Theorem topbnd 36347

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 22996	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → ((cls‘𝐽)‘(𝑋 ∖ 𝐴)) = (𝑋 ∖ ((int‘𝐽)‘𝐴)))
3	2	ineq2d 4200	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋 ∖ 𝐴))) = (((cls‘𝐽)‘𝐴) ∩ (𝑋 ∖ ((int‘𝐽)‘𝐴))))
4		indif2 4261	. . 3 ⊢ (((cls‘𝐽)‘𝐴) ∩ (𝑋 ∖ ((int‘𝐽)‘𝐴))) = ((((cls‘𝐽)‘𝐴) ∩ 𝑋) ∖ ((int‘𝐽)‘𝐴))
5	3, 4	eqtrdi 2787	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋 ∖ 𝐴))) = ((((cls‘𝐽)‘𝐴) ∩ 𝑋) ∖ ((int‘𝐽)‘𝐴)))
6	1	clsss3 23002	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → ((cls‘𝐽)‘𝐴) ⊆ 𝑋)
7		dfss2 3949	. . . 4 ⊢ (((cls‘𝐽)‘𝐴) ⊆ 𝑋 ↔ (((cls‘𝐽)‘𝐴) ∩ 𝑋) = ((cls‘𝐽)‘𝐴))
8	6, 7	sylib 218	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (((cls‘𝐽)‘𝐴) ∩ 𝑋) = ((cls‘𝐽)‘𝐴))
9	8	difeq1d 4105	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → ((((cls‘𝐽)‘𝐴) ∩ 𝑋) ∖ ((int‘𝐽)‘𝐴)) = (((cls‘𝐽)‘𝐴) ∖ ((int‘𝐽)‘𝐴)))
10	5, 9	eqtrd 2771	1 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋 ∖ 𝐴))) = (((cls‘𝐽)‘𝐴) ∖ ((int‘𝐽)‘𝐴)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ∖ cdif 3928   ∩ cin 3930   ⊆ wss 3931  ∪ cuni 4888  ‘cfv 6536  Topctop 22836  intcnt 22960  clsccl 22961

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-int 4928  df-iun 4974  df-iin 4975  df-br 5125  df-opab 5187  df-mpt 5207  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-top 22837  df-cld 22962  df-ntr 22963  df-cls 22964

This theorem is referenced by:  opnbnd  36348