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Theorem ntrclsiso 40282
Description: If (pseudo-)interior and (pseudo-)closure functions are related by the duality operator then conditions equal to claiming that either is isotonic hold equally. (Contributed by RP, 3-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
ntrclsiso (𝜑 → (∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵(𝑠𝑡 → (𝐼𝑠) ⊆ (𝐼𝑡)) ↔ ∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵(𝑠𝑡 → (𝐾𝑠) ⊆ (𝐾𝑡))))
Distinct variable groups:   𝐵,𝑖,𝑗,𝑘,𝑠,𝑡   𝑗,𝐼,𝑘,𝑠,𝑡   𝜑,𝑖,𝑗,𝑘,𝑠,𝑡
Allowed substitution hints:   𝐷(𝑡,𝑖,𝑗,𝑘,𝑠)   𝐼(𝑖)   𝐾(𝑡,𝑖,𝑗,𝑘,𝑠)   𝑂(𝑡,𝑖,𝑗,𝑘,𝑠)

Proof of Theorem ntrclsiso
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sseq1 3996 . . . . 5 (𝑠 = 𝑏 → (𝑠𝑡𝑏𝑡))
2 fveq2 6667 . . . . . 6 (𝑠 = 𝑏 → (𝐼𝑠) = (𝐼𝑏))
32sseq1d 4002 . . . . 5 (𝑠 = 𝑏 → ((𝐼𝑠) ⊆ (𝐼𝑡) ↔ (𝐼𝑏) ⊆ (𝐼𝑡)))
41, 3imbi12d 346 . . . 4 (𝑠 = 𝑏 → ((𝑠𝑡 → (𝐼𝑠) ⊆ (𝐼𝑡)) ↔ (𝑏𝑡 → (𝐼𝑏) ⊆ (𝐼𝑡))))
5 sseq2 3997 . . . . 5 (𝑡 = 𝑎 → (𝑏𝑡𝑏𝑎))
6 fveq2 6667 . . . . . 6 (𝑡 = 𝑎 → (𝐼𝑡) = (𝐼𝑎))
76sseq2d 4003 . . . . 5 (𝑡 = 𝑎 → ((𝐼𝑏) ⊆ (𝐼𝑡) ↔ (𝐼𝑏) ⊆ (𝐼𝑎)))
85, 7imbi12d 346 . . . 4 (𝑡 = 𝑎 → ((𝑏𝑡 → (𝐼𝑏) ⊆ (𝐼𝑡)) ↔ (𝑏𝑎 → (𝐼𝑏) ⊆ (𝐼𝑎))))
94, 8cbvral2v 3470 . . 3 (∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵(𝑠𝑡 → (𝐼𝑠) ⊆ (𝐼𝑡)) ↔ ∀𝑏 ∈ 𝒫 𝐵𝑎 ∈ 𝒫 𝐵(𝑏𝑎 → (𝐼𝑏) ⊆ (𝐼𝑎)))
10 ralcom 3359 . . 3 (∀𝑏 ∈ 𝒫 𝐵𝑎 ∈ 𝒫 𝐵(𝑏𝑎 → (𝐼𝑏) ⊆ (𝐼𝑎)) ↔ ∀𝑎 ∈ 𝒫 𝐵𝑏 ∈ 𝒫 𝐵(𝑏𝑎 → (𝐼𝑏) ⊆ (𝐼𝑎)))
119, 10bitri 276 . 2 (∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵(𝑠𝑡 → (𝐼𝑠) ⊆ (𝐼𝑡)) ↔ ∀𝑎 ∈ 𝒫 𝐵𝑏 ∈ 𝒫 𝐵(𝑏𝑎 → (𝐼𝑏) ⊆ (𝐼𝑎)))
12 simpl 483 . . . . 5 ((𝜑𝑠 ∈ 𝒫 𝐵) → 𝜑)
13 ntrcls.d . . . . . 6 𝐷 = (𝑂𝐵)
14 ntrcls.r . . . . . 6 (𝜑𝐼𝐷𝐾)
1513, 14ntrclsbex 40249 . . . . 5 (𝜑𝐵 ∈ V)
1612, 15syl 17 . . . 4 ((𝜑𝑠 ∈ 𝒫 𝐵) → 𝐵 ∈ V)
17 difssd 4113 . . . 4 ((𝜑𝑠 ∈ 𝒫 𝐵) → (𝐵𝑠) ⊆ 𝐵)
1816, 17sselpwd 5227 . . 3 ((𝜑𝑠 ∈ 𝒫 𝐵) → (𝐵𝑠) ∈ 𝒫 𝐵)
19 elpwi 4554 . . . 4 (𝑎 ∈ 𝒫 𝐵𝑎𝐵)
20 simpl 483 . . . . . 6 ((𝐵 ∈ V ∧ 𝑎𝐵) → 𝐵 ∈ V)
21 difssd 4113 . . . . . 6 ((𝐵 ∈ V ∧ 𝑎𝐵) → (𝐵𝑎) ⊆ 𝐵)
2220, 21sselpwd 5227 . . . . 5 ((𝐵 ∈ V ∧ 𝑎𝐵) → (𝐵𝑎) ∈ 𝒫 𝐵)
23 simpr 485 . . . . . . . 8 (((𝐵 ∈ V ∧ 𝑎𝐵) ∧ 𝑠 = (𝐵𝑎)) → 𝑠 = (𝐵𝑎))
2423difeq2d 4103 . . . . . . 7 (((𝐵 ∈ V ∧ 𝑎𝐵) ∧ 𝑠 = (𝐵𝑎)) → (𝐵𝑠) = (𝐵 ∖ (𝐵𝑎)))
2524eqeq2d 2837 . . . . . 6 (((𝐵 ∈ V ∧ 𝑎𝐵) ∧ 𝑠 = (𝐵𝑎)) → (𝑎 = (𝐵𝑠) ↔ 𝑎 = (𝐵 ∖ (𝐵𝑎))))
26 eqcom 2833 . . . . . 6 (𝑎 = (𝐵 ∖ (𝐵𝑎)) ↔ (𝐵 ∖ (𝐵𝑎)) = 𝑎)
2725, 26syl6bb 288 . . . . 5 (((𝐵 ∈ V ∧ 𝑎𝐵) ∧ 𝑠 = (𝐵𝑎)) → (𝑎 = (𝐵𝑠) ↔ (𝐵 ∖ (𝐵𝑎)) = 𝑎))
28 dfss4 4239 . . . . . . 7 (𝑎𝐵 ↔ (𝐵 ∖ (𝐵𝑎)) = 𝑎)
2928biimpi 217 . . . . . 6 (𝑎𝐵 → (𝐵 ∖ (𝐵𝑎)) = 𝑎)
3029adantl 482 . . . . 5 ((𝐵 ∈ V ∧ 𝑎𝐵) → (𝐵 ∖ (𝐵𝑎)) = 𝑎)
3122, 27, 30rspcedvd 3630 . . . 4 ((𝐵 ∈ V ∧ 𝑎𝐵) → ∃𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠))
3215, 19, 31syl2an 595 . . 3 ((𝜑𝑎 ∈ 𝒫 𝐵) → ∃𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠))
33 simpl1 1185 . . . . . 6 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵) → 𝜑)
3433, 15syl 17 . . . . 5 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵) → 𝐵 ∈ V)
35 difssd 4113 . . . . 5 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵) → (𝐵𝑡) ⊆ 𝐵)
3634, 35sselpwd 5227 . . . 4 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵) → (𝐵𝑡) ∈ 𝒫 𝐵)
37 elpwi 4554 . . . . . 6 (𝑏 ∈ 𝒫 𝐵𝑏𝐵)
38 simpl 483 . . . . . . . 8 ((𝐵 ∈ V ∧ 𝑏𝐵) → 𝐵 ∈ V)
39 difssd 4113 . . . . . . . 8 ((𝐵 ∈ V ∧ 𝑏𝐵) → (𝐵𝑏) ⊆ 𝐵)
4038, 39sselpwd 5227 . . . . . . 7 ((𝐵 ∈ V ∧ 𝑏𝐵) → (𝐵𝑏) ∈ 𝒫 𝐵)
41 simpr 485 . . . . . . . . . 10 (((𝐵 ∈ V ∧ 𝑏𝐵) ∧ 𝑡 = (𝐵𝑏)) → 𝑡 = (𝐵𝑏))
4241difeq2d 4103 . . . . . . . . 9 (((𝐵 ∈ V ∧ 𝑏𝐵) ∧ 𝑡 = (𝐵𝑏)) → (𝐵𝑡) = (𝐵 ∖ (𝐵𝑏)))
4342eqeq2d 2837 . . . . . . . 8 (((𝐵 ∈ V ∧ 𝑏𝐵) ∧ 𝑡 = (𝐵𝑏)) → (𝑏 = (𝐵𝑡) ↔ 𝑏 = (𝐵 ∖ (𝐵𝑏))))
44 eqcom 2833 . . . . . . . 8 (𝑏 = (𝐵 ∖ (𝐵𝑏)) ↔ (𝐵 ∖ (𝐵𝑏)) = 𝑏)
4543, 44syl6bb 288 . . . . . . 7 (((𝐵 ∈ V ∧ 𝑏𝐵) ∧ 𝑡 = (𝐵𝑏)) → (𝑏 = (𝐵𝑡) ↔ (𝐵 ∖ (𝐵𝑏)) = 𝑏))
46 dfss4 4239 . . . . . . . . 9 (𝑏𝐵 ↔ (𝐵 ∖ (𝐵𝑏)) = 𝑏)
4746biimpi 217 . . . . . . . 8 (𝑏𝐵 → (𝐵 ∖ (𝐵𝑏)) = 𝑏)
4847adantl 482 . . . . . . 7 ((𝐵 ∈ V ∧ 𝑏𝐵) → (𝐵 ∖ (𝐵𝑏)) = 𝑏)
4940, 45, 48rspcedvd 3630 . . . . . 6 ((𝐵 ∈ V ∧ 𝑏𝐵) → ∃𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡))
5015, 37, 49syl2an 595 . . . . 5 ((𝜑𝑏 ∈ 𝒫 𝐵) → ∃𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡))
51503ad2antl1 1179 . . . 4 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑏 ∈ 𝒫 𝐵) → ∃𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡))
52 simp12 1198 . . . . . . . . 9 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → 𝑠 ∈ 𝒫 𝐵)
5352elpwid 4556 . . . . . . . 8 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → 𝑠𝐵)
54 simp2 1131 . . . . . . . . 9 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → 𝑡 ∈ 𝒫 𝐵)
5554elpwid 4556 . . . . . . . 8 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → 𝑡𝐵)
56 sscon34b 40234 . . . . . . . 8 ((𝑠𝐵𝑡𝐵) → (𝑠𝑡 ↔ (𝐵𝑡) ⊆ (𝐵𝑠)))
5753, 55, 56syl2anc 584 . . . . . . 7 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → (𝑠𝑡 ↔ (𝐵𝑡) ⊆ (𝐵𝑠)))
5857bicomd 224 . . . . . 6 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → ((𝐵𝑡) ⊆ (𝐵𝑠) ↔ 𝑠𝑡))
59 simp11 1197 . . . . . . . . . . 11 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → 𝜑)
60 ntrcls.o . . . . . . . . . . . 12 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖𝑗))))))
6160, 13, 14ntrclsiex 40268 . . . . . . . . . . 11 (𝜑𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵))
6259, 61syl 17 . . . . . . . . . 10 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → 𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵))
63 elmapi 8418 . . . . . . . . . 10 (𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵) → 𝐼:𝒫 𝐵⟶𝒫 𝐵)
6462, 63syl 17 . . . . . . . . 9 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → 𝐼:𝒫 𝐵⟶𝒫 𝐵)
6559, 15syl 17 . . . . . . . . . 10 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → 𝐵 ∈ V)
66 difssd 4113 . . . . . . . . . 10 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → (𝐵𝑡) ⊆ 𝐵)
6765, 66sselpwd 5227 . . . . . . . . 9 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → (𝐵𝑡) ∈ 𝒫 𝐵)
6864, 67ffvelrnd 6848 . . . . . . . 8 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → (𝐼‘(𝐵𝑡)) ∈ 𝒫 𝐵)
6968elpwid 4556 . . . . . . 7 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → (𝐼‘(𝐵𝑡)) ⊆ 𝐵)
70 difssd 4113 . . . . . . . . . 10 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → (𝐵𝑠) ⊆ 𝐵)
7165, 70sselpwd 5227 . . . . . . . . 9 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → (𝐵𝑠) ∈ 𝒫 𝐵)
7264, 71ffvelrnd 6848 . . . . . . . 8 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → (𝐼‘(𝐵𝑠)) ∈ 𝒫 𝐵)
7372elpwid 4556 . . . . . . 7 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → (𝐼‘(𝐵𝑠)) ⊆ 𝐵)
74 sscon34b 40234 . . . . . . 7 (((𝐼‘(𝐵𝑡)) ⊆ 𝐵 ∧ (𝐼‘(𝐵𝑠)) ⊆ 𝐵) → ((𝐼‘(𝐵𝑡)) ⊆ (𝐼‘(𝐵𝑠)) ↔ (𝐵 ∖ (𝐼‘(𝐵𝑠))) ⊆ (𝐵 ∖ (𝐼‘(𝐵𝑡)))))
7569, 73, 74syl2anc 584 . . . . . 6 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → ((𝐼‘(𝐵𝑡)) ⊆ (𝐼‘(𝐵𝑠)) ↔ (𝐵 ∖ (𝐼‘(𝐵𝑠))) ⊆ (𝐵 ∖ (𝐼‘(𝐵𝑡)))))
7658, 75imbi12d 346 . . . . 5 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → (((𝐵𝑡) ⊆ (𝐵𝑠) → (𝐼‘(𝐵𝑡)) ⊆ (𝐼‘(𝐵𝑠))) ↔ (𝑠𝑡 → (𝐵 ∖ (𝐼‘(𝐵𝑠))) ⊆ (𝐵 ∖ (𝐼‘(𝐵𝑡))))))
77 simp3 1132 . . . . . . 7 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → 𝑏 = (𝐵𝑡))
78 simp13 1199 . . . . . . 7 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → 𝑎 = (𝐵𝑠))
7977, 78sseq12d 4004 . . . . . 6 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → (𝑏𝑎 ↔ (𝐵𝑡) ⊆ (𝐵𝑠)))
8077fveq2d 6671 . . . . . . 7 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → (𝐼𝑏) = (𝐼‘(𝐵𝑡)))
8178fveq2d 6671 . . . . . . 7 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → (𝐼𝑎) = (𝐼‘(𝐵𝑠)))
8280, 81sseq12d 4004 . . . . . 6 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → ((𝐼𝑏) ⊆ (𝐼𝑎) ↔ (𝐼‘(𝐵𝑡)) ⊆ (𝐼‘(𝐵𝑠))))
8379, 82imbi12d 346 . . . . 5 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → ((𝑏𝑎 → (𝐼𝑏) ⊆ (𝐼𝑎)) ↔ ((𝐵𝑡) ⊆ (𝐵𝑠) → (𝐼‘(𝐵𝑡)) ⊆ (𝐼‘(𝐵𝑠)))))
8460, 13, 14ntrclsfv1 40270 . . . . . . . . . 10 (𝜑 → (𝐷𝐼) = 𝐾)
8559, 84syl 17 . . . . . . . . 9 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → (𝐷𝐼) = 𝐾)
8685fveq1d 6669 . . . . . . . 8 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → ((𝐷𝐼)‘𝑠) = (𝐾𝑠))
87 eqid 2826 . . . . . . . . 9 (𝐷𝐼) = (𝐷𝐼)
88 eqid 2826 . . . . . . . . 9 ((𝐷𝐼)‘𝑠) = ((𝐷𝐼)‘𝑠)
8960, 13, 65, 62, 87, 52, 88dssmapfv3d 40230 . . . . . . . 8 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → ((𝐷𝐼)‘𝑠) = (𝐵 ∖ (𝐼‘(𝐵𝑠))))
9086, 89eqtr3d 2863 . . . . . . 7 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → (𝐾𝑠) = (𝐵 ∖ (𝐼‘(𝐵𝑠))))
9159, 14syl 17 . . . . . . . . . 10 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → 𝐼𝐷𝐾)
9260, 13, 91ntrclsfv1 40270 . . . . . . . . 9 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → (𝐷𝐼) = 𝐾)
9392fveq1d 6669 . . . . . . . 8 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → ((𝐷𝐼)‘𝑡) = (𝐾𝑡))
94 eqid 2826 . . . . . . . . 9 ((𝐷𝐼)‘𝑡) = ((𝐷𝐼)‘𝑡)
9560, 13, 65, 62, 87, 54, 94dssmapfv3d 40230 . . . . . . . 8 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → ((𝐷𝐼)‘𝑡) = (𝐵 ∖ (𝐼‘(𝐵𝑡))))
9693, 95eqtr3d 2863 . . . . . . 7 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → (𝐾𝑡) = (𝐵 ∖ (𝐼‘(𝐵𝑡))))
9790, 96sseq12d 4004 . . . . . 6 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → ((𝐾𝑠) ⊆ (𝐾𝑡) ↔ (𝐵 ∖ (𝐼‘(𝐵𝑠))) ⊆ (𝐵 ∖ (𝐼‘(𝐵𝑡)))))
9897imbi2d 342 . . . . 5 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → ((𝑠𝑡 → (𝐾𝑠) ⊆ (𝐾𝑡)) ↔ (𝑠𝑡 → (𝐵 ∖ (𝐼‘(𝐵𝑠))) ⊆ (𝐵 ∖ (𝐼‘(𝐵𝑡))))))
9976, 83, 983bitr4d 312 . . . 4 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → ((𝑏𝑎 → (𝐼𝑏) ⊆ (𝐼𝑎)) ↔ (𝑠𝑡 → (𝐾𝑠) ⊆ (𝐾𝑡))))
10036, 51, 99ralxfrd2 5309 . . 3 ((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) → (∀𝑏 ∈ 𝒫 𝐵(𝑏𝑎 → (𝐼𝑏) ⊆ (𝐼𝑎)) ↔ ∀𝑡 ∈ 𝒫 𝐵(𝑠𝑡 → (𝐾𝑠) ⊆ (𝐾𝑡))))
10118, 32, 100ralxfrd2 5309 . 2 (𝜑 → (∀𝑎 ∈ 𝒫 𝐵𝑏 ∈ 𝒫 𝐵(𝑏𝑎 → (𝐼𝑏) ⊆ (𝐼𝑎)) ↔ ∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵(𝑠𝑡 → (𝐾𝑠) ⊆ (𝐾𝑡))))
10211, 101syl5bb 284 1 (𝜑 → (∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵(𝑠𝑡 → (𝐼𝑠) ⊆ (𝐼𝑡)) ↔ ∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵(𝑠𝑡 → (𝐾𝑠) ⊆ (𝐾𝑡))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1081   = wceq 1530  wcel 2107  wral 3143  wrex 3144  Vcvv 3500  cdif 3937  wss 3940  𝒫 cpw 4542   class class class wbr 5063  cmpt 5143  wf 6348  cfv 6352  (class class class)co 7148  m cmap 8396
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-13 2385  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-reu 3150  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-ov 7151  df-oprab 7152  df-mpo 7153  df-1st 7680  df-2nd 7681  df-map 8398
This theorem is referenced by: (None)
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