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Theorem ntrclsk2 44058
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 6907 . . . 4 (𝑠 = 𝑡 → (𝐼𝑠) = (𝐼𝑡))
2 id 22 . . . 4 (𝑠 = 𝑡𝑠 = 𝑡)
31, 2sseq12d 4029 . . 3 (𝑠 = 𝑡 → ((𝐼𝑠) ⊆ 𝑠 ↔ (𝐼𝑡) ⊆ 𝑡))
43cbvralvw 3235 . 2 (∀𝑠 ∈ 𝒫 𝐵(𝐼𝑠) ⊆ 𝑠 ↔ ∀𝑡 ∈ 𝒫 𝐵(𝐼𝑡) ⊆ 𝑡)
5 ntrcls.d . . . . 5 𝐷 = (𝑂𝐵)
6 ntrcls.r . . . . 5 (𝜑𝐼𝐷𝐾)
75, 6ntrclsrcomplex 44025 . . . 4 (𝜑 → (𝐵𝑠) ∈ 𝒫 𝐵)
87adantr 480 . . 3 ((𝜑𝑠 ∈ 𝒫 𝐵) → (𝐵𝑠) ∈ 𝒫 𝐵)
95, 6ntrclsrcomplex 44025 . . . . 5 (𝜑 → (𝐵𝑡) ∈ 𝒫 𝐵)
109adantr 480 . . . 4 ((𝜑𝑡 ∈ 𝒫 𝐵) → (𝐵𝑡) ∈ 𝒫 𝐵)
11 difeq2 4130 . . . . . 6 (𝑠 = (𝐵𝑡) → (𝐵𝑠) = (𝐵 ∖ (𝐵𝑡)))
1211eqeq2d 2746 . . . . 5 (𝑠 = (𝐵𝑡) → (𝑡 = (𝐵𝑠) ↔ 𝑡 = (𝐵 ∖ (𝐵𝑡))))
1312adantl 481 . . . 4 (((𝜑𝑡 ∈ 𝒫 𝐵) ∧ 𝑠 = (𝐵𝑡)) → (𝑡 = (𝐵𝑠) ↔ 𝑡 = (𝐵 ∖ (𝐵𝑡))))
14 elpwi 4612 . . . . . . 7 (𝑡 ∈ 𝒫 𝐵𝑡𝐵)
15 dfss4 4275 . . . . . . 7 (𝑡𝐵 ↔ (𝐵 ∖ (𝐵𝑡)) = 𝑡)
1614, 15sylib 218 . . . . . 6 (𝑡 ∈ 𝒫 𝐵 → (𝐵 ∖ (𝐵𝑡)) = 𝑡)
1716adantl 481 . . . . 5 ((𝜑𝑡 ∈ 𝒫 𝐵) → (𝐵 ∖ (𝐵𝑡)) = 𝑡)
1817eqcomd 2741 . . . 4 ((𝜑𝑡 ∈ 𝒫 𝐵) → 𝑡 = (𝐵 ∖ (𝐵𝑡)))
1910, 13, 18rspcedvd 3624 . . 3 ((𝜑𝑡 ∈ 𝒫 𝐵) → ∃𝑠 ∈ 𝒫 𝐵𝑡 = (𝐵𝑠))
20 fveq2 6907 . . . . . 6 (𝑡 = (𝐵𝑠) → (𝐼𝑡) = (𝐼‘(𝐵𝑠)))
21 id 22 . . . . . 6 (𝑡 = (𝐵𝑠) → 𝑡 = (𝐵𝑠))
2220, 21sseq12d 4029 . . . . 5 (𝑡 = (𝐵𝑠) → ((𝐼𝑡) ⊆ 𝑡 ↔ (𝐼‘(𝐵𝑠)) ⊆ (𝐵𝑠)))
23223ad2ant3 1134 . . . 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 44043 . . . . . . . . . . 11 (𝜑𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵))
26 elmapi 8888 . . . . . . . . . . 11 (𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵) → 𝐼:𝒫 𝐵⟶𝒫 𝐵)
2725, 26syl 17 . . . . . . . . . 10 (𝜑𝐼:𝒫 𝐵⟶𝒫 𝐵)
28273ad2ant1 1132 . . . . . . . . 9 ((𝜑𝑠 ∈ 𝒫 𝐵𝑡 = (𝐵𝑠)) → 𝐼:𝒫 𝐵⟶𝒫 𝐵)
2973ad2ant1 1132 . . . . . . . . 9 ((𝜑𝑠 ∈ 𝒫 𝐵𝑡 = (𝐵𝑠)) → (𝐵𝑠) ∈ 𝒫 𝐵)
3028, 29ffvelcdmd 7105 . . . . . . . 8 ((𝜑𝑠 ∈ 𝒫 𝐵𝑡 = (𝐵𝑠)) → (𝐼‘(𝐵𝑠)) ∈ 𝒫 𝐵)
3130elpwid 4614 . . . . . . 7 ((𝜑𝑠 ∈ 𝒫 𝐵𝑡 = (𝐵𝑠)) → (𝐼‘(𝐵𝑠)) ⊆ 𝐵)
32 difssd 4147 . . . . . . 7 ((𝜑𝑠 ∈ 𝒫 𝐵𝑡 = (𝐵𝑠)) → (𝐵𝑠) ⊆ 𝐵)
33 sscon34b 4310 . . . . . . 7 (((𝐼‘(𝐵𝑠)) ⊆ 𝐵 ∧ (𝐵𝑠) ⊆ 𝐵) → ((𝐼‘(𝐵𝑠)) ⊆ (𝐵𝑠) ↔ (𝐵 ∖ (𝐵𝑠)) ⊆ (𝐵 ∖ (𝐼‘(𝐵𝑠)))))
3431, 32, 33syl2anc 584 . . . . . 6 ((𝜑𝑠 ∈ 𝒫 𝐵𝑡 = (𝐵𝑠)) → ((𝐼‘(𝐵𝑠)) ⊆ (𝐵𝑠) ↔ (𝐵 ∖ (𝐵𝑠)) ⊆ (𝐵 ∖ (𝐼‘(𝐵𝑠)))))
35 simp2 1136 . . . . . . 7 ((𝜑𝑠 ∈ 𝒫 𝐵𝑡 = (𝐵𝑠)) → 𝑠 ∈ 𝒫 𝐵)
36 elpwi 4612 . . . . . . . . 9 (𝑠 ∈ 𝒫 𝐵𝑠𝐵)
37 dfss4 4275 . . . . . . . . 9 (𝑠𝐵 ↔ (𝐵 ∖ (𝐵𝑠)) = 𝑠)
3836, 37sylib 218 . . . . . . . 8 (𝑠 ∈ 𝒫 𝐵 → (𝐵 ∖ (𝐵𝑠)) = 𝑠)
3938sseq1d 4027 . . . . . . 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 279 . . . . 5 ((𝜑𝑠 ∈ 𝒫 𝐵𝑡 = (𝐵𝑠)) → ((𝐼‘(𝐵𝑠)) ⊆ (𝐵𝑠) ↔ 𝑠 ⊆ (𝐵 ∖ (𝐼‘(𝐵𝑠)))))
425, 6ntrclsbex 44024 . . . . . . . 8 (𝜑𝐵 ∈ V)
43423ad2ant1 1132 . . . . . . 7 ((𝜑𝑠 ∈ 𝒫 𝐵𝑡 = (𝐵𝑠)) → 𝐵 ∈ V)
44253ad2ant1 1132 . . . . . . 7 ((𝜑𝑠 ∈ 𝒫 𝐵𝑡 = (𝐵𝑠)) → 𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵))
45 eqid 2735 . . . . . . 7 (𝐷𝐼) = (𝐷𝐼)
46 eqid 2735 . . . . . . 7 ((𝐷𝐼)‘𝑠) = ((𝐷𝐼)‘𝑠)
4724, 5, 43, 44, 45, 35, 46dssmapfv3d 44009 . . . . . 6 ((𝜑𝑠 ∈ 𝒫 𝐵𝑡 = (𝐵𝑠)) → ((𝐷𝐼)‘𝑠) = (𝐵 ∖ (𝐼‘(𝐵𝑠))))
4847sseq2d 4028 . . . . 5 ((𝜑𝑠 ∈ 𝒫 𝐵𝑡 = (𝐵𝑠)) → (𝑠 ⊆ ((𝐷𝐼)‘𝑠) ↔ 𝑠 ⊆ (𝐵 ∖ (𝐼‘(𝐵𝑠)))))
4924, 5, 6ntrclsfv1 44045 . . . . . . . 8 (𝜑 → (𝐷𝐼) = 𝐾)
5049fveq1d 6909 . . . . . . 7 (𝜑 → ((𝐷𝐼)‘𝑠) = (𝐾𝑠))
5150sseq2d 4028 . . . . . 6 (𝜑 → (𝑠 ⊆ ((𝐷𝐼)‘𝑠) ↔ 𝑠 ⊆ (𝐾𝑠)))
52513ad2ant1 1132 . . . . 5 ((𝜑𝑠 ∈ 𝒫 𝐵𝑡 = (𝐵𝑠)) → (𝑠 ⊆ ((𝐷𝐼)‘𝑠) ↔ 𝑠 ⊆ (𝐾𝑠)))
5341, 48, 523bitr2d 307 . . . 4 ((𝜑𝑠 ∈ 𝒫 𝐵𝑡 = (𝐵𝑠)) → ((𝐼‘(𝐵𝑠)) ⊆ (𝐵𝑠) ↔ 𝑠 ⊆ (𝐾𝑠)))
5423, 53bitrd 279 . . 3 ((𝜑𝑠 ∈ 𝒫 𝐵𝑡 = (𝐵𝑠)) → ((𝐼𝑡) ⊆ 𝑡𝑠 ⊆ (𝐾𝑠)))
558, 19, 54ralxfrd2 5418 . 2 (𝜑 → (∀𝑡 ∈ 𝒫 𝐵(𝐼𝑡) ⊆ 𝑡 ↔ ∀𝑠 ∈ 𝒫 𝐵𝑠 ⊆ (𝐾𝑠)))
564, 55bitrid 283 1 (𝜑 → (∀𝑠 ∈ 𝒫 𝐵(𝐼𝑠) ⊆ 𝑠 ↔ ∀𝑠 ∈ 𝒫 𝐵𝑠 ⊆ (𝐾𝑠)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  wral 3059  Vcvv 3478  cdif 3960  wss 3963  𝒫 cpw 4605   class class class wbr 5148  cmpt 5231  wf 6559  cfv 6563  (class class class)co 7431  m cmap 8865
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-map 8867
This theorem is referenced by: (None)
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