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Theorem ntrclsfveq2 38883
Description: If interior and closure functions are related then specific function values are complementary. (Contributed by RP, 27-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)
ntrclsfv.c (𝜑𝐶 ∈ 𝒫 𝐵)
Assertion
Ref Expression
ntrclsfveq2 (𝜑 → ((𝐼‘(𝐵𝑆)) = 𝐶 ↔ (𝐾𝑆) = (𝐵𝐶)))
Distinct variable groups:   𝐵,𝑖,𝑗,𝑘   𝑗,𝐼,𝑘   𝑆,𝑗   𝜑,𝑖,𝑗,𝑘
Allowed substitution hints:   𝐶(𝑖,𝑗,𝑘)   𝐷(𝑖,𝑗,𝑘)   𝑆(𝑖,𝑘)   𝐼(𝑖)   𝐾(𝑖,𝑗,𝑘)   𝑂(𝑖,𝑗,𝑘)

Proof of Theorem ntrclsfveq2
StepHypRef Expression
1 ntrcls.o . . . . . . 7 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖𝑚 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖𝑗))))))
2 ntrcls.d . . . . . . 7 𝐷 = (𝑂𝐵)
3 ntrcls.r . . . . . . 7 (𝜑𝐼𝐷𝐾)
41, 2, 3ntrclsiex 38875 . . . . . 6 (𝜑𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵))
5 elmapi 8035 . . . . . 6 (𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵) → 𝐼:𝒫 𝐵⟶𝒫 𝐵)
64, 5syl 17 . . . . 5 (𝜑𝐼:𝒫 𝐵⟶𝒫 𝐵)
72, 3ntrclsrcomplex 38857 . . . . 5 (𝜑 → (𝐵𝑆) ∈ 𝒫 𝐵)
86, 7ffvelrnd 6505 . . . 4 (𝜑 → (𝐼‘(𝐵𝑆)) ∈ 𝒫 𝐵)
98elpwid 4310 . . 3 (𝜑 → (𝐼‘(𝐵𝑆)) ⊆ 𝐵)
10 ntrclsfv.c . . . 4 (𝜑𝐶 ∈ 𝒫 𝐵)
1110elpwid 4310 . . 3 (𝜑𝐶𝐵)
12 rcompleq 38842 . . 3 (((𝐼‘(𝐵𝑆)) ⊆ 𝐵𝐶𝐵) → ((𝐼‘(𝐵𝑆)) = 𝐶 ↔ (𝐵 ∖ (𝐼‘(𝐵𝑆))) = (𝐵𝐶)))
139, 11, 12syl2anc 573 . 2 (𝜑 → ((𝐼‘(𝐵𝑆)) = 𝐶 ↔ (𝐵 ∖ (𝐼‘(𝐵𝑆))) = (𝐵𝐶)))
141, 2, 3ntrclsnvobr 38874 . . . 4 (𝜑𝐾𝐷𝐼)
15 ntrclsfv.s . . . 4 (𝜑𝑆 ∈ 𝒫 𝐵)
161, 2, 14, 15ntrclsfv 38881 . . 3 (𝜑 → (𝐾𝑆) = (𝐵 ∖ (𝐼‘(𝐵𝑆))))
1716eqeq1d 2773 . 2 (𝜑 → ((𝐾𝑆) = (𝐵𝐶) ↔ (𝐵 ∖ (𝐼‘(𝐵𝑆))) = (𝐵𝐶)))
1813, 17bitr4d 271 1 (𝜑 → ((𝐼‘(𝐵𝑆)) = 𝐶 ↔ (𝐾𝑆) = (𝐵𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1631  wcel 2145  Vcvv 3351  cdif 3720  wss 3723  𝒫 cpw 4298   class class class wbr 4787  cmpt 4864  wf 6026  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: (None)
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