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Theorem ntrclsfv 38881
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 38878 . . 3 (𝜑 → (𝐷𝐾) = 𝐼)
54fveq1d 6335 . 2 (𝜑 → ((𝐷𝐾)‘𝑆) = (𝐼𝑆))
62, 3ntrclsbex 38856 . . 3 (𝜑𝐵 ∈ V)
71, 2, 3ntrclskex 38876 . . 3 (𝜑𝐾 ∈ (𝒫 𝐵𝑚 𝒫 𝐵))
8 eqid 2771 . . 3 (𝐷𝐾) = (𝐷𝐾)
9 ntrclsfv.s . . 3 (𝜑𝑆 ∈ 𝒫 𝐵)
10 eqid 2771 . . 3 ((𝐷𝐾)‘𝑆) = ((𝐷𝐾)‘𝑆)
111, 2, 6, 7, 8, 9, 10dssmapfv3d 38837 . 2 (𝜑 → ((𝐷𝐾)‘𝑆) = (𝐵 ∖ (𝐾‘(𝐵𝑆))))
125, 11eqtr3d 2807 1 (𝜑 → (𝐼𝑆) = (𝐵 ∖ (𝐾‘(𝐵𝑆))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1631  wcel 2145  Vcvv 3351  cdif 3720  𝒫 cpw 4298   class class class wbr 4787  cmpt 4864  cfv 6030  (class class class)co 6796  𝑚 cmap 8013
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-1st 7319  df-2nd 7320  df-map 8015
This theorem is referenced by:  ntrclsfveq1  38882  ntrclsfveq2  38883  ntrclsfveq  38884  ntrclsss  38885  ntrclscls00  38888  ntrclsk4  38894
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