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Theorem topbnd 32625
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 21059 . . . 4 ((𝐽 ∈ Top ∧ 𝐴𝑋) → ((cls‘𝐽)‘(𝑋𝐴)) = (𝑋 ∖ ((int‘𝐽)‘𝐴)))
32ineq2d 3957 . . 3 ((𝐽 ∈ Top ∧ 𝐴𝑋) → (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋𝐴))) = (((cls‘𝐽)‘𝐴) ∩ (𝑋 ∖ ((int‘𝐽)‘𝐴))))
4 indif2 4013 . . 3 (((cls‘𝐽)‘𝐴) ∩ (𝑋 ∖ ((int‘𝐽)‘𝐴))) = ((((cls‘𝐽)‘𝐴) ∩ 𝑋) ∖ ((int‘𝐽)‘𝐴))
53, 4syl6eq 2810 . 2 ((𝐽 ∈ Top ∧ 𝐴𝑋) → (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋𝐴))) = ((((cls‘𝐽)‘𝐴) ∩ 𝑋) ∖ ((int‘𝐽)‘𝐴)))
61clsss3 21065 . . . 4 ((𝐽 ∈ Top ∧ 𝐴𝑋) → ((cls‘𝐽)‘𝐴) ⊆ 𝑋)
7 df-ss 3729 . . . 4 (((cls‘𝐽)‘𝐴) ⊆ 𝑋 ↔ (((cls‘𝐽)‘𝐴) ∩ 𝑋) = ((cls‘𝐽)‘𝐴))
86, 7sylib 208 . . 3 ((𝐽 ∈ Top ∧ 𝐴𝑋) → (((cls‘𝐽)‘𝐴) ∩ 𝑋) = ((cls‘𝐽)‘𝐴))
98difeq1d 3870 . 2 ((𝐽 ∈ Top ∧ 𝐴𝑋) → ((((cls‘𝐽)‘𝐴) ∩ 𝑋) ∖ ((int‘𝐽)‘𝐴)) = (((cls‘𝐽)‘𝐴) ∖ ((int‘𝐽)‘𝐴)))
105, 9eqtrd 2794 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 383   = wceq 1632  wcel 2139  cdif 3712  cin 3714  wss 3715   cuni 4588  cfv 6049  Topctop 20900  intcnt 21023  clsccl 21024
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-iin 4675  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-top 20901  df-cld 21025  df-ntr 21026  df-cls 21027
This theorem is referenced by:  opnbnd  32626
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