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Theorem kur14lem2 33865
Description: Lemma for kur14 33874. Write interior in terms of closure and complement: 𝑖𝐴 = π‘π‘˜π‘π΄ where 𝑐 is complement and π‘˜ is closure. (Contributed by Mario Carneiro, 11-Feb-2015.)
Hypotheses
Ref Expression
kur14lem.j 𝐽 ∈ Top
kur14lem.x 𝑋 = βˆͺ 𝐽
kur14lem.k 𝐾 = (clsβ€˜π½)
kur14lem.i 𝐼 = (intβ€˜π½)
kur14lem.a 𝐴 βŠ† 𝑋
Assertion
Ref Expression
kur14lem2 (πΌβ€˜π΄) = (𝑋 βˆ– (πΎβ€˜(𝑋 βˆ– 𝐴)))

Proof of Theorem kur14lem2
StepHypRef Expression
1 kur14lem.j . . 3 𝐽 ∈ Top
2 kur14lem.a . . 3 𝐴 βŠ† 𝑋
3 kur14lem.x . . . 4 𝑋 = βˆͺ 𝐽
43ntrval2 22425 . . 3 ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋) β†’ ((intβ€˜π½)β€˜π΄) = (𝑋 βˆ– ((clsβ€˜π½)β€˜(𝑋 βˆ– 𝐴))))
51, 2, 4mp2an 691 . 2 ((intβ€˜π½)β€˜π΄) = (𝑋 βˆ– ((clsβ€˜π½)β€˜(𝑋 βˆ– 𝐴)))
6 kur14lem.i . . 3 𝐼 = (intβ€˜π½)
76fveq1i 6847 . 2 (πΌβ€˜π΄) = ((intβ€˜π½)β€˜π΄)
8 kur14lem.k . . . 4 𝐾 = (clsβ€˜π½)
98fveq1i 6847 . . 3 (πΎβ€˜(𝑋 βˆ– 𝐴)) = ((clsβ€˜π½)β€˜(𝑋 βˆ– 𝐴))
109difeq2i 4083 . 2 (𝑋 βˆ– (πΎβ€˜(𝑋 βˆ– 𝐴))) = (𝑋 βˆ– ((clsβ€˜π½)β€˜(𝑋 βˆ– 𝐴)))
115, 7, 103eqtr4i 2771 1 (πΌβ€˜π΄) = (𝑋 βˆ– (πΎβ€˜(𝑋 βˆ– 𝐴)))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542   ∈ wcel 2107   βˆ– cdif 3911   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:  kur14lem6  33869  kur14lem7  33870
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