Theorem topbnd 37208

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 23380	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → ((cls‘𝐽)‘(𝑋 ∖ 𝐴)) = (𝑋 ∖ ((int‘𝐽)‘𝐴)))
3	2	ineq2d 4116	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋 ∖ 𝐴))) = (((cls‘𝐽)‘𝐴) ∩ (𝑋 ∖ ((int‘𝐽)‘𝐴))))
4		indif2 4225	. . 3 ⊢ (((cls‘𝐽)‘𝐴) ∩ (𝑋 ∖ ((int‘𝐽)‘𝐴))) = ((((cls‘𝐽)‘𝐴) ∩ 𝑋) ∖ ((int‘𝐽)‘𝐴))
5	3, 4	eqtrdi 2811	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋 ∖ 𝐴))) = ((((cls‘𝐽)‘𝐴) ∩ 𝑋) ∖ ((int‘𝐽)‘𝐴)))
6	1	clsss3 23386	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → ((cls‘𝐽)‘𝐴) ⊆ 𝑋)
7		dfss2 4093	. . . 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 4061	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → ((((cls‘𝐽)‘𝐴) ∩ 𝑋) ∖ ((int‘𝐽)‘𝐴)) = (((cls‘𝐽)‘𝐴) ∖ ((int‘𝐽)‘𝐴)))
10	5, 9	eqtrd 2795	1 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋 ∖ 𝐴))) = (((cls‘𝐽)‘𝐴) ∖ ((int‘𝐽)‘𝐴)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144   ⊆ wss 3896   ∖ cdif 4044   ∩ cin 4088  ∪ cuni 4866  ‘cfv 6543  Topctop 23220  intcnt 23344  clsccl 23345

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5327  ax-pr 5391  ax-un 7755

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-int 4907  df-iun 4952  df-iin 4953  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6497  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-top 23221  df-cld 23346  df-ntr 23347  df-cls 23348

This theorem is used by:  opnbnd  37209