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Theorem clsun 32621
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 4014 . . . . . 6 (𝑋 ∖ (𝐴𝐵)) = ((𝑋𝐴) ∩ (𝑋𝐵))
21fveq2i 6347 . . . . 5 ((int‘𝐽)‘(𝑋 ∖ (𝐴𝐵))) = ((int‘𝐽)‘((𝑋𝐴) ∩ (𝑋𝐵)))
3 difss 3872 . . . . . . 7 (𝑋𝐴) ⊆ 𝑋
4 difss 3872 . . . . . . 7 (𝑋𝐵) ⊆ 𝑋
5 clsun.1 . . . . . . . 8 𝑋 = 𝐽
65ntrin 21059 . . . . . . 7 ((𝐽 ∈ Top ∧ (𝑋𝐴) ⊆ 𝑋 ∧ (𝑋𝐵) ⊆ 𝑋) → ((int‘𝐽)‘((𝑋𝐴) ∩ (𝑋𝐵))) = (((int‘𝐽)‘(𝑋𝐴)) ∩ ((int‘𝐽)‘(𝑋𝐵))))
73, 4, 6mp3an23 1557 . . . . . 6 (𝐽 ∈ Top → ((int‘𝐽)‘((𝑋𝐴) ∩ (𝑋𝐵))) = (((int‘𝐽)‘(𝑋𝐴)) ∩ ((int‘𝐽)‘(𝑋𝐵))))
873ad2ant1 1127 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → ((int‘𝐽)‘((𝑋𝐴) ∩ (𝑋𝐵))) = (((int‘𝐽)‘(𝑋𝐴)) ∩ ((int‘𝐽)‘(𝑋𝐵))))
92, 8syl5eq 2798 . . . 4 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → ((int‘𝐽)‘(𝑋 ∖ (𝐴𝐵))) = (((int‘𝐽)‘(𝑋𝐴)) ∩ ((int‘𝐽)‘(𝑋𝐵))))
10 simp1 1130 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → 𝐽 ∈ Top)
11 unss 3922 . . . . . . 7 ((𝐴𝑋𝐵𝑋) ↔ (𝐴𝐵) ⊆ 𝑋)
1211biimpi 206 . . . . . 6 ((𝐴𝑋𝐵𝑋) → (𝐴𝐵) ⊆ 𝑋)
13123adant1 1124 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐵) ⊆ 𝑋)
145ntrdif 21050 . . . . 5 ((𝐽 ∈ Top ∧ (𝐴𝐵) ⊆ 𝑋) → ((int‘𝐽)‘(𝑋 ∖ (𝐴𝐵))) = (𝑋 ∖ ((cls‘𝐽)‘(𝐴𝐵))))
1510, 13, 14syl2anc 696 . . . 4 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → ((int‘𝐽)‘(𝑋 ∖ (𝐴𝐵))) = (𝑋 ∖ ((cls‘𝐽)‘(𝐴𝐵))))
165ntrdif 21050 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝐴𝑋) → ((int‘𝐽)‘(𝑋𝐴)) = (𝑋 ∖ ((cls‘𝐽)‘𝐴)))
17163adant3 1126 . . . . . 6 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → ((int‘𝐽)‘(𝑋𝐴)) = (𝑋 ∖ ((cls‘𝐽)‘𝐴)))
185ntrdif 21050 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝐵𝑋) → ((int‘𝐽)‘(𝑋𝐵)) = (𝑋 ∖ ((cls‘𝐽)‘𝐵)))
19183adant2 1125 . . . . . 6 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → ((int‘𝐽)‘(𝑋𝐵)) = (𝑋 ∖ ((cls‘𝐽)‘𝐵)))
2017, 19ineq12d 3950 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → (((int‘𝐽)‘(𝑋𝐴)) ∩ ((int‘𝐽)‘(𝑋𝐵))) = ((𝑋 ∖ ((cls‘𝐽)‘𝐴)) ∩ (𝑋 ∖ ((cls‘𝐽)‘𝐵))))
21 difundi 4014 . . . . 5 (𝑋 ∖ (((cls‘𝐽)‘𝐴) ∪ ((cls‘𝐽)‘𝐵))) = ((𝑋 ∖ ((cls‘𝐽)‘𝐴)) ∩ (𝑋 ∖ ((cls‘𝐽)‘𝐵)))
2220, 21syl6eqr 2804 . . . 4 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → (((int‘𝐽)‘(𝑋𝐴)) ∩ ((int‘𝐽)‘(𝑋𝐵))) = (𝑋 ∖ (((cls‘𝐽)‘𝐴) ∪ ((cls‘𝐽)‘𝐵))))
239, 15, 223eqtr3d 2794 . . 3 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → (𝑋 ∖ ((cls‘𝐽)‘(𝐴𝐵))) = (𝑋 ∖ (((cls‘𝐽)‘𝐴) ∪ ((cls‘𝐽)‘𝐵))))
2423difeq2d 3863 . 2 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → (𝑋 ∖ (𝑋 ∖ ((cls‘𝐽)‘(𝐴𝐵)))) = (𝑋 ∖ (𝑋 ∖ (((cls‘𝐽)‘𝐴) ∪ ((cls‘𝐽)‘𝐵)))))
255clscld 21045 . . . . 5 ((𝐽 ∈ Top ∧ (𝐴𝐵) ⊆ 𝑋) → ((cls‘𝐽)‘(𝐴𝐵)) ∈ (Clsd‘𝐽))
2610, 13, 25syl2anc 696 . . . 4 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → ((cls‘𝐽)‘(𝐴𝐵)) ∈ (Clsd‘𝐽))
275cldss 21027 . . . 4 (((cls‘𝐽)‘(𝐴𝐵)) ∈ (Clsd‘𝐽) → ((cls‘𝐽)‘(𝐴𝐵)) ⊆ 𝑋)
2826, 27syl 17 . . 3 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → ((cls‘𝐽)‘(𝐴𝐵)) ⊆ 𝑋)
29 dfss4 3993 . . 3 (((cls‘𝐽)‘(𝐴𝐵)) ⊆ 𝑋 ↔ (𝑋 ∖ (𝑋 ∖ ((cls‘𝐽)‘(𝐴𝐵)))) = ((cls‘𝐽)‘(𝐴𝐵)))
3028, 29sylib 208 . 2 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → (𝑋 ∖ (𝑋 ∖ ((cls‘𝐽)‘(𝐴𝐵)))) = ((cls‘𝐽)‘(𝐴𝐵)))
315clsss3 21057 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴𝑋) → ((cls‘𝐽)‘𝐴) ⊆ 𝑋)
32313adant3 1126 . . . 4 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → ((cls‘𝐽)‘𝐴) ⊆ 𝑋)
335clsss3 21057 . . . . 5 ((𝐽 ∈ Top ∧ 𝐵𝑋) → ((cls‘𝐽)‘𝐵) ⊆ 𝑋)
34333adant2 1125 . . . 4 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → ((cls‘𝐽)‘𝐵) ⊆ 𝑋)
3532, 34jca 555 . . 3 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → (((cls‘𝐽)‘𝐴) ⊆ 𝑋 ∧ ((cls‘𝐽)‘𝐵) ⊆ 𝑋))
36 unss 3922 . . . 4 ((((cls‘𝐽)‘𝐴) ⊆ 𝑋 ∧ ((cls‘𝐽)‘𝐵) ⊆ 𝑋) ↔ (((cls‘𝐽)‘𝐴) ∪ ((cls‘𝐽)‘𝐵)) ⊆ 𝑋)
37 dfss4 3993 . . . 4 ((((cls‘𝐽)‘𝐴) ∪ ((cls‘𝐽)‘𝐵)) ⊆ 𝑋 ↔ (𝑋 ∖ (𝑋 ∖ (((cls‘𝐽)‘𝐴) ∪ ((cls‘𝐽)‘𝐵)))) = (((cls‘𝐽)‘𝐴) ∪ ((cls‘𝐽)‘𝐵)))
3836, 37bitri 264 . . 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 208 . 2 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → (𝑋 ∖ (𝑋 ∖ (((cls‘𝐽)‘𝐴) ∪ ((cls‘𝐽)‘𝐵)))) = (((cls‘𝐽)‘𝐴) ∪ ((cls‘𝐽)‘𝐵)))
4024, 30, 393eqtr3d 2794 1 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → ((cls‘𝐽)‘(𝐴𝐵)) = (((cls‘𝐽)‘𝐴) ∪ ((cls‘𝐽)‘𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1072   = wceq 1624  wcel 2131  cdif 3704  cun 3705  cin 3706  wss 3707   cuni 4580  cfv 6041  Topctop 20892  Clsdccld 21014  intcnt 21015  clsccl 21016
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-rep 4915  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-ral 3047  df-rex 3048  df-reu 3049  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4581  df-int 4620  df-iun 4666  df-iin 4667  df-br 4797  df-opab 4857  df-mpt 4874  df-id 5166  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-f1 6046  df-fo 6047  df-f1o 6048  df-fv 6049  df-top 20893  df-cld 21017  df-ntr 21018  df-cls 21019
This theorem is referenced by: (None)
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