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Theorem ntrclsiso 38971
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 3786 . . . . 5 (𝑠 = 𝑏 → (𝑠𝑡𝑏𝑡))
2 fveq2 6375 . . . . . 6 (𝑠 = 𝑏 → (𝐼𝑠) = (𝐼𝑏))
32sseq1d 3792 . . . . 5 (𝑠 = 𝑏 → ((𝐼𝑠) ⊆ (𝐼𝑡) ↔ (𝐼𝑏) ⊆ (𝐼𝑡)))
41, 3imbi12d 335 . . . 4 (𝑠 = 𝑏 → ((𝑠𝑡 → (𝐼𝑠) ⊆ (𝐼𝑡)) ↔ (𝑏𝑡 → (𝐼𝑏) ⊆ (𝐼𝑡))))
5 sseq2 3787 . . . . 5 (𝑡 = 𝑎 → (𝑏𝑡𝑏𝑎))
6 fveq2 6375 . . . . . 6 (𝑡 = 𝑎 → (𝐼𝑡) = (𝐼𝑎))
76sseq2d 3793 . . . . 5 (𝑡 = 𝑎 → ((𝐼𝑏) ⊆ (𝐼𝑡) ↔ (𝐼𝑏) ⊆ (𝐼𝑎)))
85, 7imbi12d 335 . . . 4 (𝑡 = 𝑎 → ((𝑏𝑡 → (𝐼𝑏) ⊆ (𝐼𝑡)) ↔ (𝑏𝑎 → (𝐼𝑏) ⊆ (𝐼𝑎))))
94, 8cbvral2v 3327 . . 3 (∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵(𝑠𝑡 → (𝐼𝑠) ⊆ (𝐼𝑡)) ↔ ∀𝑏 ∈ 𝒫 𝐵𝑎 ∈ 𝒫 𝐵(𝑏𝑎 → (𝐼𝑏) ⊆ (𝐼𝑎)))
10 ralcom 3245 . . 3 (∀𝑏 ∈ 𝒫 𝐵𝑎 ∈ 𝒫 𝐵(𝑏𝑎 → (𝐼𝑏) ⊆ (𝐼𝑎)) ↔ ∀𝑎 ∈ 𝒫 𝐵𝑏 ∈ 𝒫 𝐵(𝑏𝑎 → (𝐼𝑏) ⊆ (𝐼𝑎)))
119, 10bitri 266 . 2 (∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵(𝑠𝑡 → (𝐼𝑠) ⊆ (𝐼𝑡)) ↔ ∀𝑎 ∈ 𝒫 𝐵𝑏 ∈ 𝒫 𝐵(𝑏𝑎 → (𝐼𝑏) ⊆ (𝐼𝑎)))
12 simpl 474 . . . . 5 ((𝜑𝑠 ∈ 𝒫 𝐵) → 𝜑)
13 ntrcls.d . . . . . 6 𝐷 = (𝑂𝐵)
14 ntrcls.r . . . . . 6 (𝜑𝐼𝐷𝐾)
1513, 14ntrclsbex 38938 . . . . 5 (𝜑𝐵 ∈ V)
1612, 15syl 17 . . . 4 ((𝜑𝑠 ∈ 𝒫 𝐵) → 𝐵 ∈ V)
17 difssd 3900 . . . 4 ((𝜑𝑠 ∈ 𝒫 𝐵) → (𝐵𝑠) ⊆ 𝐵)
1816, 17sselpwd 4968 . . 3 ((𝜑𝑠 ∈ 𝒫 𝐵) → (𝐵𝑠) ∈ 𝒫 𝐵)
19 elpwi 4325 . . . 4 (𝑎 ∈ 𝒫 𝐵𝑎𝐵)
20 simpl 474 . . . . . 6 ((𝐵 ∈ V ∧ 𝑎𝐵) → 𝐵 ∈ V)
21 difssd 3900 . . . . . 6 ((𝐵 ∈ V ∧ 𝑎𝐵) → (𝐵𝑎) ⊆ 𝐵)
2220, 21sselpwd 4968 . . . . 5 ((𝐵 ∈ V ∧ 𝑎𝐵) → (𝐵𝑎) ∈ 𝒫 𝐵)
23 simpr 477 . . . . . . . 8 (((𝐵 ∈ V ∧ 𝑎𝐵) ∧ 𝑠 = (𝐵𝑎)) → 𝑠 = (𝐵𝑎))
2423difeq2d 3890 . . . . . . 7 (((𝐵 ∈ V ∧ 𝑎𝐵) ∧ 𝑠 = (𝐵𝑎)) → (𝐵𝑠) = (𝐵 ∖ (𝐵𝑎)))
2524eqeq2d 2775 . . . . . 6 (((𝐵 ∈ V ∧ 𝑎𝐵) ∧ 𝑠 = (𝐵𝑎)) → (𝑎 = (𝐵𝑠) ↔ 𝑎 = (𝐵 ∖ (𝐵𝑎))))
26 eqcom 2772 . . . . . 6 (𝑎 = (𝐵 ∖ (𝐵𝑎)) ↔ (𝐵 ∖ (𝐵𝑎)) = 𝑎)
2725, 26syl6bb 278 . . . . 5 (((𝐵 ∈ V ∧ 𝑎𝐵) ∧ 𝑠 = (𝐵𝑎)) → (𝑎 = (𝐵𝑠) ↔ (𝐵 ∖ (𝐵𝑎)) = 𝑎))
28 dfss4 4023 . . . . . . 7 (𝑎𝐵 ↔ (𝐵 ∖ (𝐵𝑎)) = 𝑎)
2928biimpi 207 . . . . . 6 (𝑎𝐵 → (𝐵 ∖ (𝐵𝑎)) = 𝑎)
3029adantl 473 . . . . 5 ((𝐵 ∈ V ∧ 𝑎𝐵) → (𝐵 ∖ (𝐵𝑎)) = 𝑎)
3122, 27, 30rspcedvd 3468 . . . 4 ((𝐵 ∈ V ∧ 𝑎𝐵) → ∃𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠))
3215, 19, 31syl2an 589 . . 3 ((𝜑𝑎 ∈ 𝒫 𝐵) → ∃𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠))
33 simpl1 1242 . . . . . 6 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵) → 𝜑)
3433, 15syl 17 . . . . 5 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵) → 𝐵 ∈ V)
35 difssd 3900 . . . . 5 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵) → (𝐵𝑡) ⊆ 𝐵)
3634, 35sselpwd 4968 . . . 4 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵) → (𝐵𝑡) ∈ 𝒫 𝐵)
37 elpwi 4325 . . . . . 6 (𝑏 ∈ 𝒫 𝐵𝑏𝐵)
38 simpl 474 . . . . . . . 8 ((𝐵 ∈ V ∧ 𝑏𝐵) → 𝐵 ∈ V)
39 difssd 3900 . . . . . . . 8 ((𝐵 ∈ V ∧ 𝑏𝐵) → (𝐵𝑏) ⊆ 𝐵)
4038, 39sselpwd 4968 . . . . . . 7 ((𝐵 ∈ V ∧ 𝑏𝐵) → (𝐵𝑏) ∈ 𝒫 𝐵)
41 simpr 477 . . . . . . . . . 10 (((𝐵 ∈ V ∧ 𝑏𝐵) ∧ 𝑡 = (𝐵𝑏)) → 𝑡 = (𝐵𝑏))
4241difeq2d 3890 . . . . . . . . 9 (((𝐵 ∈ V ∧ 𝑏𝐵) ∧ 𝑡 = (𝐵𝑏)) → (𝐵𝑡) = (𝐵 ∖ (𝐵𝑏)))
4342eqeq2d 2775 . . . . . . . 8 (((𝐵 ∈ V ∧ 𝑏𝐵) ∧ 𝑡 = (𝐵𝑏)) → (𝑏 = (𝐵𝑡) ↔ 𝑏 = (𝐵 ∖ (𝐵𝑏))))
44 eqcom 2772 . . . . . . . 8 (𝑏 = (𝐵 ∖ (𝐵𝑏)) ↔ (𝐵 ∖ (𝐵𝑏)) = 𝑏)
4543, 44syl6bb 278 . . . . . . 7 (((𝐵 ∈ V ∧ 𝑏𝐵) ∧ 𝑡 = (𝐵𝑏)) → (𝑏 = (𝐵𝑡) ↔ (𝐵 ∖ (𝐵𝑏)) = 𝑏))
46 dfss4 4023 . . . . . . . . 9 (𝑏𝐵 ↔ (𝐵 ∖ (𝐵𝑏)) = 𝑏)
4746biimpi 207 . . . . . . . 8 (𝑏𝐵 → (𝐵 ∖ (𝐵𝑏)) = 𝑏)
4847adantl 473 . . . . . . 7 ((𝐵 ∈ V ∧ 𝑏𝐵) → (𝐵 ∖ (𝐵𝑏)) = 𝑏)
4940, 45, 48rspcedvd 3468 . . . . . 6 ((𝐵 ∈ V ∧ 𝑏𝐵) → ∃𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡))
5015, 37, 49syl2an 589 . . . . 5 ((𝜑𝑏 ∈ 𝒫 𝐵) → ∃𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡))
51503ad2antl1 1236 . . . 4 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑏 ∈ 𝒫 𝐵) → ∃𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡))
52 simp12 1261 . . . . . . . . 9 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → 𝑠 ∈ 𝒫 𝐵)
5352elpwid 4327 . . . . . . . 8 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → 𝑠𝐵)
54 simp2 1167 . . . . . . . . 9 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → 𝑡 ∈ 𝒫 𝐵)
5554elpwid 4327 . . . . . . . 8 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → 𝑡𝐵)
56 sscon34b 38923 . . . . . . . 8 ((𝑠𝐵𝑡𝐵) → (𝑠𝑡 ↔ (𝐵𝑡) ⊆ (𝐵𝑠)))
5753, 55, 56syl2anc 579 . . . . . . 7 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → (𝑠𝑡 ↔ (𝐵𝑡) ⊆ (𝐵𝑠)))
5857bicomd 214 . . . . . 6 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → ((𝐵𝑡) ⊆ (𝐵𝑠) ↔ 𝑠𝑡))
59 simp11 1260 . . . . . . . . . . 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 38957 . . . . . . . . . . 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 8082 . . . . . . . . . 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 3900 . . . . . . . . . 10 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → (𝐵𝑡) ⊆ 𝐵)
6765, 66sselpwd 4968 . . . . . . . . 9 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → (𝐵𝑡) ∈ 𝒫 𝐵)
6864, 67ffvelrnd 6550 . . . . . . . 8 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → (𝐼‘(𝐵𝑡)) ∈ 𝒫 𝐵)
6968elpwid 4327 . . . . . . 7 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → (𝐼‘(𝐵𝑡)) ⊆ 𝐵)
70 difssd 3900 . . . . . . . . . 10 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → (𝐵𝑠) ⊆ 𝐵)
7165, 70sselpwd 4968 . . . . . . . . 9 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → (𝐵𝑠) ∈ 𝒫 𝐵)
7264, 71ffvelrnd 6550 . . . . . . . 8 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → (𝐼‘(𝐵𝑠)) ∈ 𝒫 𝐵)
7372elpwid 4327 . . . . . . 7 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → (𝐼‘(𝐵𝑠)) ⊆ 𝐵)
74 sscon34b 38923 . . . . . . 7 (((𝐼‘(𝐵𝑡)) ⊆ 𝐵 ∧ (𝐼‘(𝐵𝑠)) ⊆ 𝐵) → ((𝐼‘(𝐵𝑡)) ⊆ (𝐼‘(𝐵𝑠)) ↔ (𝐵 ∖ (𝐼‘(𝐵𝑠))) ⊆ (𝐵 ∖ (𝐼‘(𝐵𝑡)))))
7569, 73, 74syl2anc 579 . . . . . 6 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → ((𝐼‘(𝐵𝑡)) ⊆ (𝐼‘(𝐵𝑠)) ↔ (𝐵 ∖ (𝐼‘(𝐵𝑠))) ⊆ (𝐵 ∖ (𝐼‘(𝐵𝑡)))))
7658, 75imbi12d 335 . . . . 5 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → (((𝐵𝑡) ⊆ (𝐵𝑠) → (𝐼‘(𝐵𝑡)) ⊆ (𝐼‘(𝐵𝑠))) ↔ (𝑠𝑡 → (𝐵 ∖ (𝐼‘(𝐵𝑠))) ⊆ (𝐵 ∖ (𝐼‘(𝐵𝑡))))))
77 simp3 1168 . . . . . . 7 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → 𝑏 = (𝐵𝑡))
78 simp13 1262 . . . . . . 7 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → 𝑎 = (𝐵𝑠))
7977, 78sseq12d 3794 . . . . . 6 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → (𝑏𝑎 ↔ (𝐵𝑡) ⊆ (𝐵𝑠)))
8077fveq2d 6379 . . . . . . 7 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → (𝐼𝑏) = (𝐼‘(𝐵𝑡)))
8178fveq2d 6379 . . . . . . 7 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → (𝐼𝑎) = (𝐼‘(𝐵𝑠)))
8280, 81sseq12d 3794 . . . . . 6 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → ((𝐼𝑏) ⊆ (𝐼𝑎) ↔ (𝐼‘(𝐵𝑡)) ⊆ (𝐼‘(𝐵𝑠))))
8379, 82imbi12d 335 . . . . 5 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → ((𝑏𝑎 → (𝐼𝑏) ⊆ (𝐼𝑎)) ↔ ((𝐵𝑡) ⊆ (𝐵𝑠) → (𝐼‘(𝐵𝑡)) ⊆ (𝐼‘(𝐵𝑠)))))
8460, 13, 14ntrclsfv1 38959 . . . . . . . . . 10 (𝜑 → (𝐷𝐼) = 𝐾)
8559, 84syl 17 . . . . . . . . 9 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → (𝐷𝐼) = 𝐾)
8685fveq1d 6377 . . . . . . . 8 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → ((𝐷𝐼)‘𝑠) = (𝐾𝑠))
87 eqid 2765 . . . . . . . . 9 (𝐷𝐼) = (𝐷𝐼)
88 eqid 2765 . . . . . . . . 9 ((𝐷𝐼)‘𝑠) = ((𝐷𝐼)‘𝑠)
8960, 13, 65, 62, 87, 52, 88dssmapfv3d 38919 . . . . . . . 8 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → ((𝐷𝐼)‘𝑠) = (𝐵 ∖ (𝐼‘(𝐵𝑠))))
9086, 89eqtr3d 2801 . . . . . . 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 38959 . . . . . . . . 9 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → (𝐷𝐼) = 𝐾)
9392fveq1d 6377 . . . . . . . 8 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → ((𝐷𝐼)‘𝑡) = (𝐾𝑡))
94 eqid 2765 . . . . . . . . 9 ((𝐷𝐼)‘𝑡) = ((𝐷𝐼)‘𝑡)
9560, 13, 65, 62, 87, 54, 94dssmapfv3d 38919 . . . . . . . 8 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → ((𝐷𝐼)‘𝑡) = (𝐵 ∖ (𝐼‘(𝐵𝑡))))
9693, 95eqtr3d 2801 . . . . . . 7 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → (𝐾𝑡) = (𝐵 ∖ (𝐼‘(𝐵𝑡))))
9790, 96sseq12d 3794 . . . . . 6 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → ((𝐾𝑠) ⊆ (𝐾𝑡) ↔ (𝐵 ∖ (𝐼‘(𝐵𝑠))) ⊆ (𝐵 ∖ (𝐼‘(𝐵𝑡)))))
9897imbi2d 331 . . . . 5 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → ((𝑠𝑡 → (𝐾𝑠) ⊆ (𝐾𝑡)) ↔ (𝑠𝑡 → (𝐵 ∖ (𝐼‘(𝐵𝑠))) ⊆ (𝐵 ∖ (𝐼‘(𝐵𝑡))))))
9976, 83, 983bitr4d 302 . . . 4 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → ((𝑏𝑎 → (𝐼𝑏) ⊆ (𝐼𝑎)) ↔ (𝑠𝑡 → (𝐾𝑠) ⊆ (𝐾𝑡))))
10036, 51, 99ralxfrd2 5047 . . 3 ((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) → (∀𝑏 ∈ 𝒫 𝐵(𝑏𝑎 → (𝐼𝑏) ⊆ (𝐼𝑎)) ↔ ∀𝑡 ∈ 𝒫 𝐵(𝑠𝑡 → (𝐾𝑠) ⊆ (𝐾𝑡))))
10118, 32, 100ralxfrd2 5047 . 2 (𝜑 → (∀𝑎 ∈ 𝒫 𝐵𝑏 ∈ 𝒫 𝐵(𝑏𝑎 → (𝐼𝑏) ⊆ (𝐼𝑎)) ↔ ∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵(𝑠𝑡 → (𝐾𝑠) ⊆ (𝐾𝑡))))
10211, 101syl5bb 274 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 197  wa 384  w3a 1107   = wceq 1652  wcel 2155  wral 3055  wrex 3056  Vcvv 3350  cdif 3729  wss 3732  𝒫 cpw 4315   class class class wbr 4809  cmpt 4888  wf 6064  cfv 6068  (class class class)co 6842  𝑚 cmap 8060
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 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
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 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-1st 7366  df-2nd 7367  df-map 8062
This theorem is referenced by: (None)
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