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Theorem ntrclsk4 38789
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 fveq2 6304 . . . . 5 (𝑠 = 𝑡 → (𝐼𝑠) = (𝐼𝑡))
21fveq2d 6308 . . . 4 (𝑠 = 𝑡 → (𝐼‘(𝐼𝑠)) = (𝐼‘(𝐼𝑡)))
32, 1eqeq12d 2739 . . 3 (𝑠 = 𝑡 → ((𝐼‘(𝐼𝑠)) = (𝐼𝑠) ↔ (𝐼‘(𝐼𝑡)) = (𝐼𝑡)))
43cbvralv 3274 . 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 38752 . . . 4 (𝜑 → (𝐵𝑠) ∈ 𝒫 𝐵)
87adantr 472 . . 3 ((𝜑𝑠 ∈ 𝒫 𝐵) → (𝐵𝑠) ∈ 𝒫 𝐵)
95, 6ntrclsrcomplex 38752 . . . . 5 (𝜑 → (𝐵𝑡) ∈ 𝒫 𝐵)
109adantr 472 . . . 4 ((𝜑𝑡 ∈ 𝒫 𝐵) → (𝐵𝑡) ∈ 𝒫 𝐵)
11 difeq2 3830 . . . . . 6 (𝑠 = (𝐵𝑡) → (𝐵𝑠) = (𝐵 ∖ (𝐵𝑡)))
1211eqeq2d 2734 . . . . 5 (𝑠 = (𝐵𝑡) → (𝑡 = (𝐵𝑠) ↔ 𝑡 = (𝐵 ∖ (𝐵𝑡))))
1312adantl 473 . . . 4 (((𝜑𝑡 ∈ 𝒫 𝐵) ∧ 𝑠 = (𝐵𝑡)) → (𝑡 = (𝐵𝑠) ↔ 𝑡 = (𝐵 ∖ (𝐵𝑡))))
14 elpwi 4276 . . . . . . 7 (𝑡 ∈ 𝒫 𝐵𝑡𝐵)
15 dfss4 3966 . . . . . . 7 (𝑡𝐵 ↔ (𝐵 ∖ (𝐵𝑡)) = 𝑡)
1614, 15sylib 208 . . . . . 6 (𝑡 ∈ 𝒫 𝐵 → (𝐵 ∖ (𝐵𝑡)) = 𝑡)
1716eqcomd 2730 . . . . 5 (𝑡 ∈ 𝒫 𝐵𝑡 = (𝐵 ∖ (𝐵𝑡)))
1817adantl 473 . . . 4 ((𝜑𝑡 ∈ 𝒫 𝐵) → 𝑡 = (𝐵 ∖ (𝐵𝑡)))
1910, 13, 18rspcedvd 3421 . . 3 ((𝜑𝑡 ∈ 𝒫 𝐵) → ∃𝑠 ∈ 𝒫 𝐵𝑡 = (𝐵𝑠))
20 fveq2 6304 . . . . . . 7 (𝑡 = (𝐵𝑠) → (𝐼𝑡) = (𝐼‘(𝐵𝑠)))
2120fveq2d 6308 . . . . . 6 (𝑡 = (𝐵𝑠) → (𝐼‘(𝐼𝑡)) = (𝐼‘(𝐼‘(𝐵𝑠))))
2221, 20eqeq12d 2739 . . . . 5 (𝑡 = (𝐵𝑠) → ((𝐼‘(𝐼𝑡)) = (𝐼𝑡) ↔ (𝐼‘(𝐼‘(𝐵𝑠))) = (𝐼‘(𝐵𝑠))))
23223ad2ant3 1127 . . . 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 38770 . . . . . . . . . . 11 (𝜑𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵))
26 elmapi 7996 . . . . . . . . . . 11 (𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵) → 𝐼:𝒫 𝐵⟶𝒫 𝐵)
2725, 26syl 17 . . . . . . . . . 10 (𝜑𝐼:𝒫 𝐵⟶𝒫 𝐵)
2827, 7ffvelrnd 6475 . . . . . . . . . 10 (𝜑 → (𝐼‘(𝐵𝑠)) ∈ 𝒫 𝐵)
2927, 28ffvelrnd 6475 . . . . . . . . 9 (𝜑 → (𝐼‘(𝐼‘(𝐵𝑠))) ∈ 𝒫 𝐵)
3029elpwid 4278 . . . . . . . 8 (𝜑 → (𝐼‘(𝐼‘(𝐵𝑠))) ⊆ 𝐵)
3128elpwid 4278 . . . . . . . 8 (𝜑 → (𝐼‘(𝐵𝑠)) ⊆ 𝐵)
32 rcompleq 38737 . . . . . . . 8 (((𝐼‘(𝐼‘(𝐵𝑠))) ⊆ 𝐵 ∧ (𝐼‘(𝐵𝑠)) ⊆ 𝐵) → ((𝐼‘(𝐼‘(𝐵𝑠))) = (𝐼‘(𝐵𝑠)) ↔ (𝐵 ∖ (𝐼‘(𝐼‘(𝐵𝑠)))) = (𝐵 ∖ (𝐼‘(𝐵𝑠)))))
3330, 31, 32syl2anc 696 . . . . . . 7 (𝜑 → ((𝐼‘(𝐼‘(𝐵𝑠))) = (𝐼‘(𝐵𝑠)) ↔ (𝐵 ∖ (𝐼‘(𝐼‘(𝐵𝑠)))) = (𝐵 ∖ (𝐼‘(𝐵𝑠)))))
3433adantr 472 . . . . . 6 ((𝜑𝑠 ∈ 𝒫 𝐵) → ((𝐼‘(𝐼‘(𝐵𝑠))) = (𝐼‘(𝐵𝑠)) ↔ (𝐵 ∖ (𝐼‘(𝐼‘(𝐵𝑠)))) = (𝐵 ∖ (𝐼‘(𝐵𝑠)))))
3524, 5, 6ntrclsnvobr 38769 . . . . . . . . . 10 (𝜑𝐾𝐷𝐼)
3635adantr 472 . . . . . . . . 9 ((𝜑𝑠 ∈ 𝒫 𝐵) → 𝐾𝐷𝐼)
3724, 5, 35ntrclsiex 38770 . . . . . . . . . . 11 (𝜑𝐾 ∈ (𝒫 𝐵𝑚 𝒫 𝐵))
38 elmapi 7996 . . . . . . . . . . 11 (𝐾 ∈ (𝒫 𝐵𝑚 𝒫 𝐵) → 𝐾:𝒫 𝐵⟶𝒫 𝐵)
3937, 38syl 17 . . . . . . . . . 10 (𝜑𝐾:𝒫 𝐵⟶𝒫 𝐵)
4039ffvelrnda 6474 . . . . . . . . 9 ((𝜑𝑠 ∈ 𝒫 𝐵) → (𝐾𝑠) ∈ 𝒫 𝐵)
4124, 5, 36, 40ntrclsfv 38776 . . . . . . . 8 ((𝜑𝑠 ∈ 𝒫 𝐵) → (𝐾‘(𝐾𝑠)) = (𝐵 ∖ (𝐼‘(𝐵 ∖ (𝐾𝑠)))))
42 simpr 479 . . . . . . . . . . . . 13 ((𝜑𝑠 ∈ 𝒫 𝐵) → 𝑠 ∈ 𝒫 𝐵)
4324, 5, 36, 42ntrclsfv 38776 . . . . . . . . . . . 12 ((𝜑𝑠 ∈ 𝒫 𝐵) → (𝐾𝑠) = (𝐵 ∖ (𝐼‘(𝐵𝑠))))
4443difeq2d 3836 . . . . . . . . . . 11 ((𝜑𝑠 ∈ 𝒫 𝐵) → (𝐵 ∖ (𝐾𝑠)) = (𝐵 ∖ (𝐵 ∖ (𝐼‘(𝐵𝑠)))))
45 dfss4 3966 . . . . . . . . . . . . 13 ((𝐼‘(𝐵𝑠)) ⊆ 𝐵 ↔ (𝐵 ∖ (𝐵 ∖ (𝐼‘(𝐵𝑠)))) = (𝐼‘(𝐵𝑠)))
4631, 45sylib 208 . . . . . . . . . . . 12 (𝜑 → (𝐵 ∖ (𝐵 ∖ (𝐼‘(𝐵𝑠)))) = (𝐼‘(𝐵𝑠)))
4746adantr 472 . . . . . . . . . . 11 ((𝜑𝑠 ∈ 𝒫 𝐵) → (𝐵 ∖ (𝐵 ∖ (𝐼‘(𝐵𝑠)))) = (𝐼‘(𝐵𝑠)))
4844, 47eqtrd 2758 . . . . . . . . . 10 ((𝜑𝑠 ∈ 𝒫 𝐵) → (𝐵 ∖ (𝐾𝑠)) = (𝐼‘(𝐵𝑠)))
4948fveq2d 6308 . . . . . . . . 9 ((𝜑𝑠 ∈ 𝒫 𝐵) → (𝐼‘(𝐵 ∖ (𝐾𝑠))) = (𝐼‘(𝐼‘(𝐵𝑠))))
5049difeq2d 3836 . . . . . . . 8 ((𝜑𝑠 ∈ 𝒫 𝐵) → (𝐵 ∖ (𝐼‘(𝐵 ∖ (𝐾𝑠)))) = (𝐵 ∖ (𝐼‘(𝐼‘(𝐵𝑠)))))
5141, 50eqtrd 2758 . . . . . . 7 ((𝜑𝑠 ∈ 𝒫 𝐵) → (𝐾‘(𝐾𝑠)) = (𝐵 ∖ (𝐼‘(𝐼‘(𝐵𝑠)))))
5251, 43eqeq12d 2739 . . . . . 6 ((𝜑𝑠 ∈ 𝒫 𝐵) → ((𝐾‘(𝐾𝑠)) = (𝐾𝑠) ↔ (𝐵 ∖ (𝐼‘(𝐼‘(𝐵𝑠)))) = (𝐵 ∖ (𝐼‘(𝐵𝑠)))))
5334, 52bitr4d 271 . . . . 5 ((𝜑𝑠 ∈ 𝒫 𝐵) → ((𝐼‘(𝐼‘(𝐵𝑠))) = (𝐼‘(𝐵𝑠)) ↔ (𝐾‘(𝐾𝑠)) = (𝐾𝑠)))
54533adant3 1124 . . . 4 ((𝜑𝑠 ∈ 𝒫 𝐵𝑡 = (𝐵𝑠)) → ((𝐼‘(𝐼‘(𝐵𝑠))) = (𝐼‘(𝐵𝑠)) ↔ (𝐾‘(𝐾𝑠)) = (𝐾𝑠)))
5523, 54bitrd 268 . . 3 ((𝜑𝑠 ∈ 𝒫 𝐵𝑡 = (𝐵𝑠)) → ((𝐼‘(𝐼𝑡)) = (𝐼𝑡) ↔ (𝐾‘(𝐾𝑠)) = (𝐾𝑠)))
568, 19, 55ralxfrd2 4989 . 2 (𝜑 → (∀𝑡 ∈ 𝒫 𝐵(𝐼‘(𝐼𝑡)) = (𝐼𝑡) ↔ ∀𝑠 ∈ 𝒫 𝐵(𝐾‘(𝐾𝑠)) = (𝐾𝑠)))
574, 56syl5bb 272 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 196  wa 383  w3a 1072   = wceq 1596  wcel 2103  wral 3014  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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