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Theorem ntrclsfv 40399
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 40396 . . 3 (𝜑 → (𝐷𝐾) = 𝐼)
54fveq1d 6665 . 2 (𝜑 → ((𝐷𝐾)‘𝑆) = (𝐼𝑆))
62, 3ntrclsbex 40374 . . 3 (𝜑𝐵 ∈ V)
71, 2, 3ntrclskex 40394 . . 3 (𝜑𝐾 ∈ (𝒫 𝐵m 𝒫 𝐵))
8 eqid 2819 . . 3 (𝐷𝐾) = (𝐷𝐾)
9 ntrclsfv.s . . 3 (𝜑𝑆 ∈ 𝒫 𝐵)
10 eqid 2819 . . 3 ((𝐷𝐾)‘𝑆) = ((𝐷𝐾)‘𝑆)
111, 2, 6, 7, 8, 9, 10dssmapfv3d 40355 . 2 (𝜑 → ((𝐷𝐾)‘𝑆) = (𝐵 ∖ (𝐾‘(𝐵𝑆))))
125, 11eqtr3d 2856 1 (𝜑 → (𝐼𝑆) = (𝐵 ∖ (𝐾‘(𝐵𝑆))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1531  wcel 2108  Vcvv 3493  cdif 3931  𝒫 cpw 4537   class class class wbr 5057  cmpt 5137  cfv 6348  (class class class)co 7148  m cmap 8398
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-reu 3143  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7151  df-oprab 7152  df-mpo 7153  df-1st 7681  df-2nd 7682  df-map 8400
This theorem is referenced by:  ntrclsfveq1  40400  ntrclsfveq2  40401  ntrclsfveq  40402  ntrclsss  40403  ntrclscls00  40406  ntrclsk4  40412
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