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Theorem difopn 20761
Description: The difference of a closed set with an open set is open. (Contributed by Mario Carneiro, 6-Jan-2014.)
Hypothesis
Ref Expression
iscld.1 𝑋 = 𝐽
Assertion
Ref Expression
difopn ((𝐴𝐽𝐵 ∈ (Clsd‘𝐽)) → (𝐴𝐵) ∈ 𝐽)

Proof of Theorem difopn
StepHypRef Expression
1 elssuni 4438 . . . . . 6 (𝐴𝐽𝐴 𝐽)
2 iscld.1 . . . . . 6 𝑋 = 𝐽
31, 2syl6sseqr 3636 . . . . 5 (𝐴𝐽𝐴𝑋)
43adantr 481 . . . 4 ((𝐴𝐽𝐵 ∈ (Clsd‘𝐽)) → 𝐴𝑋)
5 df-ss 3573 . . . 4 (𝐴𝑋 ↔ (𝐴𝑋) = 𝐴)
64, 5sylib 208 . . 3 ((𝐴𝐽𝐵 ∈ (Clsd‘𝐽)) → (𝐴𝑋) = 𝐴)
76difeq1d 3710 . 2 ((𝐴𝐽𝐵 ∈ (Clsd‘𝐽)) → ((𝐴𝑋) ∖ 𝐵) = (𝐴𝐵))
8 indif2 3851 . . 3 (𝐴 ∩ (𝑋𝐵)) = ((𝐴𝑋) ∖ 𝐵)
9 cldrcl 20753 . . . . 5 (𝐵 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top)
109adantl 482 . . . 4 ((𝐴𝐽𝐵 ∈ (Clsd‘𝐽)) → 𝐽 ∈ Top)
11 simpl 473 . . . 4 ((𝐴𝐽𝐵 ∈ (Clsd‘𝐽)) → 𝐴𝐽)
122cldopn 20758 . . . . 5 (𝐵 ∈ (Clsd‘𝐽) → (𝑋𝐵) ∈ 𝐽)
1312adantl 482 . . . 4 ((𝐴𝐽𝐵 ∈ (Clsd‘𝐽)) → (𝑋𝐵) ∈ 𝐽)
14 inopn 20636 . . . 4 ((𝐽 ∈ Top ∧ 𝐴𝐽 ∧ (𝑋𝐵) ∈ 𝐽) → (𝐴 ∩ (𝑋𝐵)) ∈ 𝐽)
1510, 11, 13, 14syl3anc 1323 . . 3 ((𝐴𝐽𝐵 ∈ (Clsd‘𝐽)) → (𝐴 ∩ (𝑋𝐵)) ∈ 𝐽)
168, 15syl5eqelr 2703 . 2 ((𝐴𝐽𝐵 ∈ (Clsd‘𝐽)) → ((𝐴𝑋) ∖ 𝐵) ∈ 𝐽)
177, 16eqeltrrd 2699 1 ((𝐴𝐽𝐵 ∈ (Clsd‘𝐽)) → (𝐴𝐵) ∈ 𝐽)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  cdif 3556  cin 3558  wss 3559   cuni 4407  cfv 5852  Topctop 20630  Clsdccld 20743
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-iota 5815  df-fun 5854  df-fn 5855  df-fv 5860  df-top 20631  df-cld 20746
This theorem is referenced by:  bcthlem5  23048  cldssbrsiga  30055  poimirlem30  33106  dirkercncflem2  39654  fourierdlem62  39718
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