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Theorem topbnd 35699
Description: Two equivalent expressions for the boundary of a topology. (Contributed by Jeff Hankins, 23-Sep-2009.)
Hypothesis
Ref Expression
topbnd.1 𝑋 = βˆͺ 𝐽
Assertion
Ref Expression
topbnd ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋) β†’ (((clsβ€˜π½)β€˜π΄) ∩ ((clsβ€˜π½)β€˜(𝑋 βˆ– 𝐴))) = (((clsβ€˜π½)β€˜π΄) βˆ– ((intβ€˜π½)β€˜π΄)))

Proof of Theorem topbnd
StepHypRef Expression
1 topbnd.1 . . . . 5 𝑋 = βˆͺ 𝐽
21clsdif 22879 . . . 4 ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋) β†’ ((clsβ€˜π½)β€˜(𝑋 βˆ– 𝐴)) = (𝑋 βˆ– ((intβ€˜π½)β€˜π΄)))
32ineq2d 4204 . . 3 ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋) β†’ (((clsβ€˜π½)β€˜π΄) ∩ ((clsβ€˜π½)β€˜(𝑋 βˆ– 𝐴))) = (((clsβ€˜π½)β€˜π΄) ∩ (𝑋 βˆ– ((intβ€˜π½)β€˜π΄))))
4 indif2 4262 . . 3 (((clsβ€˜π½)β€˜π΄) ∩ (𝑋 βˆ– ((intβ€˜π½)β€˜π΄))) = ((((clsβ€˜π½)β€˜π΄) ∩ 𝑋) βˆ– ((intβ€˜π½)β€˜π΄))
53, 4eqtrdi 2780 . 2 ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋) β†’ (((clsβ€˜π½)β€˜π΄) ∩ ((clsβ€˜π½)β€˜(𝑋 βˆ– 𝐴))) = ((((clsβ€˜π½)β€˜π΄) ∩ 𝑋) βˆ– ((intβ€˜π½)β€˜π΄)))
61clsss3 22885 . . . 4 ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋) β†’ ((clsβ€˜π½)β€˜π΄) βŠ† 𝑋)
7 df-ss 3957 . . . 4 (((clsβ€˜π½)β€˜π΄) βŠ† 𝑋 ↔ (((clsβ€˜π½)β€˜π΄) ∩ 𝑋) = ((clsβ€˜π½)β€˜π΄))
86, 7sylib 217 . . 3 ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋) β†’ (((clsβ€˜π½)β€˜π΄) ∩ 𝑋) = ((clsβ€˜π½)β€˜π΄))
98difeq1d 4113 . 2 ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋) β†’ ((((clsβ€˜π½)β€˜π΄) ∩ 𝑋) βˆ– ((intβ€˜π½)β€˜π΄)) = (((clsβ€˜π½)β€˜π΄) βˆ– ((intβ€˜π½)β€˜π΄)))
105, 9eqtrd 2764 1 ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋) β†’ (((clsβ€˜π½)β€˜π΄) ∩ ((clsβ€˜π½)β€˜(𝑋 βˆ– 𝐴))) = (((clsβ€˜π½)β€˜π΄) βˆ– ((intβ€˜π½)β€˜π΄)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098   βˆ– cdif 3937   ∩ cin 3939   βŠ† wss 3940  βˆͺ cuni 4899  β€˜cfv 6533  Topctop 22717  intcnt 22843  clsccl 22844
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-int 4941  df-iun 4989  df-iin 4990  df-br 5139  df-opab 5201  df-mpt 5222  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-top 22718  df-cld 22845  df-ntr 22846  df-cls 22847
This theorem is referenced by:  opnbnd  35700
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