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Theorem ntrclsk4 39141
Description: Idempotence of the interior function is equivalent to idempotence of the closure function. (Contributed by RP, 10-Jul-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)
Assertion
Ref Expression
ntrclsk4 (𝜑 → (∀𝑠 ∈ 𝒫 𝐵(𝐼‘(𝐼𝑠)) = (𝐼𝑠) ↔ ∀𝑠 ∈ 𝒫 𝐵(𝐾‘(𝐾𝑠)) = (𝐾𝑠)))
Distinct variable groups:   𝐵,𝑖,𝑗,𝑘,𝑠   𝑗,𝐼,𝑘,𝑠   𝑗,𝐾   𝜑,𝑖,𝑗,𝑘,𝑠
Allowed substitution hints:   𝐷(𝑖,𝑗,𝑘,𝑠)   𝐼(𝑖)   𝐾(𝑖,𝑘,𝑠)   𝑂(𝑖,𝑗,𝑘,𝑠)

Proof of Theorem ntrclsk4
Dummy variable 𝑡 is distinct from all other variables.
StepHypRef Expression
1 2fveq3 6415 . . . 4 (𝑠 = 𝑡 → (𝐼‘(𝐼𝑠)) = (𝐼‘(𝐼𝑡)))
2 fveq2 6410 . . . 4 (𝑠 = 𝑡 → (𝐼𝑠) = (𝐼𝑡))
31, 2eqeq12d 2813 . . 3 (𝑠 = 𝑡 → ((𝐼‘(𝐼𝑠)) = (𝐼𝑠) ↔ (𝐼‘(𝐼𝑡)) = (𝐼𝑡)))
43cbvralv 3353 . 2 (∀𝑠 ∈ 𝒫 𝐵(𝐼‘(𝐼𝑠)) = (𝐼𝑠) ↔ ∀𝑡 ∈ 𝒫 𝐵(𝐼‘(𝐼𝑡)) = (𝐼𝑡))
5 ntrcls.d . . . . 5 𝐷 = (𝑂𝐵)
6 ntrcls.r . . . . 5 (𝜑𝐼𝐷𝐾)
75, 6ntrclsrcomplex 39104 . . . 4 (𝜑 → (𝐵𝑠) ∈ 𝒫 𝐵)
87adantr 473 . . 3 ((𝜑𝑠 ∈ 𝒫 𝐵) → (𝐵𝑠) ∈ 𝒫 𝐵)
95, 6ntrclsrcomplex 39104 . . . . 5 (𝜑 → (𝐵𝑡) ∈ 𝒫 𝐵)
109adantr 473 . . . 4 ((𝜑𝑡 ∈ 𝒫 𝐵) → (𝐵𝑡) ∈ 𝒫 𝐵)
11 difeq2 3919 . . . . . 6 (𝑠 = (𝐵𝑡) → (𝐵𝑠) = (𝐵 ∖ (𝐵𝑡)))
1211eqeq2d 2808 . . . . 5 (𝑠 = (𝐵𝑡) → (𝑡 = (𝐵𝑠) ↔ 𝑡 = (𝐵 ∖ (𝐵𝑡))))
1312adantl 474 . . . 4 (((𝜑𝑡 ∈ 𝒫 𝐵) ∧ 𝑠 = (𝐵𝑡)) → (𝑡 = (𝐵𝑠) ↔ 𝑡 = (𝐵 ∖ (𝐵𝑡))))
14 elpwi 4358 . . . . . . 7 (𝑡 ∈ 𝒫 𝐵𝑡𝐵)
15 dfss4 4058 . . . . . . 7 (𝑡𝐵 ↔ (𝐵 ∖ (𝐵𝑡)) = 𝑡)
1614, 15sylib 210 . . . . . 6 (𝑡 ∈ 𝒫 𝐵 → (𝐵 ∖ (𝐵𝑡)) = 𝑡)
1716eqcomd 2804 . . . . 5 (𝑡 ∈ 𝒫 𝐵𝑡 = (𝐵 ∖ (𝐵𝑡)))
1817adantl 474 . . . 4 ((𝜑𝑡 ∈ 𝒫 𝐵) → 𝑡 = (𝐵 ∖ (𝐵𝑡)))
1910, 13, 18rspcedvd 3503 . . 3 ((𝜑𝑡 ∈ 𝒫 𝐵) → ∃𝑠 ∈ 𝒫 𝐵𝑡 = (𝐵𝑠))
20 2fveq3 6415 . . . . . 6 (𝑡 = (𝐵𝑠) → (𝐼‘(𝐼𝑡)) = (𝐼‘(𝐼‘(𝐵𝑠))))
21 fveq2 6410 . . . . . 6 (𝑡 = (𝐵𝑠) → (𝐼𝑡) = (𝐼‘(𝐵𝑠)))
2220, 21eqeq12d 2813 . . . . 5 (𝑡 = (𝐵𝑠) → ((𝐼‘(𝐼𝑡)) = (𝐼𝑡) ↔ (𝐼‘(𝐼‘(𝐵𝑠))) = (𝐼‘(𝐵𝑠))))
23223ad2ant3 1166 . . . 4 ((𝜑𝑠 ∈ 𝒫 𝐵𝑡 = (𝐵𝑠)) → ((𝐼‘(𝐼𝑡)) = (𝐼𝑡) ↔ (𝐼‘(𝐼‘(𝐵𝑠))) = (𝐼‘(𝐵𝑠))))
24 ntrcls.o . . . . . . . . . . . 12 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖𝑚 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖𝑗))))))
2524, 5, 6ntrclsiex 39122 . . . . . . . . . . 11 (𝜑𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵))
26 elmapi 8116 . . . . . . . . . . 11 (𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵) → 𝐼:𝒫 𝐵⟶𝒫 𝐵)
2725, 26syl 17 . . . . . . . . . 10 (𝜑𝐼:𝒫 𝐵⟶𝒫 𝐵)
2827, 7ffvelrnd 6585 . . . . . . . . . 10 (𝜑 → (𝐼‘(𝐵𝑠)) ∈ 𝒫 𝐵)
2927, 28ffvelrnd 6585 . . . . . . . . 9 (𝜑 → (𝐼‘(𝐼‘(𝐵𝑠))) ∈ 𝒫 𝐵)
3029elpwid 4360 . . . . . . . 8 (𝜑 → (𝐼‘(𝐼‘(𝐵𝑠))) ⊆ 𝐵)
3128elpwid 4360 . . . . . . . 8 (𝜑 → (𝐼‘(𝐵𝑠)) ⊆ 𝐵)
32 rcompleq 39089 . . . . . . . 8 (((𝐼‘(𝐼‘(𝐵𝑠))) ⊆ 𝐵 ∧ (𝐼‘(𝐵𝑠)) ⊆ 𝐵) → ((𝐼‘(𝐼‘(𝐵𝑠))) = (𝐼‘(𝐵𝑠)) ↔ (𝐵 ∖ (𝐼‘(𝐼‘(𝐵𝑠)))) = (𝐵 ∖ (𝐼‘(𝐵𝑠)))))
3330, 31, 32syl2anc 580 . . . . . . 7 (𝜑 → ((𝐼‘(𝐼‘(𝐵𝑠))) = (𝐼‘(𝐵𝑠)) ↔ (𝐵 ∖ (𝐼‘(𝐼‘(𝐵𝑠)))) = (𝐵 ∖ (𝐼‘(𝐵𝑠)))))
3433adantr 473 . . . . . 6 ((𝜑𝑠 ∈ 𝒫 𝐵) → ((𝐼‘(𝐼‘(𝐵𝑠))) = (𝐼‘(𝐵𝑠)) ↔ (𝐵 ∖ (𝐼‘(𝐼‘(𝐵𝑠)))) = (𝐵 ∖ (𝐼‘(𝐵𝑠)))))
3524, 5, 6ntrclsnvobr 39121 . . . . . . . . . 10 (𝜑𝐾𝐷𝐼)
3635adantr 473 . . . . . . . . 9 ((𝜑𝑠 ∈ 𝒫 𝐵) → 𝐾𝐷𝐼)
3724, 5, 35ntrclsiex 39122 . . . . . . . . . . 11 (𝜑𝐾 ∈ (𝒫 𝐵𝑚 𝒫 𝐵))
38 elmapi 8116 . . . . . . . . . . 11 (𝐾 ∈ (𝒫 𝐵𝑚 𝒫 𝐵) → 𝐾:𝒫 𝐵⟶𝒫 𝐵)
3937, 38syl 17 . . . . . . . . . 10 (𝜑𝐾:𝒫 𝐵⟶𝒫 𝐵)
4039ffvelrnda 6584 . . . . . . . . 9 ((𝜑𝑠 ∈ 𝒫 𝐵) → (𝐾𝑠) ∈ 𝒫 𝐵)
4124, 5, 36, 40ntrclsfv 39128 . . . . . . . 8 ((𝜑𝑠 ∈ 𝒫 𝐵) → (𝐾‘(𝐾𝑠)) = (𝐵 ∖ (𝐼‘(𝐵 ∖ (𝐾𝑠)))))
42 simpr 478 . . . . . . . . . . . . 13 ((𝜑𝑠 ∈ 𝒫 𝐵) → 𝑠 ∈ 𝒫 𝐵)
4324, 5, 36, 42ntrclsfv 39128 . . . . . . . . . . . 12 ((𝜑𝑠 ∈ 𝒫 𝐵) → (𝐾𝑠) = (𝐵 ∖ (𝐼‘(𝐵𝑠))))
4443difeq2d 3925 . . . . . . . . . . 11 ((𝜑𝑠 ∈ 𝒫 𝐵) → (𝐵 ∖ (𝐾𝑠)) = (𝐵 ∖ (𝐵 ∖ (𝐼‘(𝐵𝑠)))))
45 dfss4 4058 . . . . . . . . . . . . 13 ((𝐼‘(𝐵𝑠)) ⊆ 𝐵 ↔ (𝐵 ∖ (𝐵 ∖ (𝐼‘(𝐵𝑠)))) = (𝐼‘(𝐵𝑠)))
4631, 45sylib 210 . . . . . . . . . . . 12 (𝜑 → (𝐵 ∖ (𝐵 ∖ (𝐼‘(𝐵𝑠)))) = (𝐼‘(𝐵𝑠)))
4746adantr 473 . . . . . . . . . . 11 ((𝜑𝑠 ∈ 𝒫 𝐵) → (𝐵 ∖ (𝐵 ∖ (𝐼‘(𝐵𝑠)))) = (𝐼‘(𝐵𝑠)))
4844, 47eqtrd 2832 . . . . . . . . . 10 ((𝜑𝑠 ∈ 𝒫 𝐵) → (𝐵 ∖ (𝐾𝑠)) = (𝐼‘(𝐵𝑠)))
4948fveq2d 6414 . . . . . . . . 9 ((𝜑𝑠 ∈ 𝒫 𝐵) → (𝐼‘(𝐵 ∖ (𝐾𝑠))) = (𝐼‘(𝐼‘(𝐵𝑠))))
5049difeq2d 3925 . . . . . . . 8 ((𝜑𝑠 ∈ 𝒫 𝐵) → (𝐵 ∖ (𝐼‘(𝐵 ∖ (𝐾𝑠)))) = (𝐵 ∖ (𝐼‘(𝐼‘(𝐵𝑠)))))
5141, 50eqtrd 2832 . . . . . . 7 ((𝜑𝑠 ∈ 𝒫 𝐵) → (𝐾‘(𝐾𝑠)) = (𝐵 ∖ (𝐼‘(𝐼‘(𝐵𝑠)))))
5251, 43eqeq12d 2813 . . . . . 6 ((𝜑𝑠 ∈ 𝒫 𝐵) → ((𝐾‘(𝐾𝑠)) = (𝐾𝑠) ↔ (𝐵 ∖ (𝐼‘(𝐼‘(𝐵𝑠)))) = (𝐵 ∖ (𝐼‘(𝐵𝑠)))))
5334, 52bitr4d 274 . . . . 5 ((𝜑𝑠 ∈ 𝒫 𝐵) → ((𝐼‘(𝐼‘(𝐵𝑠))) = (𝐼‘(𝐵𝑠)) ↔ (𝐾‘(𝐾𝑠)) = (𝐾𝑠)))
54533adant3 1163 . . . 4 ((𝜑𝑠 ∈ 𝒫 𝐵𝑡 = (𝐵𝑠)) → ((𝐼‘(𝐼‘(𝐵𝑠))) = (𝐼‘(𝐵𝑠)) ↔ (𝐾‘(𝐾𝑠)) = (𝐾𝑠)))
5523, 54bitrd 271 . . 3 ((𝜑𝑠 ∈ 𝒫 𝐵𝑡 = (𝐵𝑠)) → ((𝐼‘(𝐼𝑡)) = (𝐼𝑡) ↔ (𝐾‘(𝐾𝑠)) = (𝐾𝑠)))
568, 19, 55ralxfrd2 5081 . 2 (𝜑 → (∀𝑡 ∈ 𝒫 𝐵(𝐼‘(𝐼𝑡)) = (𝐼𝑡) ↔ ∀𝑠 ∈ 𝒫 𝐵(𝐾‘(𝐾𝑠)) = (𝐾𝑠)))
574, 56syl5bb 275 1 (𝜑 → (∀𝑠 ∈ 𝒫 𝐵(𝐼‘(𝐼𝑠)) = (𝐼𝑠) ↔ ∀𝑠 ∈ 𝒫 𝐵(𝐾‘(𝐾𝑠)) = (𝐾𝑠)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385  w3a 1108   = wceq 1653  wcel 2157  wral 3088  Vcvv 3384  cdif 3765  wss 3768  𝒫 cpw 4348   class class class wbr 4842  cmpt 4921  wf 6096  cfv 6100  (class class class)co 6877  𝑚 cmap 8094
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2776  ax-rep 4963  ax-sep 4974  ax-nul 4982  ax-pow 5034  ax-pr 5096  ax-un 7182
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2785  df-cleq 2791  df-clel 2794  df-nfc 2929  df-ne 2971  df-ral 3093  df-rex 3094  df-reu 3095  df-rab 3097  df-v 3386  df-sbc 3633  df-csb 3728  df-dif 3771  df-un 3773  df-in 3775  df-ss 3782  df-nul 4115  df-if 4277  df-pw 4350  df-sn 4368  df-pr 4370  df-op 4374  df-uni 4628  df-iun 4711  df-br 4843  df-opab 4905  df-mpt 4922  df-id 5219  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-iota 6063  df-fun 6102  df-fn 6103  df-f 6104  df-f1 6105  df-fo 6106  df-f1o 6107  df-fv 6108  df-ov 6880  df-oprab 6881  df-mpt2 6882  df-1st 7400  df-2nd 7401  df-map 8096
This theorem is referenced by: (None)
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