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Theorem ntrclsk2 39064
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 6379 . . . 4 (𝑠 = 𝑡 → (𝐼𝑠) = (𝐼𝑡))
2 id 22 . . . 4 (𝑠 = 𝑡𝑠 = 𝑡)
31, 2sseq12d 3796 . . 3 (𝑠 = 𝑡 → ((𝐼𝑠) ⊆ 𝑠 ↔ (𝐼𝑡) ⊆ 𝑡))
43cbvralv 3319 . 2 (∀𝑠 ∈ 𝒫 𝐵(𝐼𝑠) ⊆ 𝑠 ↔ ∀𝑡 ∈ 𝒫 𝐵(𝐼𝑡) ⊆ 𝑡)
5 ntrcls.d . . . . 5 𝐷 = (𝑂𝐵)
6 ntrcls.r . . . . 5 (𝜑𝐼𝐷𝐾)
75, 6ntrclsrcomplex 39031 . . . 4 (𝜑 → (𝐵𝑠) ∈ 𝒫 𝐵)
87adantr 472 . . 3 ((𝜑𝑠 ∈ 𝒫 𝐵) → (𝐵𝑠) ∈ 𝒫 𝐵)
95, 6ntrclsrcomplex 39031 . . . . 5 (𝜑 → (𝐵𝑡) ∈ 𝒫 𝐵)
109adantr 472 . . . 4 ((𝜑𝑡 ∈ 𝒫 𝐵) → (𝐵𝑡) ∈ 𝒫 𝐵)
11 difeq2 3886 . . . . . 6 (𝑠 = (𝐵𝑡) → (𝐵𝑠) = (𝐵 ∖ (𝐵𝑡)))
1211eqeq2d 2775 . . . . 5 (𝑠 = (𝐵𝑡) → (𝑡 = (𝐵𝑠) ↔ 𝑡 = (𝐵 ∖ (𝐵𝑡))))
1312adantl 473 . . . 4 (((𝜑𝑡 ∈ 𝒫 𝐵) ∧ 𝑠 = (𝐵𝑡)) → (𝑡 = (𝐵𝑠) ↔ 𝑡 = (𝐵 ∖ (𝐵𝑡))))
14 elpwi 4327 . . . . . . 7 (𝑡 ∈ 𝒫 𝐵𝑡𝐵)
15 dfss4 4025 . . . . . . 7 (𝑡𝐵 ↔ (𝐵 ∖ (𝐵𝑡)) = 𝑡)
1614, 15sylib 209 . . . . . 6 (𝑡 ∈ 𝒫 𝐵 → (𝐵 ∖ (𝐵𝑡)) = 𝑡)
1716adantl 473 . . . . 5 ((𝜑𝑡 ∈ 𝒫 𝐵) → (𝐵 ∖ (𝐵𝑡)) = 𝑡)
1817eqcomd 2771 . . . 4 ((𝜑𝑡 ∈ 𝒫 𝐵) → 𝑡 = (𝐵 ∖ (𝐵𝑡)))
1910, 13, 18rspcedvd 3469 . . 3 ((𝜑𝑡 ∈ 𝒫 𝐵) → ∃𝑠 ∈ 𝒫 𝐵𝑡 = (𝐵𝑠))
20 fveq2 6379 . . . . . 6 (𝑡 = (𝐵𝑠) → (𝐼𝑡) = (𝐼‘(𝐵𝑠)))
21 id 22 . . . . . 6 (𝑡 = (𝐵𝑠) → 𝑡 = (𝐵𝑠))
2220, 21sseq12d 3796 . . . . 5 (𝑡 = (𝐵𝑠) → ((𝐼𝑡) ⊆ 𝑡 ↔ (𝐼‘(𝐵𝑠)) ⊆ (𝐵𝑠)))
23223ad2ant3 1165 . . . 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 39049 . . . . . . . . . . 11 (𝜑𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵))
26 elmapi 8086 . . . . . . . . . . 11 (𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵) → 𝐼:𝒫 𝐵⟶𝒫 𝐵)
2725, 26syl 17 . . . . . . . . . 10 (𝜑𝐼:𝒫 𝐵⟶𝒫 𝐵)
28273ad2ant1 1163 . . . . . . . . 9 ((𝜑𝑠 ∈ 𝒫 𝐵𝑡 = (𝐵𝑠)) → 𝐼:𝒫 𝐵⟶𝒫 𝐵)
2973ad2ant1 1163 . . . . . . . . 9 ((𝜑𝑠 ∈ 𝒫 𝐵𝑡 = (𝐵𝑠)) → (𝐵𝑠) ∈ 𝒫 𝐵)
3028, 29ffvelrnd 6554 . . . . . . . 8 ((𝜑𝑠 ∈ 𝒫 𝐵𝑡 = (𝐵𝑠)) → (𝐼‘(𝐵𝑠)) ∈ 𝒫 𝐵)
3130elpwid 4329 . . . . . . 7 ((𝜑𝑠 ∈ 𝒫 𝐵𝑡 = (𝐵𝑠)) → (𝐼‘(𝐵𝑠)) ⊆ 𝐵)
32 difssd 3902 . . . . . . 7 ((𝜑𝑠 ∈ 𝒫 𝐵𝑡 = (𝐵𝑠)) → (𝐵𝑠) ⊆ 𝐵)
33 sscon34b 39015 . . . . . . 7 (((𝐼‘(𝐵𝑠)) ⊆ 𝐵 ∧ (𝐵𝑠) ⊆ 𝐵) → ((𝐼‘(𝐵𝑠)) ⊆ (𝐵𝑠) ↔ (𝐵 ∖ (𝐵𝑠)) ⊆ (𝐵 ∖ (𝐼‘(𝐵𝑠)))))
3431, 32, 33syl2anc 579 . . . . . 6 ((𝜑𝑠 ∈ 𝒫 𝐵𝑡 = (𝐵𝑠)) → ((𝐼‘(𝐵𝑠)) ⊆ (𝐵𝑠) ↔ (𝐵 ∖ (𝐵𝑠)) ⊆ (𝐵 ∖ (𝐼‘(𝐵𝑠)))))
35 simp2 1167 . . . . . . 7 ((𝜑𝑠 ∈ 𝒫 𝐵𝑡 = (𝐵𝑠)) → 𝑠 ∈ 𝒫 𝐵)
36 elpwi 4327 . . . . . . . . 9 (𝑠 ∈ 𝒫 𝐵𝑠𝐵)
37 dfss4 4025 . . . . . . . . 9 (𝑠𝐵 ↔ (𝐵 ∖ (𝐵𝑠)) = 𝑠)
3836, 37sylib 209 . . . . . . . 8 (𝑠 ∈ 𝒫 𝐵 → (𝐵 ∖ (𝐵𝑠)) = 𝑠)
3938sseq1d 3794 . . . . . . 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 270 . . . . 5 ((𝜑𝑠 ∈ 𝒫 𝐵𝑡 = (𝐵𝑠)) → ((𝐼‘(𝐵𝑠)) ⊆ (𝐵𝑠) ↔ 𝑠 ⊆ (𝐵 ∖ (𝐼‘(𝐵𝑠)))))
425, 6ntrclsbex 39030 . . . . . . . 8 (𝜑𝐵 ∈ V)
43423ad2ant1 1163 . . . . . . 7 ((𝜑𝑠 ∈ 𝒫 𝐵𝑡 = (𝐵𝑠)) → 𝐵 ∈ V)
44253ad2ant1 1163 . . . . . . 7 ((𝜑𝑠 ∈ 𝒫 𝐵𝑡 = (𝐵𝑠)) → 𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵))
45 eqid 2765 . . . . . . 7 (𝐷𝐼) = (𝐷𝐼)
46 eqid 2765 . . . . . . 7 ((𝐷𝐼)‘𝑠) = ((𝐷𝐼)‘𝑠)
4724, 5, 43, 44, 45, 35, 46dssmapfv3d 39011 . . . . . 6 ((𝜑𝑠 ∈ 𝒫 𝐵𝑡 = (𝐵𝑠)) → ((𝐷𝐼)‘𝑠) = (𝐵 ∖ (𝐼‘(𝐵𝑠))))
4847sseq2d 3795 . . . . 5 ((𝜑𝑠 ∈ 𝒫 𝐵𝑡 = (𝐵𝑠)) → (𝑠 ⊆ ((𝐷𝐼)‘𝑠) ↔ 𝑠 ⊆ (𝐵 ∖ (𝐼‘(𝐵𝑠)))))
4924, 5, 6ntrclsfv1 39051 . . . . . . . 8 (𝜑 → (𝐷𝐼) = 𝐾)
5049fveq1d 6381 . . . . . . 7 (𝜑 → ((𝐷𝐼)‘𝑠) = (𝐾𝑠))
5150sseq2d 3795 . . . . . 6 (𝜑 → (𝑠 ⊆ ((𝐷𝐼)‘𝑠) ↔ 𝑠 ⊆ (𝐾𝑠)))
52513ad2ant1 1163 . . . . 5 ((𝜑𝑠 ∈ 𝒫 𝐵𝑡 = (𝐵𝑠)) → (𝑠 ⊆ ((𝐷𝐼)‘𝑠) ↔ 𝑠 ⊆ (𝐾𝑠)))
5341, 48, 523bitr2d 298 . . . 4 ((𝜑𝑠 ∈ 𝒫 𝐵𝑡 = (𝐵𝑠)) → ((𝐼‘(𝐵𝑠)) ⊆ (𝐵𝑠) ↔ 𝑠 ⊆ (𝐾𝑠)))
5423, 53bitrd 270 . . 3 ((𝜑𝑠 ∈ 𝒫 𝐵𝑡 = (𝐵𝑠)) → ((𝐼𝑡) ⊆ 𝑡𝑠 ⊆ (𝐾𝑠)))
558, 19, 54ralxfrd2 5049 . 2 (𝜑 → (∀𝑡 ∈ 𝒫 𝐵(𝐼𝑡) ⊆ 𝑡 ↔ ∀𝑠 ∈ 𝒫 𝐵𝑠 ⊆ (𝐾𝑠)))
564, 55syl5bb 274 1 (𝜑 → (∀𝑠 ∈ 𝒫 𝐵(𝐼𝑠) ⊆ 𝑠 ↔ ∀𝑠 ∈ 𝒫 𝐵𝑠 ⊆ (𝐾𝑠)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  w3a 1107   = wceq 1652  wcel 2155  wral 3055  Vcvv 3350  cdif 3731  wss 3734  𝒫 cpw 4317   class class class wbr 4811  cmpt 4890  wf 6066  cfv 6070  (class class class)co 6846  𝑚 cmap 8064
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4932  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7151
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3599  df-csb 3694  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-nul 4082  df-if 4246  df-pw 4319  df-sn 4337  df-pr 4339  df-op 4343  df-uni 4597  df-iun 4680  df-br 4812  df-opab 4874  df-mpt 4891  df-id 5187  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-f1 6075  df-fo 6076  df-f1o 6077  df-fv 6078  df-ov 6849  df-oprab 6850  df-mpt2 6851  df-1st 7370  df-2nd 7371  df-map 8066
This theorem is referenced by: (None)
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