenced by: (None)
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