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Theorem ntrclsk4 40772
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 6654 . . . 4 (𝑠 = 𝑡 → (𝐼‘(𝐼𝑠)) = (𝐼‘(𝐼𝑡)))
2 fveq2 6649 . . . 4 (𝑠 = 𝑡 → (𝐼𝑠) = (𝐼𝑡))
31, 2eqeq12d 2817 . . 3 (𝑠 = 𝑡 → ((𝐼‘(𝐼𝑠)) = (𝐼𝑠) ↔ (𝐼‘(𝐼𝑡)) = (𝐼𝑡)))
43cbvralvw 3399 . 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 40735 . . . 4 (𝜑 → (𝐵𝑠) ∈ 𝒫 𝐵)
87adantr 484 . . 3 ((𝜑𝑠 ∈ 𝒫 𝐵) → (𝐵𝑠) ∈ 𝒫 𝐵)
95, 6ntrclsrcomplex 40735 . . . . 5 (𝜑 → (𝐵𝑡) ∈ 𝒫 𝐵)
109adantr 484 . . . 4 ((𝜑𝑡 ∈ 𝒫 𝐵) → (𝐵𝑡) ∈ 𝒫 𝐵)
11 difeq2 4047 . . . . . 6 (𝑠 = (𝐵𝑡) → (𝐵𝑠) = (𝐵 ∖ (𝐵𝑡)))
1211eqeq2d 2812 . . . . 5 (𝑠 = (𝐵𝑡) → (𝑡 = (𝐵𝑠) ↔ 𝑡 = (𝐵 ∖ (𝐵𝑡))))
1312adantl 485 . . . 4 (((𝜑𝑡 ∈ 𝒫 𝐵) ∧ 𝑠 = (𝐵𝑡)) → (𝑡 = (𝐵𝑠) ↔ 𝑡 = (𝐵 ∖ (𝐵𝑡))))
14 elpwi 4509 . . . . . . 7 (𝑡 ∈ 𝒫 𝐵𝑡𝐵)
15 dfss4 4188 . . . . . . 7 (𝑡𝐵 ↔ (𝐵 ∖ (𝐵𝑡)) = 𝑡)
1614, 15sylib 221 . . . . . 6 (𝑡 ∈ 𝒫 𝐵 → (𝐵 ∖ (𝐵𝑡)) = 𝑡)
1716eqcomd 2807 . . . . 5 (𝑡 ∈ 𝒫 𝐵𝑡 = (𝐵 ∖ (𝐵𝑡)))
1817adantl 485 . . . 4 ((𝜑𝑡 ∈ 𝒫 𝐵) → 𝑡 = (𝐵 ∖ (𝐵𝑡)))
1910, 13, 18rspcedvd 3577 . . 3 ((𝜑𝑡 ∈ 𝒫 𝐵) → ∃𝑠 ∈ 𝒫 𝐵𝑡 = (𝐵𝑠))
20 2fveq3 6654 . . . . . 6 (𝑡 = (𝐵𝑠) → (𝐼‘(𝐼𝑡)) = (𝐼‘(𝐼‘(𝐵𝑠))))
21 fveq2 6649 . . . . . 6 (𝑡 = (𝐵𝑠) → (𝐼𝑡) = (𝐼‘(𝐵𝑠)))
2220, 21eqeq12d 2817 . . . . 5 (𝑡 = (𝐵𝑠) → ((𝐼‘(𝐼𝑡)) = (𝐼𝑡) ↔ (𝐼‘(𝐼‘(𝐵𝑠))) = (𝐼‘(𝐵𝑠))))
23223ad2ant3 1132 . . . 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 40753 . . . . . . . . . . 11 (𝜑𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵))
26 elmapi 8415 . . . . . . . . . . 11 (𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵) → 𝐼:𝒫 𝐵⟶𝒫 𝐵)
2725, 26syl 17 . . . . . . . . . 10 (𝜑𝐼:𝒫 𝐵⟶𝒫 𝐵)
2827, 7ffvelrnd 6833 . . . . . . . . . 10 (𝜑 → (𝐼‘(𝐵𝑠)) ∈ 𝒫 𝐵)
2927, 28ffvelrnd 6833 . . . . . . . . 9 (𝜑 → (𝐼‘(𝐼‘(𝐵𝑠))) ∈ 𝒫 𝐵)
3029elpwid 4511 . . . . . . . 8 (𝜑 → (𝐼‘(𝐼‘(𝐵𝑠))) ⊆ 𝐵)
3128elpwid 4511 . . . . . . . 8 (𝜑 → (𝐼‘(𝐵𝑠)) ⊆ 𝐵)
32 rcompleq 4223 . . . . . . . 8 (((𝐼‘(𝐼‘(𝐵𝑠))) ⊆ 𝐵 ∧ (𝐼‘(𝐵𝑠)) ⊆ 𝐵) → ((𝐼‘(𝐼‘(𝐵𝑠))) = (𝐼‘(𝐵𝑠)) ↔ (𝐵 ∖ (𝐼‘(𝐼‘(𝐵𝑠)))) = (𝐵 ∖ (𝐼‘(𝐵𝑠)))))
3330, 31, 32syl2anc 587 . . . . . . 7 (𝜑 → ((𝐼‘(𝐼‘(𝐵𝑠))) = (𝐼‘(𝐵𝑠)) ↔ (𝐵 ∖ (𝐼‘(𝐼‘(𝐵𝑠)))) = (𝐵 ∖ (𝐼‘(𝐵𝑠)))))
3433adantr 484 . . . . . 6 ((𝜑𝑠 ∈ 𝒫 𝐵) → ((𝐼‘(𝐼‘(𝐵𝑠))) = (𝐼‘(𝐵𝑠)) ↔ (𝐵 ∖ (𝐼‘(𝐼‘(𝐵𝑠)))) = (𝐵 ∖ (𝐼‘(𝐵𝑠)))))
3524, 5, 6ntrclsnvobr 40752 . . . . . . . . . 10 (𝜑𝐾𝐷𝐼)
3635adantr 484 . . . . . . . . 9 ((𝜑𝑠 ∈ 𝒫 𝐵) → 𝐾𝐷𝐼)
3724, 5, 35ntrclsiex 40753 . . . . . . . . . . 11 (𝜑𝐾 ∈ (𝒫 𝐵m 𝒫 𝐵))
38 elmapi 8415 . . . . . . . . . . 11 (𝐾 ∈ (𝒫 𝐵m 𝒫 𝐵) → 𝐾:𝒫 𝐵⟶𝒫 𝐵)
3937, 38syl 17 . . . . . . . . . 10 (𝜑𝐾:𝒫 𝐵⟶𝒫 𝐵)
4039ffvelrnda 6832 . . . . . . . . 9 ((𝜑𝑠 ∈ 𝒫 𝐵) → (𝐾𝑠) ∈ 𝒫 𝐵)
4124, 5, 36, 40ntrclsfv 40759 . . . . . . . 8 ((𝜑𝑠 ∈ 𝒫 𝐵) → (𝐾‘(𝐾𝑠)) = (𝐵 ∖ (𝐼‘(𝐵 ∖ (𝐾𝑠)))))
42 simpr 488 . . . . . . . . . . . . 13 ((𝜑𝑠 ∈ 𝒫 𝐵) → 𝑠 ∈ 𝒫 𝐵)
4324, 5, 36, 42ntrclsfv 40759 . . . . . . . . . . . 12 ((𝜑𝑠 ∈ 𝒫 𝐵) → (𝐾𝑠) = (𝐵 ∖ (𝐼‘(𝐵𝑠))))
4443difeq2d 4053 . . . . . . . . . . 11 ((𝜑𝑠 ∈ 𝒫 𝐵) → (𝐵 ∖ (𝐾𝑠)) = (𝐵 ∖ (𝐵 ∖ (𝐼‘(𝐵𝑠)))))
45 dfss4 4188 . . . . . . . . . . . . 13 ((𝐼‘(𝐵𝑠)) ⊆ 𝐵 ↔ (𝐵 ∖ (𝐵 ∖ (𝐼‘(𝐵𝑠)))) = (𝐼‘(𝐵𝑠)))
4631, 45sylib 221 . . . . . . . . . . . 12 (𝜑 → (𝐵 ∖ (𝐵 ∖ (𝐼‘(𝐵𝑠)))) = (𝐼‘(𝐵𝑠)))
4746adantr 484 . . . . . . . . . . 11 ((𝜑𝑠 ∈ 𝒫 𝐵) → (𝐵 ∖ (𝐵 ∖ (𝐼‘(𝐵𝑠)))) = (𝐼‘(𝐵𝑠)))
4844, 47eqtrd 2836 . . . . . . . . . 10 ((𝜑𝑠 ∈ 𝒫 𝐵) → (𝐵 ∖ (𝐾𝑠)) = (𝐼‘(𝐵𝑠)))
4948fveq2d 6653 . . . . . . . . 9 ((𝜑𝑠 ∈ 𝒫 𝐵) → (𝐼‘(𝐵 ∖ (𝐾𝑠))) = (𝐼‘(𝐼‘(𝐵𝑠))))
5049difeq2d 4053 . . . . . . . 8 ((𝜑𝑠 ∈ 𝒫 𝐵) → (𝐵 ∖ (𝐼‘(𝐵 ∖ (𝐾𝑠)))) = (𝐵 ∖ (𝐼‘(𝐼‘(𝐵𝑠)))))
5141, 50eqtrd 2836 . . . . . . 7 ((𝜑𝑠 ∈ 𝒫 𝐵) → (𝐾‘(𝐾𝑠)) = (𝐵 ∖ (𝐼‘(𝐼‘(𝐵𝑠)))))
5251, 43eqeq12d 2817 . . . . . 6 ((𝜑𝑠 ∈ 𝒫 𝐵) → ((𝐾‘(𝐾𝑠)) = (𝐾𝑠) ↔ (𝐵 ∖ (𝐼‘(𝐼‘(𝐵𝑠)))) = (𝐵 ∖ (𝐼‘(𝐵𝑠)))))
5334, 52bitr4d 285 . . . . 5 ((𝜑𝑠 ∈ 𝒫 𝐵) → ((𝐼‘(𝐼‘(𝐵𝑠))) = (𝐼‘(𝐵𝑠)) ↔ (𝐾‘(𝐾𝑠)) = (𝐾𝑠)))
54533adant3 1129 . . . 4 ((𝜑𝑠 ∈ 𝒫 𝐵𝑡 = (𝐵𝑠)) → ((𝐼‘(𝐼‘(𝐵𝑠))) = (𝐼‘(𝐵𝑠)) ↔ (𝐾‘(𝐾𝑠)) = (𝐾𝑠)))
5523, 54bitrd 282 . . 3 ((𝜑𝑠 ∈ 𝒫 𝐵𝑡 = (𝐵𝑠)) → ((𝐼‘(𝐼𝑡)) = (𝐼𝑡) ↔ (𝐾‘(𝐾𝑠)) = (𝐾𝑠)))
568, 19, 55ralxfrd2 5281 . 2 (𝜑 → (∀𝑡 ∈ 𝒫 𝐵(𝐼‘(𝐼𝑡)) = (𝐼𝑡) ↔ ∀𝑠 ∈ 𝒫 𝐵(𝐾‘(𝐾𝑠)) = (𝐾𝑠)))
574, 56syl5bb 286 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 209  wa 399  w3a 1084   = wceq 1538  wcel 2112  wral 3109  Vcvv 3444  cdif 3881  wss 3884  𝒫 cpw 4500   class class class wbr 5033  cmpt 5113  wf 6324  cfv 6328  (class class class)co 7139  m cmap 8393
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-1st 7675  df-2nd 7676  df-map 8395
This theorem is referenced by: (None)
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