Theorem topbnd 36723

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 ∧ 𝐴 ⊆ 𝑋) → (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋 ∖ 𝐴))) = (((cls‘𝐽)‘𝐴) ∖ ((int‘𝐽)‘𝐴)))

Proof of Theorem topbnd

Step	Hyp	Ref	Expression
1		topbnd.1	. . . . 5 ⊢ 𝑋 = ∪ 𝐽
2	1	clsdif 23178	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → ((cls‘𝐽)‘(𝑋 ∖ 𝐴)) = (𝑋 ∖ ((int‘𝐽)‘𝐴)))
3	2	ineq2d 4181	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋 ∖ 𝐴))) = (((cls‘𝐽)‘𝐴) ∩ (𝑋 ∖ ((int‘𝐽)‘𝐴))))
4		indif2 4242	. . 3 ⊢ (((cls‘𝐽)‘𝐴) ∩ (𝑋 ∖ ((int‘𝐽)‘𝐴))) = ((((cls‘𝐽)‘𝐴) ∩ 𝑋) ∖ ((int‘𝐽)‘𝐴))
5	3, 4	eqtrdi 2820	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋 ∖ 𝐴))) = ((((cls‘𝐽)‘𝐴) ∩ 𝑋) ∖ ((int‘𝐽)‘𝐴)))
6	1	clsss3 23184	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → ((cls‘𝐽)‘𝐴) ⊆ 𝑋)
7		dfss2 3931	. . . 4 ⊢ (((cls‘𝐽)‘𝐴) ⊆ 𝑋 ↔ (((cls‘𝐽)‘𝐴) ∩ 𝑋) = ((cls‘𝐽)‘𝐴))
8	6, 7	sylib 221	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (((cls‘𝐽)‘𝐴) ∩ 𝑋) = ((cls‘𝐽)‘𝐴))
9	8	difeq1d 4088	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → ((((cls‘𝐽)‘𝐴) ∩ 𝑋) ∖ ((int‘𝐽)‘𝐴)) = (((cls‘𝐽)‘𝐴) ∖ ((int‘𝐽)‘𝐴)))
10	5, 9	eqtrd 2804	1 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋 ∖ 𝐴))) = (((cls‘𝐽)‘𝐴) ∖ ((int‘𝐽)‘𝐴)))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1567   ∈ wcel 2149   ∖ cdif 3910   ∩ cin 3912   ⊆ wss 3913  ∪ cuni 4876  ‘cfv 6537  Topctop 23018  intcnt 23142  clsccl 23143

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-int 4917  df-iun 4962  df-iin 4963  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-top 23019  df-cld 23144  df-ntr 23145  df-cls 23146

This theorem is referenced by:  opnbnd  36724