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Theorem topbnd 33672
Description: Two equivalent expressions for the boundary of a topology. (Contributed by Jeff Hankins, 23-Sep-2009.)
Hypothesis
Ref Expression
topbnd.1 𝑋 = 𝐽
Assertion
Ref Expression
topbnd ((𝐽 ∈ Top ∧ 𝐴𝑋) → (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋𝐴))) = (((cls‘𝐽)‘𝐴) ∖ ((int‘𝐽)‘𝐴)))

Proof of Theorem topbnd
StepHypRef Expression
1 topbnd.1 . . . . 5 𝑋 = 𝐽
21clsdif 21660 . . . 4 ((𝐽 ∈ Top ∧ 𝐴𝑋) → ((cls‘𝐽)‘(𝑋𝐴)) = (𝑋 ∖ ((int‘𝐽)‘𝐴)))
32ineq2d 4188 . . 3 ((𝐽 ∈ Top ∧ 𝐴𝑋) → (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋𝐴))) = (((cls‘𝐽)‘𝐴) ∩ (𝑋 ∖ ((int‘𝐽)‘𝐴))))
4 indif2 4246 . . 3 (((cls‘𝐽)‘𝐴) ∩ (𝑋 ∖ ((int‘𝐽)‘𝐴))) = ((((cls‘𝐽)‘𝐴) ∩ 𝑋) ∖ ((int‘𝐽)‘𝐴))
53, 4syl6eq 2872 . 2 ((𝐽 ∈ Top ∧ 𝐴𝑋) → (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋𝐴))) = ((((cls‘𝐽)‘𝐴) ∩ 𝑋) ∖ ((int‘𝐽)‘𝐴)))
61clsss3 21666 . . . 4 ((𝐽 ∈ Top ∧ 𝐴𝑋) → ((cls‘𝐽)‘𝐴) ⊆ 𝑋)
7 df-ss 3951 . . . 4 (((cls‘𝐽)‘𝐴) ⊆ 𝑋 ↔ (((cls‘𝐽)‘𝐴) ∩ 𝑋) = ((cls‘𝐽)‘𝐴))
86, 7sylib 220 . . 3 ((𝐽 ∈ Top ∧ 𝐴𝑋) → (((cls‘𝐽)‘𝐴) ∩ 𝑋) = ((cls‘𝐽)‘𝐴))
98difeq1d 4097 . 2 ((𝐽 ∈ Top ∧ 𝐴𝑋) → ((((cls‘𝐽)‘𝐴) ∩ 𝑋) ∖ ((int‘𝐽)‘𝐴)) = (((cls‘𝐽)‘𝐴) ∖ ((int‘𝐽)‘𝐴)))
105, 9eqtrd 2856 1 ((𝐽 ∈ Top ∧ 𝐴𝑋) → (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋𝐴))) = (((cls‘𝐽)‘𝐴) ∖ ((int‘𝐽)‘𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398   = wceq 1533  wcel 2110  cdif 3932  cin 3934  wss 3935   cuni 4837  cfv 6354  Topctop 21500  intcnt 21624  clsccl 21625
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5189  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4838  df-int 4876  df-iun 4920  df-iin 4921  df-br 5066  df-opab 5128  df-mpt 5146  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-top 21501  df-cld 21626  df-ntr 21627  df-cls 21628
This theorem is referenced by:  opnbnd  33673
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