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Theorem ntrclsk2 37887
Description: An interior function is contracting if and only if the closure function is expansive. (Contributed by RP, 9-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)
Assertion
Ref Expression
ntrclsk2 (𝜑 → (∀𝑠 ∈ 𝒫 𝐵(𝐼𝑠) ⊆ 𝑠 ↔ ∀𝑠 ∈ 𝒫 𝐵𝑠 ⊆ (𝐾𝑠)))
Distinct variable groups:   𝐵,𝑖,𝑗,𝑘,𝑠   𝑗,𝐼,𝑘,𝑠   𝜑,𝑖,𝑗,𝑘,𝑠
Allowed substitution hints:   𝐷(𝑖,𝑗,𝑘,𝑠)   𝐼(𝑖)   𝐾(𝑖,𝑗,𝑘,𝑠)   𝑂(𝑖,𝑗,𝑘,𝑠)

Proof of Theorem ntrclsk2
Dummy variable 𝑡 is distinct from all other variables.
StepHypRef Expression
1 fveq2 6158 . . . 4 (𝑠 = 𝑡 → (𝐼𝑠) = (𝐼𝑡))
2 id 22 . . . 4 (𝑠 = 𝑡𝑠 = 𝑡)
31, 2sseq12d 3619 . . 3 (𝑠 = 𝑡 → ((𝐼𝑠) ⊆ 𝑠 ↔ (𝐼𝑡) ⊆ 𝑡))
43cbvralv 3163 . 2 (∀𝑠 ∈ 𝒫 𝐵(𝐼𝑠) ⊆ 𝑠 ↔ ∀𝑡 ∈ 𝒫 𝐵(𝐼𝑡) ⊆ 𝑡)
5 ntrcls.d . . . . 5 𝐷 = (𝑂𝐵)
6 ntrcls.r . . . . 5 (𝜑𝐼𝐷𝐾)
75, 6ntrclsrcomplex 37854 . . . 4 (𝜑 → (𝐵𝑠) ∈ 𝒫 𝐵)
87adantr 481 . . 3 ((𝜑𝑠 ∈ 𝒫 𝐵) → (𝐵𝑠) ∈ 𝒫 𝐵)
95, 6ntrclsrcomplex 37854 . . . . 5 (𝜑 → (𝐵𝑡) ∈ 𝒫 𝐵)
109adantr 481 . . . 4 ((𝜑𝑡 ∈ 𝒫 𝐵) → (𝐵𝑡) ∈ 𝒫 𝐵)
11 difeq2 3706 . . . . . 6 (𝑠 = (𝐵𝑡) → (𝐵𝑠) = (𝐵 ∖ (𝐵𝑡)))
1211eqeq2d 2631 . . . . 5 (𝑠 = (𝐵𝑡) → (𝑡 = (𝐵𝑠) ↔ 𝑡 = (𝐵 ∖ (𝐵𝑡))))
1312adantl 482 . . . 4 (((𝜑𝑡 ∈ 𝒫 𝐵) ∧ 𝑠 = (𝐵𝑡)) → (𝑡 = (𝐵𝑠) ↔ 𝑡 = (𝐵 ∖ (𝐵𝑡))))
14 elpwi 4146 . . . . . . 7 (𝑡 ∈ 𝒫 𝐵𝑡𝐵)
15 dfss4 3842 . . . . . . 7 (𝑡𝐵 ↔ (𝐵 ∖ (𝐵𝑡)) = 𝑡)
1614, 15sylib 208 . . . . . 6 (𝑡 ∈ 𝒫 𝐵 → (𝐵 ∖ (𝐵𝑡)) = 𝑡)
1716adantl 482 . . . . 5 ((𝜑𝑡 ∈ 𝒫 𝐵) → (𝐵 ∖ (𝐵𝑡)) = 𝑡)
1817eqcomd 2627 . . . 4 ((𝜑𝑡 ∈ 𝒫 𝐵) → 𝑡 = (𝐵 ∖ (𝐵𝑡)))
1910, 13, 18rspcedvd 3306 . . 3 ((𝜑𝑡 ∈ 𝒫 𝐵) → ∃𝑠 ∈ 𝒫 𝐵𝑡 = (𝐵𝑠))
20 fveq2 6158 . . . . . 6 (𝑡 = (𝐵𝑠) → (𝐼𝑡) = (𝐼‘(𝐵𝑠)))
21 id 22 . . . . . 6 (𝑡 = (𝐵𝑠) → 𝑡 = (𝐵𝑠))
2220, 21sseq12d 3619 . . . . 5 (𝑡 = (𝐵𝑠) → ((𝐼𝑡) ⊆ 𝑡 ↔ (𝐼‘(𝐵𝑠)) ⊆ (𝐵𝑠)))
23223ad2ant3 1082 . . . 4 ((𝜑𝑠 ∈ 𝒫 𝐵𝑡 = (𝐵𝑠)) → ((𝐼𝑡) ⊆ 𝑡 ↔ (𝐼‘(𝐵𝑠)) ⊆ (𝐵𝑠)))
24 ntrcls.o . . . . . . . . . . . 12 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖𝑚 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖𝑗))))))
2524, 5, 6ntrclsiex 37872 . . . . . . . . . . 11 (𝜑𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵))
26 elmapi 7839 . . . . . . . . . . 11 (𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵) → 𝐼:𝒫 𝐵⟶𝒫 𝐵)
2725, 26syl 17 . . . . . . . . . 10 (𝜑𝐼:𝒫 𝐵⟶𝒫 𝐵)
28273ad2ant1 1080 . . . . . . . . 9 ((𝜑𝑠 ∈ 𝒫 𝐵𝑡 = (𝐵𝑠)) → 𝐼:𝒫 𝐵⟶𝒫 𝐵)
2973ad2ant1 1080 . . . . . . . . 9 ((𝜑𝑠 ∈ 𝒫 𝐵𝑡 = (𝐵𝑠)) → (𝐵𝑠) ∈ 𝒫 𝐵)
3028, 29ffvelrnd 6326 . . . . . . . 8 ((𝜑𝑠 ∈ 𝒫 𝐵𝑡 = (𝐵𝑠)) → (𝐼‘(𝐵𝑠)) ∈ 𝒫 𝐵)
3130elpwid 4148 . . . . . . 7 ((𝜑𝑠 ∈ 𝒫 𝐵𝑡 = (𝐵𝑠)) → (𝐼‘(𝐵𝑠)) ⊆ 𝐵)
32 difssd 3722 . . . . . . 7 ((𝜑𝑠 ∈ 𝒫 𝐵𝑡 = (𝐵𝑠)) → (𝐵𝑠) ⊆ 𝐵)
33 sscon34b 37838 . . . . . . 7 (((𝐼‘(𝐵𝑠)) ⊆ 𝐵 ∧ (𝐵𝑠) ⊆ 𝐵) → ((𝐼‘(𝐵𝑠)) ⊆ (𝐵𝑠) ↔ (𝐵 ∖ (𝐵𝑠)) ⊆ (𝐵 ∖ (𝐼‘(𝐵𝑠)))))
3431, 32, 33syl2anc 692 . . . . . 6 ((𝜑𝑠 ∈ 𝒫 𝐵𝑡 = (𝐵𝑠)) → ((𝐼‘(𝐵𝑠)) ⊆ (𝐵𝑠) ↔ (𝐵 ∖ (𝐵𝑠)) ⊆ (𝐵 ∖ (𝐼‘(𝐵𝑠)))))
35 simp2 1060 . . . . . . 7 ((𝜑𝑠 ∈ 𝒫 𝐵𝑡 = (𝐵𝑠)) → 𝑠 ∈ 𝒫 𝐵)
36 elpwi 4146 . . . . . . . . 9 (𝑠 ∈ 𝒫 𝐵𝑠𝐵)
37 dfss4 3842 . . . . . . . . 9 (𝑠𝐵 ↔ (𝐵 ∖ (𝐵𝑠)) = 𝑠)
3836, 37sylib 208 . . . . . . . 8 (𝑠 ∈ 𝒫 𝐵 → (𝐵 ∖ (𝐵𝑠)) = 𝑠)
3938sseq1d 3617 . . . . . . 7 (𝑠 ∈ 𝒫 𝐵 → ((𝐵 ∖ (𝐵𝑠)) ⊆ (𝐵 ∖ (𝐼‘(𝐵𝑠))) ↔ 𝑠 ⊆ (𝐵 ∖ (𝐼‘(𝐵𝑠)))))
4035, 39syl 17 . . . . . 6 ((𝜑𝑠 ∈ 𝒫 𝐵𝑡 = (𝐵𝑠)) → ((𝐵 ∖ (𝐵𝑠)) ⊆ (𝐵 ∖ (𝐼‘(𝐵𝑠))) ↔ 𝑠 ⊆ (𝐵 ∖ (𝐼‘(𝐵𝑠)))))
4134, 40bitrd 268 . . . . 5 ((𝜑𝑠 ∈ 𝒫 𝐵𝑡 = (𝐵𝑠)) → ((𝐼‘(𝐵𝑠)) ⊆ (𝐵𝑠) ↔ 𝑠 ⊆ (𝐵 ∖ (𝐼‘(𝐵𝑠)))))
425, 6ntrclsbex 37853 . . . . . . . 8 (𝜑𝐵 ∈ V)
43423ad2ant1 1080 . . . . . . 7 ((𝜑𝑠 ∈ 𝒫 𝐵𝑡 = (𝐵𝑠)) → 𝐵 ∈ V)
44253ad2ant1 1080 . . . . . . 7 ((𝜑𝑠 ∈ 𝒫 𝐵𝑡 = (𝐵𝑠)) → 𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵))
45 eqid 2621 . . . . . . 7 (𝐷𝐼) = (𝐷𝐼)
46 eqid 2621 . . . . . . 7 ((𝐷𝐼)‘𝑠) = ((𝐷𝐼)‘𝑠)
4724, 5, 43, 44, 45, 35, 46dssmapfv3d 37834 . . . . . 6 ((𝜑𝑠 ∈ 𝒫 𝐵𝑡 = (𝐵𝑠)) → ((𝐷𝐼)‘𝑠) = (𝐵 ∖ (𝐼‘(𝐵𝑠))))
4847sseq2d 3618 . . . . 5 ((𝜑𝑠 ∈ 𝒫 𝐵𝑡 = (𝐵𝑠)) → (𝑠 ⊆ ((𝐷𝐼)‘𝑠) ↔ 𝑠 ⊆ (𝐵 ∖ (𝐼‘(𝐵𝑠)))))
4924, 5, 6ntrclsfv1 37874 . . . . . . . 8 (𝜑 → (𝐷𝐼) = 𝐾)
5049fveq1d 6160 . . . . . . 7 (𝜑 → ((𝐷𝐼)‘𝑠) = (𝐾𝑠))
5150sseq2d 3618 . . . . . 6 (𝜑 → (𝑠 ⊆ ((𝐷𝐼)‘𝑠) ↔ 𝑠 ⊆ (𝐾𝑠)))
52513ad2ant1 1080 . . . . 5 ((𝜑𝑠 ∈ 𝒫 𝐵𝑡 = (𝐵𝑠)) → (𝑠 ⊆ ((𝐷𝐼)‘𝑠) ↔ 𝑠 ⊆ (𝐾𝑠)))
5341, 48, 523bitr2d 296 . . . 4 ((𝜑𝑠 ∈ 𝒫 𝐵𝑡 = (𝐵𝑠)) → ((𝐼‘(𝐵𝑠)) ⊆ (𝐵𝑠) ↔ 𝑠 ⊆ (𝐾𝑠)))
5423, 53bitrd 268 . . 3 ((𝜑𝑠 ∈ 𝒫 𝐵𝑡 = (𝐵𝑠)) → ((𝐼𝑡) ⊆ 𝑡𝑠 ⊆ (𝐾𝑠)))
558, 19, 54ralxfrd2 4854 . 2 (𝜑 → (∀𝑡 ∈ 𝒫 𝐵(𝐼𝑡) ⊆ 𝑡 ↔ ∀𝑠 ∈ 𝒫 𝐵𝑠 ⊆ (𝐾𝑠)))
564, 55syl5bb 272 1 (𝜑 → (∀𝑠 ∈ 𝒫 𝐵(𝐼𝑠) ⊆ 𝑠 ↔ ∀𝑠 ∈ 𝒫 𝐵𝑠 ⊆ (𝐾𝑠)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2908  Vcvv 3190  cdif 3557  wss 3560  𝒫 cpw 4136   class class class wbr 4623  cmpt 4683  wf 5853  cfv 5857  (class class class)co 6615  𝑚 cmap 7817
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-1st 7128  df-2nd 7129  df-map 7819
This theorem is referenced by: (None)
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