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Theorem topbnd 34849
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 22427 . . . 4 ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋) β†’ ((clsβ€˜π½)β€˜(𝑋 βˆ– 𝐴)) = (𝑋 βˆ– ((intβ€˜π½)β€˜π΄)))
32ineq2d 4176 . . 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 4234 . . 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 2789 . 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 22433 . . . 4 ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋) β†’ ((clsβ€˜π½)β€˜π΄) βŠ† 𝑋)
7 df-ss 3931 . . . 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 4085 . 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 2773 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 397   = wceq 1542   ∈ wcel 2107   βˆ– cdif 3911   ∩ cin 3913   βŠ† wss 3914  βˆͺ cuni 4869  β€˜cfv 6500  Topctop 22265  intcnt 22391  clsccl 22392
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-int 4912  df-iun 4960  df-iin 4961  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-top 22266  df-cld 22393  df-ntr 22394  df-cls 22395
This theorem is referenced by:  opnbnd  34850
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