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Theorem ntrclsfveq2 40396
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 40388 . . . . . 6 (𝜑𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵))
5 elmapi 8420 . . . . . 6 (𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵) → 𝐼:𝒫 𝐵⟶𝒫 𝐵)
64, 5syl 17 . . . . 5 (𝜑𝐼:𝒫 𝐵⟶𝒫 𝐵)
72, 3ntrclsrcomplex 40370 . . . . 5 (𝜑 → (𝐵𝑆) ∈ 𝒫 𝐵)
86, 7ffvelrnd 6845 . . . 4 (𝜑 → (𝐼‘(𝐵𝑆)) ∈ 𝒫 𝐵)
98elpwid 4551 . . 3 (𝜑 → (𝐼‘(𝐵𝑆)) ⊆ 𝐵)
10 ntrclsfv.c . . . 4 (𝜑𝐶 ∈ 𝒫 𝐵)
1110elpwid 4551 . . 3 (𝜑𝐶𝐵)
12 rcompleq 40355 . . 3 (((𝐼‘(𝐵𝑆)) ⊆ 𝐵𝐶𝐵) → ((𝐼‘(𝐵𝑆)) = 𝐶 ↔ (𝐵 ∖ (𝐼‘(𝐵𝑆))) = (𝐵𝐶)))
139, 11, 12syl2anc 586 . 2 (𝜑 → ((𝐼‘(𝐵𝑆)) = 𝐶 ↔ (𝐵 ∖ (𝐼‘(𝐵𝑆))) = (𝐵𝐶)))
141, 2, 3ntrclsnvobr 40387 . . . 4 (𝜑𝐾𝐷𝐼)
15 ntrclsfv.s . . . 4 (𝜑𝑆 ∈ 𝒫 𝐵)
161, 2, 14, 15ntrclsfv 40394 . . 3 (𝜑 → (𝐾𝑆) = (𝐵 ∖ (𝐼‘(𝐵𝑆))))
1716eqeq1d 2821 . 2 (𝜑 → ((𝐾𝑆) = (𝐵𝐶) ↔ (𝐵 ∖ (𝐼‘(𝐵𝑆))) = (𝐵𝐶)))
1813, 17bitr4d 284 1 (𝜑 → ((𝐼‘(𝐵𝑆)) = 𝐶 ↔ (𝐾𝑆) = (𝐵𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208   = wceq 1530  wcel 2107  Vcvv 3493  cdif 3931  wss 3934  𝒫 cpw 4537   class class class wbr 5057  cmpt 5137  wf 6344  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 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  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 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  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: (None)
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