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Theorem ntrclsk2 38683
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 6229 . . . 4 (𝑠 = 𝑡 → (𝐼𝑠) = (𝐼𝑡))
2 id 22 . . . 4 (𝑠 = 𝑡𝑠 = 𝑡)
31, 2sseq12d 3667 . . 3 (𝑠 = 𝑡 → ((𝐼𝑠) ⊆ 𝑠 ↔ (𝐼𝑡) ⊆ 𝑡))
43cbvralv 3201 . 2 (∀𝑠 ∈ 𝒫 𝐵(𝐼𝑠) ⊆ 𝑠 ↔ ∀𝑡 ∈ 𝒫 𝐵(𝐼𝑡) ⊆ 𝑡)
5 ntrcls.d . . . . 5 𝐷 = (𝑂𝐵)
6 ntrcls.r . . . . 5 (𝜑𝐼𝐷𝐾)
75, 6ntrclsrcomplex 38650 . . . 4 (𝜑 → (𝐵𝑠) ∈ 𝒫 𝐵)
87adantr 480 . . 3 ((𝜑𝑠 ∈ 𝒫 𝐵) → (𝐵𝑠) ∈ 𝒫 𝐵)
95, 6ntrclsrcomplex 38650 . . . . 5 (𝜑 → (𝐵𝑡) ∈ 𝒫 𝐵)
109adantr 480 . . . 4 ((𝜑𝑡 ∈ 𝒫 𝐵) → (𝐵𝑡) ∈ 𝒫 𝐵)
11 difeq2 3755 . . . . . 6 (𝑠 = (𝐵𝑡) → (𝐵𝑠) = (𝐵 ∖ (𝐵𝑡)))
1211eqeq2d 2661 . . . . 5 (𝑠 = (𝐵𝑡) → (𝑡 = (𝐵𝑠) ↔ 𝑡 = (𝐵 ∖ (𝐵𝑡))))
1312adantl 481 . . . 4 (((𝜑𝑡 ∈ 𝒫 𝐵) ∧ 𝑠 = (𝐵𝑡)) → (𝑡 = (𝐵𝑠) ↔ 𝑡 = (𝐵 ∖ (𝐵𝑡))))
14 elpwi 4201 . . . . . . 7 (𝑡 ∈ 𝒫 𝐵𝑡𝐵)
15 dfss4 3891 . . . . . . 7 (𝑡𝐵 ↔ (𝐵 ∖ (𝐵𝑡)) = 𝑡)
1614, 15sylib 208 . . . . . 6 (𝑡 ∈ 𝒫 𝐵 → (𝐵 ∖ (𝐵𝑡)) = 𝑡)
1716adantl 481 . . . . 5 ((𝜑𝑡 ∈ 𝒫 𝐵) → (𝐵 ∖ (𝐵𝑡)) = 𝑡)
1817eqcomd 2657 . . . 4 ((𝜑𝑡 ∈ 𝒫 𝐵) → 𝑡 = (𝐵 ∖ (𝐵𝑡)))
1910, 13, 18rspcedvd 3348 . . 3 ((𝜑𝑡 ∈ 𝒫 𝐵) → ∃𝑠 ∈ 𝒫 𝐵𝑡 = (𝐵𝑠))
20 fveq2 6229 . . . . . 6 (𝑡 = (𝐵𝑠) → (𝐼𝑡) = (𝐼‘(𝐵𝑠)))
21 id 22 . . . . . 6 (𝑡 = (𝐵𝑠) → 𝑡 = (𝐵𝑠))
2220, 21sseq12d 3667 . . . . 5 (𝑡 = (𝐵𝑠) → ((𝐼𝑡) ⊆ 𝑡 ↔ (𝐼‘(𝐵𝑠)) ⊆ (𝐵𝑠)))
23223ad2ant3 1104 . . . 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 38668 . . . . . . . . . . 11 (𝜑𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵))
26 elmapi 7921 . . . . . . . . . . 11 (𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵) → 𝐼:𝒫 𝐵⟶𝒫 𝐵)
2725, 26syl 17 . . . . . . . . . 10 (𝜑𝐼:𝒫 𝐵⟶𝒫 𝐵)
28273ad2ant1 1102 . . . . . . . . 9 ((𝜑𝑠 ∈ 𝒫 𝐵𝑡 = (𝐵𝑠)) → 𝐼:𝒫 𝐵⟶𝒫 𝐵)
2973ad2ant1 1102 . . . . . . . . 9 ((𝜑𝑠 ∈ 𝒫 𝐵𝑡 = (𝐵𝑠)) → (𝐵𝑠) ∈ 𝒫 𝐵)
3028, 29ffvelrnd 6400 . . . . . . . 8 ((𝜑𝑠 ∈ 𝒫 𝐵𝑡 = (𝐵𝑠)) → (𝐼‘(𝐵𝑠)) ∈ 𝒫 𝐵)
3130elpwid 4203 . . . . . . 7 ((𝜑𝑠 ∈ 𝒫 𝐵𝑡 = (𝐵𝑠)) → (𝐼‘(𝐵𝑠)) ⊆ 𝐵)
32 difssd 3771 . . . . . . 7 ((𝜑𝑠 ∈ 𝒫 𝐵𝑡 = (𝐵𝑠)) → (𝐵𝑠) ⊆ 𝐵)
33 sscon34b 38634 . . . . . . 7 (((𝐼‘(𝐵𝑠)) ⊆ 𝐵 ∧ (𝐵𝑠) ⊆ 𝐵) → ((𝐼‘(𝐵𝑠)) ⊆ (𝐵𝑠) ↔ (𝐵 ∖ (𝐵𝑠)) ⊆ (𝐵 ∖ (𝐼‘(𝐵𝑠)))))
3431, 32, 33syl2anc 694 . . . . . 6 ((𝜑𝑠 ∈ 𝒫 𝐵𝑡 = (𝐵𝑠)) → ((𝐼‘(𝐵𝑠)) ⊆ (𝐵𝑠) ↔ (𝐵 ∖ (𝐵𝑠)) ⊆ (𝐵 ∖ (𝐼‘(𝐵𝑠)))))
35 simp2 1082 . . . . . . 7 ((𝜑𝑠 ∈ 𝒫 𝐵𝑡 = (𝐵𝑠)) → 𝑠 ∈ 𝒫 𝐵)
36 elpwi 4201 . . . . . . . . 9 (𝑠 ∈ 𝒫 𝐵𝑠𝐵)
37 dfss4 3891 . . . . . . . . 9 (𝑠𝐵 ↔ (𝐵 ∖ (𝐵𝑠)) = 𝑠)
3836, 37sylib 208 . . . . . . . 8 (𝑠 ∈ 𝒫 𝐵 → (𝐵 ∖ (𝐵𝑠)) = 𝑠)
3938sseq1d 3665 . . . . . . 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 38649 . . . . . . . 8 (𝜑𝐵 ∈ V)
43423ad2ant1 1102 . . . . . . 7 ((𝜑𝑠 ∈ 𝒫 𝐵𝑡 = (𝐵𝑠)) → 𝐵 ∈ V)
44253ad2ant1 1102 . . . . . . 7 ((𝜑𝑠 ∈ 𝒫 𝐵𝑡 = (𝐵𝑠)) → 𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵))
45 eqid 2651 . . . . . . 7 (𝐷𝐼) = (𝐷𝐼)
46 eqid 2651 . . . . . . 7 ((𝐷𝐼)‘𝑠) = ((𝐷𝐼)‘𝑠)
4724, 5, 43, 44, 45, 35, 46dssmapfv3d 38630 . . . . . 6 ((𝜑𝑠 ∈ 𝒫 𝐵𝑡 = (𝐵𝑠)) → ((𝐷𝐼)‘𝑠) = (𝐵 ∖ (𝐼‘(𝐵𝑠))))
4847sseq2d 3666 . . . . 5 ((𝜑𝑠 ∈ 𝒫 𝐵𝑡 = (𝐵𝑠)) → (𝑠 ⊆ ((𝐷𝐼)‘𝑠) ↔ 𝑠 ⊆ (𝐵 ∖ (𝐼‘(𝐵𝑠)))))
4924, 5, 6ntrclsfv1 38670 . . . . . . . 8 (𝜑 → (𝐷𝐼) = 𝐾)
5049fveq1d 6231 . . . . . . 7 (𝜑 → ((𝐷𝐼)‘𝑠) = (𝐾𝑠))
5150sseq2d 3666 . . . . . 6 (𝜑 → (𝑠 ⊆ ((𝐷𝐼)‘𝑠) ↔ 𝑠 ⊆ (𝐾𝑠)))
52513ad2ant1 1102 . . . . 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 4914 . 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 383  w3a 1054   = wceq 1523  wcel 2030  wral 2941  Vcvv 3231  cdif 3604  wss 3607  𝒫 cpw 4191   class class class wbr 4685  cmpt 4762  wf 5922  cfv 5926  (class class class)co 6690  𝑚 cmap 7899
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-1st 7210  df-2nd 7211  df-map 7901
This theorem is referenced by: (None)
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