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Theorem ntrclsfveq1 42646
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 4606 . . . . 5 (𝜑𝐶𝐵)
3 dfss4 4255 . . . . 5 (𝐶𝐵 ↔ (𝐵 ∖ (𝐵𝐶)) = 𝐶)
42, 3sylib 217 . . . 4 (𝜑 → (𝐵 ∖ (𝐵𝐶)) = 𝐶)
54eqcomd 2738 . . 3 (𝜑𝐶 = (𝐵 ∖ (𝐵𝐶)))
65eqeq2d 2743 . 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 42645 . . 3 (𝜑 → (𝐼𝑆) = (𝐵 ∖ (𝐾‘(𝐵𝑆))))
1211eqeq1d 2734 . 2 (𝜑 → ((𝐼𝑆) = 𝐶 ↔ (𝐵 ∖ (𝐾‘(𝐵𝑆))) = 𝐶))
137, 8, 9ntrclskex 42640 . . . . . 6 (𝜑𝐾 ∈ (𝒫 𝐵m 𝒫 𝐵))
14 elmapi 8828 . . . . . 6 (𝐾 ∈ (𝒫 𝐵m 𝒫 𝐵) → 𝐾:𝒫 𝐵⟶𝒫 𝐵)
1513, 14syl 17 . . . . 5 (𝜑𝐾:𝒫 𝐵⟶𝒫 𝐵)
168, 9ntrclsrcomplex 42621 . . . . 5 (𝜑 → (𝐵𝑆) ∈ 𝒫 𝐵)
1715, 16ffvelcdmd 7073 . . . 4 (𝜑 → (𝐾‘(𝐵𝑆)) ∈ 𝒫 𝐵)
1817elpwid 4606 . . 3 (𝜑 → (𝐾‘(𝐵𝑆)) ⊆ 𝐵)
19 difssd 4129 . . 3 (𝜑 → (𝐵𝐶) ⊆ 𝐵)
20 rcompleq 4292 . . 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 310 1 (𝜑 → ((𝐼𝑆) = 𝐶 ↔ (𝐾‘(𝐵𝑆)) = (𝐵𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1541  wcel 2106  Vcvv 3474  cdif 3942  wss 3945  𝒫 cpw 4597   class class class wbr 5142  cmpt 5225  wf 6529  cfv 6533  (class class class)co 7394  m cmap 8805
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5357  ax-pr 5421  ax-un 7709
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3775  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5568  df-xp 5676  df-rel 5677  df-cnv 5678  df-co 5679  df-dm 5680  df-rn 5681  df-res 5682  df-ima 5683  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7397  df-oprab 7398  df-mpo 7399  df-1st 7959  df-2nd 7960  df-map 8807
This theorem is referenced by:  ntrclsfveq  42648
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