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Theorem clsun 33790
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 4209 . . . . . 6 (𝑋 ∖ (𝐴𝐵)) = ((𝑋𝐴) ∩ (𝑋𝐵))
21fveq2i 6652 . . . . 5 ((int‘𝐽)‘(𝑋 ∖ (𝐴𝐵))) = ((int‘𝐽)‘((𝑋𝐴) ∩ (𝑋𝐵)))
3 difss 4062 . . . . . . 7 (𝑋𝐴) ⊆ 𝑋
4 difss 4062 . . . . . . 7 (𝑋𝐵) ⊆ 𝑋
5 clsun.1 . . . . . . . 8 𝑋 = 𝐽
65ntrin 21670 . . . . . . 7 ((𝐽 ∈ Top ∧ (𝑋𝐴) ⊆ 𝑋 ∧ (𝑋𝐵) ⊆ 𝑋) → ((int‘𝐽)‘((𝑋𝐴) ∩ (𝑋𝐵))) = (((int‘𝐽)‘(𝑋𝐴)) ∩ ((int‘𝐽)‘(𝑋𝐵))))
73, 4, 6mp3an23 1450 . . . . . 6 (𝐽 ∈ Top → ((int‘𝐽)‘((𝑋𝐴) ∩ (𝑋𝐵))) = (((int‘𝐽)‘(𝑋𝐴)) ∩ ((int‘𝐽)‘(𝑋𝐵))))
873ad2ant1 1130 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → ((int‘𝐽)‘((𝑋𝐴) ∩ (𝑋𝐵))) = (((int‘𝐽)‘(𝑋𝐴)) ∩ ((int‘𝐽)‘(𝑋𝐵))))
92, 8syl5eq 2848 . . . 4 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → ((int‘𝐽)‘(𝑋 ∖ (𝐴𝐵))) = (((int‘𝐽)‘(𝑋𝐴)) ∩ ((int‘𝐽)‘(𝑋𝐵))))
10 simp1 1133 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → 𝐽 ∈ Top)
11 unss 4114 . . . . . . 7 ((𝐴𝑋𝐵𝑋) ↔ (𝐴𝐵) ⊆ 𝑋)
1211biimpi 219 . . . . . 6 ((𝐴𝑋𝐵𝑋) → (𝐴𝐵) ⊆ 𝑋)
13123adant1 1127 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐵) ⊆ 𝑋)
145ntrdif 21661 . . . . 5 ((𝐽 ∈ Top ∧ (𝐴𝐵) ⊆ 𝑋) → ((int‘𝐽)‘(𝑋 ∖ (𝐴𝐵))) = (𝑋 ∖ ((cls‘𝐽)‘(𝐴𝐵))))
1510, 13, 14syl2anc 587 . . . 4 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → ((int‘𝐽)‘(𝑋 ∖ (𝐴𝐵))) = (𝑋 ∖ ((cls‘𝐽)‘(𝐴𝐵))))
165ntrdif 21661 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝐴𝑋) → ((int‘𝐽)‘(𝑋𝐴)) = (𝑋 ∖ ((cls‘𝐽)‘𝐴)))
17163adant3 1129 . . . . . 6 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → ((int‘𝐽)‘(𝑋𝐴)) = (𝑋 ∖ ((cls‘𝐽)‘𝐴)))
185ntrdif 21661 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝐵𝑋) → ((int‘𝐽)‘(𝑋𝐵)) = (𝑋 ∖ ((cls‘𝐽)‘𝐵)))
19183adant2 1128 . . . . . 6 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → ((int‘𝐽)‘(𝑋𝐵)) = (𝑋 ∖ ((cls‘𝐽)‘𝐵)))
2017, 19ineq12d 4143 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → (((int‘𝐽)‘(𝑋𝐴)) ∩ ((int‘𝐽)‘(𝑋𝐵))) = ((𝑋 ∖ ((cls‘𝐽)‘𝐴)) ∩ (𝑋 ∖ ((cls‘𝐽)‘𝐵))))
21 difundi 4209 . . . . 5 (𝑋 ∖ (((cls‘𝐽)‘𝐴) ∪ ((cls‘𝐽)‘𝐵))) = ((𝑋 ∖ ((cls‘𝐽)‘𝐴)) ∩ (𝑋 ∖ ((cls‘𝐽)‘𝐵)))
2220, 21eqtr4di 2854 . . . 4 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → (((int‘𝐽)‘(𝑋𝐴)) ∩ ((int‘𝐽)‘(𝑋𝐵))) = (𝑋 ∖ (((cls‘𝐽)‘𝐴) ∪ ((cls‘𝐽)‘𝐵))))
239, 15, 223eqtr3d 2844 . . 3 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → (𝑋 ∖ ((cls‘𝐽)‘(𝐴𝐵))) = (𝑋 ∖ (((cls‘𝐽)‘𝐴) ∪ ((cls‘𝐽)‘𝐵))))
2423difeq2d 4053 . 2 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → (𝑋 ∖ (𝑋 ∖ ((cls‘𝐽)‘(𝐴𝐵)))) = (𝑋 ∖ (𝑋 ∖ (((cls‘𝐽)‘𝐴) ∪ ((cls‘𝐽)‘𝐵)))))
255clscld 21656 . . . . 5 ((𝐽 ∈ Top ∧ (𝐴𝐵) ⊆ 𝑋) → ((cls‘𝐽)‘(𝐴𝐵)) ∈ (Clsd‘𝐽))
2610, 13, 25syl2anc 587 . . . 4 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → ((cls‘𝐽)‘(𝐴𝐵)) ∈ (Clsd‘𝐽))
275cldss 21638 . . . 4 (((cls‘𝐽)‘(𝐴𝐵)) ∈ (Clsd‘𝐽) → ((cls‘𝐽)‘(𝐴𝐵)) ⊆ 𝑋)
2826, 27syl 17 . . 3 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → ((cls‘𝐽)‘(𝐴𝐵)) ⊆ 𝑋)
29 dfss4 4188 . . 3 (((cls‘𝐽)‘(𝐴𝐵)) ⊆ 𝑋 ↔ (𝑋 ∖ (𝑋 ∖ ((cls‘𝐽)‘(𝐴𝐵)))) = ((cls‘𝐽)‘(𝐴𝐵)))
3028, 29sylib 221 . 2 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → (𝑋 ∖ (𝑋 ∖ ((cls‘𝐽)‘(𝐴𝐵)))) = ((cls‘𝐽)‘(𝐴𝐵)))
315clsss3 21668 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴𝑋) → ((cls‘𝐽)‘𝐴) ⊆ 𝑋)
32313adant3 1129 . . . 4 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → ((cls‘𝐽)‘𝐴) ⊆ 𝑋)
335clsss3 21668 . . . . 5 ((𝐽 ∈ Top ∧ 𝐵𝑋) → ((cls‘𝐽)‘𝐵) ⊆ 𝑋)
34333adant2 1128 . . . 4 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → ((cls‘𝐽)‘𝐵) ⊆ 𝑋)
3532, 34jca 515 . . 3 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → (((cls‘𝐽)‘𝐴) ⊆ 𝑋 ∧ ((cls‘𝐽)‘𝐵) ⊆ 𝑋))
36 unss 4114 . . . 4 ((((cls‘𝐽)‘𝐴) ⊆ 𝑋 ∧ ((cls‘𝐽)‘𝐵) ⊆ 𝑋) ↔ (((cls‘𝐽)‘𝐴) ∪ ((cls‘𝐽)‘𝐵)) ⊆ 𝑋)
37 dfss4 4188 . . . 4 ((((cls‘𝐽)‘𝐴) ∪ ((cls‘𝐽)‘𝐵)) ⊆ 𝑋 ↔ (𝑋 ∖ (𝑋 ∖ (((cls‘𝐽)‘𝐴) ∪ ((cls‘𝐽)‘𝐵)))) = (((cls‘𝐽)‘𝐴) ∪ ((cls‘𝐽)‘𝐵)))
3836, 37bitri 278 . . 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 221 . 2 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → (𝑋 ∖ (𝑋 ∖ (((cls‘𝐽)‘𝐴) ∪ ((cls‘𝐽)‘𝐵)))) = (((cls‘𝐽)‘𝐴) ∪ ((cls‘𝐽)‘𝐵)))
4024, 30, 393eqtr3d 2844 1 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → ((cls‘𝐽)‘(𝐴𝐵)) = (((cls‘𝐽)‘𝐴) ∪ ((cls‘𝐽)‘𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1084   = wceq 1538  wcel 2112  cdif 3881  cun 3882  cin 3883  wss 3884   cuni 4803  cfv 6328  Topctop 21502  Clsdccld 21625  intcnt 21626  clsccl 21627
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-iin 4887  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-top 21503  df-cld 21628  df-ntr 21629  df-cls 21630
This theorem is referenced by: (None)
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