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Theorem ntrclsfveq1 44050
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
ntrclsfveq1 (𝜑 → ((𝐼𝑆) = 𝐶 ↔ (𝐾‘(𝐵𝑆)) = (𝐵𝐶)))
Distinct variable groups:   𝐵,𝑖,𝑗,𝑘   𝑗,𝐾,𝑘   𝑆,𝑗   𝜑,𝑖,𝑗,𝑘
Allowed substitution hints:   𝐶(𝑖,𝑗,𝑘)   𝐷(𝑖,𝑗,𝑘)   𝑆(𝑖,𝑘)   𝐼(𝑖,𝑗,𝑘)   𝐾(𝑖)   𝑂(𝑖,𝑗,𝑘)

Proof of Theorem ntrclsfveq1
StepHypRef Expression
1 ntrclsfv.c . . . . . 6 (𝜑𝐶 ∈ 𝒫 𝐵)
21elpwid 4614 . . . . 5 (𝜑𝐶𝐵)
3 dfss4 4275 . . . . 5 (𝐶𝐵 ↔ (𝐵 ∖ (𝐵𝐶)) = 𝐶)
42, 3sylib 218 . . . 4 (𝜑 → (𝐵 ∖ (𝐵𝐶)) = 𝐶)
54eqcomd 2741 . . 3 (𝜑𝐶 = (𝐵 ∖ (𝐵𝐶)))
65eqeq2d 2746 . 2 (𝜑 → ((𝐵 ∖ (𝐾‘(𝐵𝑆))) = 𝐶 ↔ (𝐵 ∖ (𝐾‘(𝐵𝑆))) = (𝐵 ∖ (𝐵𝐶))))
7 ntrcls.o . . . 4 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖𝑗))))))
8 ntrcls.d . . . 4 𝐷 = (𝑂𝐵)
9 ntrcls.r . . . 4 (𝜑𝐼𝐷𝐾)
10 ntrclsfv.s . . . 4 (𝜑𝑆 ∈ 𝒫 𝐵)
117, 8, 9, 10ntrclsfv 44049 . . 3 (𝜑 → (𝐼𝑆) = (𝐵 ∖ (𝐾‘(𝐵𝑆))))
1211eqeq1d 2737 . 2 (𝜑 → ((𝐼𝑆) = 𝐶 ↔ (𝐵 ∖ (𝐾‘(𝐵𝑆))) = 𝐶))
137, 8, 9ntrclskex 44044 . . . . . 6 (𝜑𝐾 ∈ (𝒫 𝐵m 𝒫 𝐵))
14 elmapi 8888 . . . . . 6 (𝐾 ∈ (𝒫 𝐵m 𝒫 𝐵) → 𝐾:𝒫 𝐵⟶𝒫 𝐵)
1513, 14syl 17 . . . . 5 (𝜑𝐾:𝒫 𝐵⟶𝒫 𝐵)
168, 9ntrclsrcomplex 44025 . . . . 5 (𝜑 → (𝐵𝑆) ∈ 𝒫 𝐵)
1715, 16ffvelcdmd 7105 . . . 4 (𝜑 → (𝐾‘(𝐵𝑆)) ∈ 𝒫 𝐵)
1817elpwid 4614 . . 3 (𝜑 → (𝐾‘(𝐵𝑆)) ⊆ 𝐵)
19 difssd 4147 . . 3 (𝜑 → (𝐵𝐶) ⊆ 𝐵)
20 rcompleq 4311 . . 3 (((𝐾‘(𝐵𝑆)) ⊆ 𝐵 ∧ (𝐵𝐶) ⊆ 𝐵) → ((𝐾‘(𝐵𝑆)) = (𝐵𝐶) ↔ (𝐵 ∖ (𝐾‘(𝐵𝑆))) = (𝐵 ∖ (𝐵𝐶))))
2118, 19, 20syl2anc 584 . 2 (𝜑 → ((𝐾‘(𝐵𝑆)) = (𝐵𝐶) ↔ (𝐵 ∖ (𝐾‘(𝐵𝑆))) = (𝐵 ∖ (𝐵𝐶))))
226, 12, 213bitr4d 311 1 (𝜑 → ((𝐼𝑆) = 𝐶 ↔ (𝐾‘(𝐵𝑆)) = (𝐵𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1537  wcel 2106  Vcvv 3478  cdif 3960  wss 3963  𝒫 cpw 4605   class class class wbr 5148  cmpt 5231  wf 6559  cfv 6563  (class class class)co 7431  m cmap 8865
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-map 8867
This theorem is referenced by:  ntrclsfveq  44052
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