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Theorem clsun 32448
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 3912 . . . . . 6 (𝑋 ∖ (𝐴𝐵)) = ((𝑋𝐴) ∩ (𝑋𝐵))
21fveq2i 6232 . . . . 5 ((int‘𝐽)‘(𝑋 ∖ (𝐴𝐵))) = ((int‘𝐽)‘((𝑋𝐴) ∩ (𝑋𝐵)))
3 difss 3770 . . . . . . 7 (𝑋𝐴) ⊆ 𝑋
4 difss 3770 . . . . . . 7 (𝑋𝐵) ⊆ 𝑋
5 clsun.1 . . . . . . . 8 𝑋 = 𝐽
65ntrin 20913 . . . . . . 7 ((𝐽 ∈ Top ∧ (𝑋𝐴) ⊆ 𝑋 ∧ (𝑋𝐵) ⊆ 𝑋) → ((int‘𝐽)‘((𝑋𝐴) ∩ (𝑋𝐵))) = (((int‘𝐽)‘(𝑋𝐴)) ∩ ((int‘𝐽)‘(𝑋𝐵))))
73, 4, 6mp3an23 1456 . . . . . 6 (𝐽 ∈ Top → ((int‘𝐽)‘((𝑋𝐴) ∩ (𝑋𝐵))) = (((int‘𝐽)‘(𝑋𝐴)) ∩ ((int‘𝐽)‘(𝑋𝐵))))
873ad2ant1 1102 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → ((int‘𝐽)‘((𝑋𝐴) ∩ (𝑋𝐵))) = (((int‘𝐽)‘(𝑋𝐴)) ∩ ((int‘𝐽)‘(𝑋𝐵))))
92, 8syl5eq 2697 . . . 4 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → ((int‘𝐽)‘(𝑋 ∖ (𝐴𝐵))) = (((int‘𝐽)‘(𝑋𝐴)) ∩ ((int‘𝐽)‘(𝑋𝐵))))
10 simp1 1081 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → 𝐽 ∈ Top)
11 unss 3820 . . . . . . 7 ((𝐴𝑋𝐵𝑋) ↔ (𝐴𝐵) ⊆ 𝑋)
1211biimpi 206 . . . . . 6 ((𝐴𝑋𝐵𝑋) → (𝐴𝐵) ⊆ 𝑋)
13123adant1 1099 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐵) ⊆ 𝑋)
145ntrdif 20904 . . . . 5 ((𝐽 ∈ Top ∧ (𝐴𝐵) ⊆ 𝑋) → ((int‘𝐽)‘(𝑋 ∖ (𝐴𝐵))) = (𝑋 ∖ ((cls‘𝐽)‘(𝐴𝐵))))
1510, 13, 14syl2anc 694 . . . 4 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → ((int‘𝐽)‘(𝑋 ∖ (𝐴𝐵))) = (𝑋 ∖ ((cls‘𝐽)‘(𝐴𝐵))))
165ntrdif 20904 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝐴𝑋) → ((int‘𝐽)‘(𝑋𝐴)) = (𝑋 ∖ ((cls‘𝐽)‘𝐴)))
17163adant3 1101 . . . . . 6 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → ((int‘𝐽)‘(𝑋𝐴)) = (𝑋 ∖ ((cls‘𝐽)‘𝐴)))
185ntrdif 20904 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝐵𝑋) → ((int‘𝐽)‘(𝑋𝐵)) = (𝑋 ∖ ((cls‘𝐽)‘𝐵)))
19183adant2 1100 . . . . . 6 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → ((int‘𝐽)‘(𝑋𝐵)) = (𝑋 ∖ ((cls‘𝐽)‘𝐵)))
2017, 19ineq12d 3848 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → (((int‘𝐽)‘(𝑋𝐴)) ∩ ((int‘𝐽)‘(𝑋𝐵))) = ((𝑋 ∖ ((cls‘𝐽)‘𝐴)) ∩ (𝑋 ∖ ((cls‘𝐽)‘𝐵))))
21 difundi 3912 . . . . 5 (𝑋 ∖ (((cls‘𝐽)‘𝐴) ∪ ((cls‘𝐽)‘𝐵))) = ((𝑋 ∖ ((cls‘𝐽)‘𝐴)) ∩ (𝑋 ∖ ((cls‘𝐽)‘𝐵)))
2220, 21syl6eqr 2703 . . . 4 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → (((int‘𝐽)‘(𝑋𝐴)) ∩ ((int‘𝐽)‘(𝑋𝐵))) = (𝑋 ∖ (((cls‘𝐽)‘𝐴) ∪ ((cls‘𝐽)‘𝐵))))
239, 15, 223eqtr3d 2693 . . 3 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → (𝑋 ∖ ((cls‘𝐽)‘(𝐴𝐵))) = (𝑋 ∖ (((cls‘𝐽)‘𝐴) ∪ ((cls‘𝐽)‘𝐵))))
2423difeq2d 3761 . 2 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → (𝑋 ∖ (𝑋 ∖ ((cls‘𝐽)‘(𝐴𝐵)))) = (𝑋 ∖ (𝑋 ∖ (((cls‘𝐽)‘𝐴) ∪ ((cls‘𝐽)‘𝐵)))))
255clscld 20899 . . . . 5 ((𝐽 ∈ Top ∧ (𝐴𝐵) ⊆ 𝑋) → ((cls‘𝐽)‘(𝐴𝐵)) ∈ (Clsd‘𝐽))
2610, 13, 25syl2anc 694 . . . 4 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → ((cls‘𝐽)‘(𝐴𝐵)) ∈ (Clsd‘𝐽))
275cldss 20881 . . . 4 (((cls‘𝐽)‘(𝐴𝐵)) ∈ (Clsd‘𝐽) → ((cls‘𝐽)‘(𝐴𝐵)) ⊆ 𝑋)
2826, 27syl 17 . . 3 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → ((cls‘𝐽)‘(𝐴𝐵)) ⊆ 𝑋)
29 dfss4 3891 . . 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 20911 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴𝑋) → ((cls‘𝐽)‘𝐴) ⊆ 𝑋)
32313adant3 1101 . . . 4 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → ((cls‘𝐽)‘𝐴) ⊆ 𝑋)
335clsss3 20911 . . . . 5 ((𝐽 ∈ Top ∧ 𝐵𝑋) → ((cls‘𝐽)‘𝐵) ⊆ 𝑋)
34333adant2 1100 . . . 4 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → ((cls‘𝐽)‘𝐵) ⊆ 𝑋)
3532, 34jca 553 . . 3 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → (((cls‘𝐽)‘𝐴) ⊆ 𝑋 ∧ ((cls‘𝐽)‘𝐵) ⊆ 𝑋))
36 unss 3820 . . . 4 ((((cls‘𝐽)‘𝐴) ⊆ 𝑋 ∧ ((cls‘𝐽)‘𝐵) ⊆ 𝑋) ↔ (((cls‘𝐽)‘𝐴) ∪ ((cls‘𝐽)‘𝐵)) ⊆ 𝑋)
37 dfss4 3891 . . . 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 2693 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 1054   = wceq 1523  wcel 2030  cdif 3604  cun 3605  cin 3606  wss 3607   cuni 4468  cfv 5926  Topctop 20746  Clsdccld 20868  intcnt 20869  clsccl 20870
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-iin 4555  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-top 20747  df-cld 20871  df-ntr 20872  df-cls 20873
This theorem is referenced by: (None)
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