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Theorem ntrclsfveq2 38778
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 38770 . . . . . 6 (𝜑𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵))
5 elmapi 7996 . . . . . 6 (𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵) → 𝐼:𝒫 𝐵⟶𝒫 𝐵)
64, 5syl 17 . . . . 5 (𝜑𝐼:𝒫 𝐵⟶𝒫 𝐵)
72, 3ntrclsrcomplex 38752 . . . . 5 (𝜑 → (𝐵𝑆) ∈ 𝒫 𝐵)
86, 7ffvelrnd 6475 . . . 4 (𝜑 → (𝐼‘(𝐵𝑆)) ∈ 𝒫 𝐵)
98elpwid 4278 . . 3 (𝜑 → (𝐼‘(𝐵𝑆)) ⊆ 𝐵)
10 ntrclsfv.c . . . 4 (𝜑𝐶 ∈ 𝒫 𝐵)
1110elpwid 4278 . . 3 (𝜑𝐶𝐵)
12 rcompleq 38737 . . 3 (((𝐼‘(𝐵𝑆)) ⊆ 𝐵𝐶𝐵) → ((𝐼‘(𝐵𝑆)) = 𝐶 ↔ (𝐵 ∖ (𝐼‘(𝐵𝑆))) = (𝐵𝐶)))
139, 11, 12syl2anc 696 . 2 (𝜑 → ((𝐼‘(𝐵𝑆)) = 𝐶 ↔ (𝐵 ∖ (𝐼‘(𝐵𝑆))) = (𝐵𝐶)))
141, 2, 3ntrclsnvobr 38769 . . . 4 (𝜑𝐾𝐷𝐼)
15 ntrclsfv.s . . . 4 (𝜑𝑆 ∈ 𝒫 𝐵)
161, 2, 14, 15ntrclsfv 38776 . . 3 (𝜑 → (𝐾𝑆) = (𝐵 ∖ (𝐼‘(𝐵𝑆))))
1716eqeq1d 2726 . 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 1596  wcel 2103  Vcvv 3304  cdif 3677  wss 3680  𝒫 cpw 4266   class class class wbr 4760  cmpt 4837  wf 5997  cfv 6001  (class class class)co 6765  𝑚 cmap 7974
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-rep 4879  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-ral 3019  df-rex 3020  df-reu 3021  df-rab 3023  df-v 3306  df-sbc 3542  df-csb 3640  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-op 4292  df-uni 4545  df-iun 4630  df-br 4761  df-opab 4821  df-mpt 4838  df-id 5128  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-f1 6006  df-fo 6007  df-f1o 6008  df-fv 6009  df-ov 6768  df-oprab 6769  df-mpt2 6770  df-1st 7285  df-2nd 7286  df-map 7976
This theorem is referenced by: (None)
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