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Theorem ntrclsfv 41637
Description: The value of the interior (closure) expressed in terms of the closure (interior). (Contributed by RP, 25-Jun-2021.)
Hypotheses
Ref Expression
ntrcls.o 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖𝑗))))))
ntrcls.d 𝐷 = (𝑂𝐵)
ntrcls.r (𝜑𝐼𝐷𝐾)
ntrclsfv.s (𝜑𝑆 ∈ 𝒫 𝐵)
Assertion
Ref Expression
ntrclsfv (𝜑 → (𝐼𝑆) = (𝐵 ∖ (𝐾‘(𝐵𝑆))))
Distinct variable groups:   𝐵,𝑖,𝑗,𝑘   𝑗,𝐾,𝑘   𝑆,𝑗   𝜑,𝑖,𝑗,𝑘
Allowed substitution hints:   𝐷(𝑖,𝑗,𝑘)   𝑆(𝑖,𝑘)   𝐼(𝑖,𝑗,𝑘)   𝐾(𝑖)   𝑂(𝑖,𝑗,𝑘)

Proof of Theorem ntrclsfv
StepHypRef Expression
1 ntrcls.o . . . 4 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖𝑗))))))
2 ntrcls.d . . . 4 𝐷 = (𝑂𝐵)
3 ntrcls.r . . . 4 (𝜑𝐼𝐷𝐾)
41, 2, 3ntrclsfv2 41634 . . 3 (𝜑 → (𝐷𝐾) = 𝐼)
54fveq1d 6771 . 2 (𝜑 → ((𝐷𝐾)‘𝑆) = (𝐼𝑆))
62, 3ntrclsbex 41612 . . 3 (𝜑𝐵 ∈ V)
71, 2, 3ntrclskex 41632 . . 3 (𝜑𝐾 ∈ (𝒫 𝐵m 𝒫 𝐵))
8 eqid 2740 . . 3 (𝐷𝐾) = (𝐷𝐾)
9 ntrclsfv.s . . 3 (𝜑𝑆 ∈ 𝒫 𝐵)
10 eqid 2740 . . 3 ((𝐷𝐾)‘𝑆) = ((𝐷𝐾)‘𝑆)
111, 2, 6, 7, 8, 9, 10dssmapfv3d 41595 . 2 (𝜑 → ((𝐷𝐾)‘𝑆) = (𝐵 ∖ (𝐾‘(𝐵𝑆))))
125, 11eqtr3d 2782 1 (𝜑 → (𝐼𝑆) = (𝐵 ∖ (𝐾‘(𝐵𝑆))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2110  Vcvv 3431  cdif 3889  𝒫 cpw 4539   class class class wbr 5079  cmpt 5162  cfv 6431  (class class class)co 7269  m cmap 8596
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-rep 5214  ax-sep 5227  ax-nul 5234  ax-pow 5292  ax-pr 5356  ax-un 7580
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-ral 3071  df-rex 3072  df-reu 3073  df-rab 3075  df-v 3433  df-sbc 3721  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-iun 4932  df-br 5080  df-opab 5142  df-mpt 5163  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6389  df-fun 6433  df-fn 6434  df-f 6435  df-f1 6436  df-fo 6437  df-f1o 6438  df-fv 6439  df-ov 7272  df-oprab 7273  df-mpo 7274  df-1st 7822  df-2nd 7823  df-map 8598
This theorem is referenced by:  ntrclsfveq1  41638  ntrclsfveq2  41639  ntrclsfveq  41640  ntrclsss  41641  ntrclscls00  41644  ntrclsk4  41650
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