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Theorem clsun 32919
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 4106 . . . . . 6 (𝑋 ∖ (𝐴𝐵)) = ((𝑋𝐴) ∩ (𝑋𝐵))
21fveq2i 6451 . . . . 5 ((int‘𝐽)‘(𝑋 ∖ (𝐴𝐵))) = ((int‘𝐽)‘((𝑋𝐴) ∩ (𝑋𝐵)))
3 difss 3960 . . . . . . 7 (𝑋𝐴) ⊆ 𝑋
4 difss 3960 . . . . . . 7 (𝑋𝐵) ⊆ 𝑋
5 clsun.1 . . . . . . . 8 𝑋 = 𝐽
65ntrin 21284 . . . . . . 7 ((𝐽 ∈ Top ∧ (𝑋𝐴) ⊆ 𝑋 ∧ (𝑋𝐵) ⊆ 𝑋) → ((int‘𝐽)‘((𝑋𝐴) ∩ (𝑋𝐵))) = (((int‘𝐽)‘(𝑋𝐴)) ∩ ((int‘𝐽)‘(𝑋𝐵))))
73, 4, 6mp3an23 1526 . . . . . 6 (𝐽 ∈ Top → ((int‘𝐽)‘((𝑋𝐴) ∩ (𝑋𝐵))) = (((int‘𝐽)‘(𝑋𝐴)) ∩ ((int‘𝐽)‘(𝑋𝐵))))
873ad2ant1 1124 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → ((int‘𝐽)‘((𝑋𝐴) ∩ (𝑋𝐵))) = (((int‘𝐽)‘(𝑋𝐴)) ∩ ((int‘𝐽)‘(𝑋𝐵))))
92, 8syl5eq 2826 . . . 4 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → ((int‘𝐽)‘(𝑋 ∖ (𝐴𝐵))) = (((int‘𝐽)‘(𝑋𝐴)) ∩ ((int‘𝐽)‘(𝑋𝐵))))
10 simp1 1127 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → 𝐽 ∈ Top)
11 unss 4010 . . . . . . 7 ((𝐴𝑋𝐵𝑋) ↔ (𝐴𝐵) ⊆ 𝑋)
1211biimpi 208 . . . . . 6 ((𝐴𝑋𝐵𝑋) → (𝐴𝐵) ⊆ 𝑋)
13123adant1 1121 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐵) ⊆ 𝑋)
145ntrdif 21275 . . . . 5 ((𝐽 ∈ Top ∧ (𝐴𝐵) ⊆ 𝑋) → ((int‘𝐽)‘(𝑋 ∖ (𝐴𝐵))) = (𝑋 ∖ ((cls‘𝐽)‘(𝐴𝐵))))
1510, 13, 14syl2anc 579 . . . 4 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → ((int‘𝐽)‘(𝑋 ∖ (𝐴𝐵))) = (𝑋 ∖ ((cls‘𝐽)‘(𝐴𝐵))))
165ntrdif 21275 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝐴𝑋) → ((int‘𝐽)‘(𝑋𝐴)) = (𝑋 ∖ ((cls‘𝐽)‘𝐴)))
17163adant3 1123 . . . . . 6 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → ((int‘𝐽)‘(𝑋𝐴)) = (𝑋 ∖ ((cls‘𝐽)‘𝐴)))
185ntrdif 21275 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝐵𝑋) → ((int‘𝐽)‘(𝑋𝐵)) = (𝑋 ∖ ((cls‘𝐽)‘𝐵)))
19183adant2 1122 . . . . . 6 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → ((int‘𝐽)‘(𝑋𝐵)) = (𝑋 ∖ ((cls‘𝐽)‘𝐵)))
2017, 19ineq12d 4038 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → (((int‘𝐽)‘(𝑋𝐴)) ∩ ((int‘𝐽)‘(𝑋𝐵))) = ((𝑋 ∖ ((cls‘𝐽)‘𝐴)) ∩ (𝑋 ∖ ((cls‘𝐽)‘𝐵))))
21 difundi 4106 . . . . 5 (𝑋 ∖ (((cls‘𝐽)‘𝐴) ∪ ((cls‘𝐽)‘𝐵))) = ((𝑋 ∖ ((cls‘𝐽)‘𝐴)) ∩ (𝑋 ∖ ((cls‘𝐽)‘𝐵)))
2220, 21syl6eqr 2832 . . . 4 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → (((int‘𝐽)‘(𝑋𝐴)) ∩ ((int‘𝐽)‘(𝑋𝐵))) = (𝑋 ∖ (((cls‘𝐽)‘𝐴) ∪ ((cls‘𝐽)‘𝐵))))
239, 15, 223eqtr3d 2822 . . 3 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → (𝑋 ∖ ((cls‘𝐽)‘(𝐴𝐵))) = (𝑋 ∖ (((cls‘𝐽)‘𝐴) ∪ ((cls‘𝐽)‘𝐵))))
2423difeq2d 3951 . 2 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → (𝑋 ∖ (𝑋 ∖ ((cls‘𝐽)‘(𝐴𝐵)))) = (𝑋 ∖ (𝑋 ∖ (((cls‘𝐽)‘𝐴) ∪ ((cls‘𝐽)‘𝐵)))))
255clscld 21270 . . . . 5 ((𝐽 ∈ Top ∧ (𝐴𝐵) ⊆ 𝑋) → ((cls‘𝐽)‘(𝐴𝐵)) ∈ (Clsd‘𝐽))
2610, 13, 25syl2anc 579 . . . 4 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → ((cls‘𝐽)‘(𝐴𝐵)) ∈ (Clsd‘𝐽))
275cldss 21252 . . . 4 (((cls‘𝐽)‘(𝐴𝐵)) ∈ (Clsd‘𝐽) → ((cls‘𝐽)‘(𝐴𝐵)) ⊆ 𝑋)
2826, 27syl 17 . . 3 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → ((cls‘𝐽)‘(𝐴𝐵)) ⊆ 𝑋)
29 dfss4 4085 . . 3 (((cls‘𝐽)‘(𝐴𝐵)) ⊆ 𝑋 ↔ (𝑋 ∖ (𝑋 ∖ ((cls‘𝐽)‘(𝐴𝐵)))) = ((cls‘𝐽)‘(𝐴𝐵)))
3028, 29sylib 210 . 2 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → (𝑋 ∖ (𝑋 ∖ ((cls‘𝐽)‘(𝐴𝐵)))) = ((cls‘𝐽)‘(𝐴𝐵)))
315clsss3 21282 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴𝑋) → ((cls‘𝐽)‘𝐴) ⊆ 𝑋)
32313adant3 1123 . . . 4 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → ((cls‘𝐽)‘𝐴) ⊆ 𝑋)
335clsss3 21282 . . . . 5 ((𝐽 ∈ Top ∧ 𝐵𝑋) → ((cls‘𝐽)‘𝐵) ⊆ 𝑋)
34333adant2 1122 . . . 4 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → ((cls‘𝐽)‘𝐵) ⊆ 𝑋)
3532, 34jca 507 . . 3 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → (((cls‘𝐽)‘𝐴) ⊆ 𝑋 ∧ ((cls‘𝐽)‘𝐵) ⊆ 𝑋))
36 unss 4010 . . . 4 ((((cls‘𝐽)‘𝐴) ⊆ 𝑋 ∧ ((cls‘𝐽)‘𝐵) ⊆ 𝑋) ↔ (((cls‘𝐽)‘𝐴) ∪ ((cls‘𝐽)‘𝐵)) ⊆ 𝑋)
37 dfss4 4085 . . . 4 ((((cls‘𝐽)‘𝐴) ∪ ((cls‘𝐽)‘𝐵)) ⊆ 𝑋 ↔ (𝑋 ∖ (𝑋 ∖ (((cls‘𝐽)‘𝐴) ∪ ((cls‘𝐽)‘𝐵)))) = (((cls‘𝐽)‘𝐴) ∪ ((cls‘𝐽)‘𝐵)))
3836, 37bitri 267 . . 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 210 . 2 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → (𝑋 ∖ (𝑋 ∖ (((cls‘𝐽)‘𝐴) ∪ ((cls‘𝐽)‘𝐵)))) = (((cls‘𝐽)‘𝐴) ∪ ((cls‘𝐽)‘𝐵)))
4024, 30, 393eqtr3d 2822 1 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → ((cls‘𝐽)‘(𝐴𝐵)) = (((cls‘𝐽)‘𝐴) ∪ ((cls‘𝐽)‘𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386  w3a 1071   = wceq 1601  wcel 2107  cdif 3789  cun 3790  cin 3791  wss 3792   cuni 4673  cfv 6137  Topctop 21116  Clsdccld 21239  intcnt 21240  clsccl 21241
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5008  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140  ax-un 7228
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4674  df-int 4713  df-iun 4757  df-iin 4758  df-br 4889  df-opab 4951  df-mpt 4968  df-id 5263  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-f1 6142  df-fo 6143  df-f1o 6144  df-fv 6145  df-top 21117  df-cld 21242  df-ntr 21243  df-cls 21244
This theorem is referenced by: (None)
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