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Theorem ntrclsiso 41263
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 3912 . . . . 5 (𝑠 = 𝑏 → (𝑠𝑡𝑏𝑡))
2 fveq2 6686 . . . . . 6 (𝑠 = 𝑏 → (𝐼𝑠) = (𝐼𝑏))
32sseq1d 3918 . . . . 5 (𝑠 = 𝑏 → ((𝐼𝑠) ⊆ (𝐼𝑡) ↔ (𝐼𝑏) ⊆ (𝐼𝑡)))
41, 3imbi12d 348 . . . 4 (𝑠 = 𝑏 → ((𝑠𝑡 → (𝐼𝑠) ⊆ (𝐼𝑡)) ↔ (𝑏𝑡 → (𝐼𝑏) ⊆ (𝐼𝑡))))
5 sseq2 3913 . . . . 5 (𝑡 = 𝑎 → (𝑏𝑡𝑏𝑎))
6 fveq2 6686 . . . . . 6 (𝑡 = 𝑎 → (𝐼𝑡) = (𝐼𝑎))
76sseq2d 3919 . . . . 5 (𝑡 = 𝑎 → ((𝐼𝑏) ⊆ (𝐼𝑡) ↔ (𝐼𝑏) ⊆ (𝐼𝑎)))
85, 7imbi12d 348 . . . 4 (𝑡 = 𝑎 → ((𝑏𝑡 → (𝐼𝑏) ⊆ (𝐼𝑡)) ↔ (𝑏𝑎 → (𝐼𝑏) ⊆ (𝐼𝑎))))
94, 8cbvral2vw 3363 . . 3 (∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵(𝑠𝑡 → (𝐼𝑠) ⊆ (𝐼𝑡)) ↔ ∀𝑏 ∈ 𝒫 𝐵𝑎 ∈ 𝒫 𝐵(𝑏𝑎 → (𝐼𝑏) ⊆ (𝐼𝑎)))
10 ralcom 3259 . . 3 (∀𝑏 ∈ 𝒫 𝐵𝑎 ∈ 𝒫 𝐵(𝑏𝑎 → (𝐼𝑏) ⊆ (𝐼𝑎)) ↔ ∀𝑎 ∈ 𝒫 𝐵𝑏 ∈ 𝒫 𝐵(𝑏𝑎 → (𝐼𝑏) ⊆ (𝐼𝑎)))
119, 10bitri 278 . 2 (∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵(𝑠𝑡 → (𝐼𝑠) ⊆ (𝐼𝑡)) ↔ ∀𝑎 ∈ 𝒫 𝐵𝑏 ∈ 𝒫 𝐵(𝑏𝑎 → (𝐼𝑏) ⊆ (𝐼𝑎)))
12 simpl 486 . . . . 5 ((𝜑𝑠 ∈ 𝒫 𝐵) → 𝜑)
13 ntrcls.d . . . . . 6 𝐷 = (𝑂𝐵)
14 ntrcls.r . . . . . 6 (𝜑𝐼𝐷𝐾)
1513, 14ntrclsbex 41230 . . . . 5 (𝜑𝐵 ∈ V)
1612, 15syl 17 . . . 4 ((𝜑𝑠 ∈ 𝒫 𝐵) → 𝐵 ∈ V)
17 difssd 4033 . . . 4 ((𝜑𝑠 ∈ 𝒫 𝐵) → (𝐵𝑠) ⊆ 𝐵)
1816, 17sselpwd 5204 . . 3 ((𝜑𝑠 ∈ 𝒫 𝐵) → (𝐵𝑠) ∈ 𝒫 𝐵)
19 elpwi 4507 . . . 4 (𝑎 ∈ 𝒫 𝐵𝑎𝐵)
20 simpl 486 . . . . . 6 ((𝐵 ∈ V ∧ 𝑎𝐵) → 𝐵 ∈ V)
21 difssd 4033 . . . . . 6 ((𝐵 ∈ V ∧ 𝑎𝐵) → (𝐵𝑎) ⊆ 𝐵)
2220, 21sselpwd 5204 . . . . 5 ((𝐵 ∈ V ∧ 𝑎𝐵) → (𝐵𝑎) ∈ 𝒫 𝐵)
23 simpr 488 . . . . . . . 8 (((𝐵 ∈ V ∧ 𝑎𝐵) ∧ 𝑠 = (𝐵𝑎)) → 𝑠 = (𝐵𝑎))
2423difeq2d 4023 . . . . . . 7 (((𝐵 ∈ V ∧ 𝑎𝐵) ∧ 𝑠 = (𝐵𝑎)) → (𝐵𝑠) = (𝐵 ∖ (𝐵𝑎)))
2524eqeq2d 2750 . . . . . 6 (((𝐵 ∈ V ∧ 𝑎𝐵) ∧ 𝑠 = (𝐵𝑎)) → (𝑎 = (𝐵𝑠) ↔ 𝑎 = (𝐵 ∖ (𝐵𝑎))))
26 eqcom 2746 . . . . . 6 (𝑎 = (𝐵 ∖ (𝐵𝑎)) ↔ (𝐵 ∖ (𝐵𝑎)) = 𝑎)
2725, 26bitrdi 290 . . . . 5 (((𝐵 ∈ V ∧ 𝑎𝐵) ∧ 𝑠 = (𝐵𝑎)) → (𝑎 = (𝐵𝑠) ↔ (𝐵 ∖ (𝐵𝑎)) = 𝑎))
28 dfss4 4159 . . . . . . 7 (𝑎𝐵 ↔ (𝐵 ∖ (𝐵𝑎)) = 𝑎)
2928biimpi 219 . . . . . 6 (𝑎𝐵 → (𝐵 ∖ (𝐵𝑎)) = 𝑎)
3029adantl 485 . . . . 5 ((𝐵 ∈ V ∧ 𝑎𝐵) → (𝐵 ∖ (𝐵𝑎)) = 𝑎)
3122, 27, 30rspcedvd 3532 . . . 4 ((𝐵 ∈ V ∧ 𝑎𝐵) → ∃𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠))
3215, 19, 31syl2an 599 . . 3 ((𝜑𝑎 ∈ 𝒫 𝐵) → ∃𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠))
33 simpl1 1192 . . . . . 6 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵) → 𝜑)
3433, 15syl 17 . . . . 5 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵) → 𝐵 ∈ V)
35 difssd 4033 . . . . 5 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵) → (𝐵𝑡) ⊆ 𝐵)
3634, 35sselpwd 5204 . . . 4 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵) → (𝐵𝑡) ∈ 𝒫 𝐵)
37 elpwi 4507 . . . . . 6 (𝑏 ∈ 𝒫 𝐵𝑏𝐵)
38 simpl 486 . . . . . . . 8 ((𝐵 ∈ V ∧ 𝑏𝐵) → 𝐵 ∈ V)
39 difssd 4033 . . . . . . . 8 ((𝐵 ∈ V ∧ 𝑏𝐵) → (𝐵𝑏) ⊆ 𝐵)
4038, 39sselpwd 5204 . . . . . . 7 ((𝐵 ∈ V ∧ 𝑏𝐵) → (𝐵𝑏) ∈ 𝒫 𝐵)
41 simpr 488 . . . . . . . . . 10 (((𝐵 ∈ V ∧ 𝑏𝐵) ∧ 𝑡 = (𝐵𝑏)) → 𝑡 = (𝐵𝑏))
4241difeq2d 4023 . . . . . . . . 9 (((𝐵 ∈ V ∧ 𝑏𝐵) ∧ 𝑡 = (𝐵𝑏)) → (𝐵𝑡) = (𝐵 ∖ (𝐵𝑏)))
4342eqeq2d 2750 . . . . . . . 8 (((𝐵 ∈ V ∧ 𝑏𝐵) ∧ 𝑡 = (𝐵𝑏)) → (𝑏 = (𝐵𝑡) ↔ 𝑏 = (𝐵 ∖ (𝐵𝑏))))
44 eqcom 2746 . . . . . . . 8 (𝑏 = (𝐵 ∖ (𝐵𝑏)) ↔ (𝐵 ∖ (𝐵𝑏)) = 𝑏)
4543, 44bitrdi 290 . . . . . . 7 (((𝐵 ∈ V ∧ 𝑏𝐵) ∧ 𝑡 = (𝐵𝑏)) → (𝑏 = (𝐵𝑡) ↔ (𝐵 ∖ (𝐵𝑏)) = 𝑏))
46 dfss4 4159 . . . . . . . . 9 (𝑏𝐵 ↔ (𝐵 ∖ (𝐵𝑏)) = 𝑏)
4746biimpi 219 . . . . . . . 8 (𝑏𝐵 → (𝐵 ∖ (𝐵𝑏)) = 𝑏)
4847adantl 485 . . . . . . 7 ((𝐵 ∈ V ∧ 𝑏𝐵) → (𝐵 ∖ (𝐵𝑏)) = 𝑏)
4940, 45, 48rspcedvd 3532 . . . . . 6 ((𝐵 ∈ V ∧ 𝑏𝐵) → ∃𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡))
5015, 37, 49syl2an 599 . . . . 5 ((𝜑𝑏 ∈ 𝒫 𝐵) → ∃𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡))
51503ad2antl1 1186 . . . 4 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑏 ∈ 𝒫 𝐵) → ∃𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡))
52 simp12 1205 . . . . . . . . 9 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → 𝑠 ∈ 𝒫 𝐵)
5352elpwid 4509 . . . . . . . 8 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → 𝑠𝐵)
54 simp2 1138 . . . . . . . . 9 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → 𝑡 ∈ 𝒫 𝐵)
5554elpwid 4509 . . . . . . . 8 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → 𝑡𝐵)
56 sscon34b 4195 . . . . . . . 8 ((𝑠𝐵𝑡𝐵) → (𝑠𝑡 ↔ (𝐵𝑡) ⊆ (𝐵𝑠)))
5753, 55, 56syl2anc 587 . . . . . . 7 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → (𝑠𝑡 ↔ (𝐵𝑡) ⊆ (𝐵𝑠)))
5857bicomd 226 . . . . . 6 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → ((𝐵𝑡) ⊆ (𝐵𝑠) ↔ 𝑠𝑡))
59 simp11 1204 . . . . . . . . . . 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 41249 . . . . . . . . . . 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 8471 . . . . . . . . . 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 4033 . . . . . . . . . 10 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → (𝐵𝑡) ⊆ 𝐵)
6765, 66sselpwd 5204 . . . . . . . . 9 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → (𝐵𝑡) ∈ 𝒫 𝐵)
6864, 67ffvelrnd 6874 . . . . . . . 8 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → (𝐼‘(𝐵𝑡)) ∈ 𝒫 𝐵)
6968elpwid 4509 . . . . . . 7 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → (𝐼‘(𝐵𝑡)) ⊆ 𝐵)
70 difssd 4033 . . . . . . . . . 10 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → (𝐵𝑠) ⊆ 𝐵)
7165, 70sselpwd 5204 . . . . . . . . 9 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → (𝐵𝑠) ∈ 𝒫 𝐵)
7264, 71ffvelrnd 6874 . . . . . . . 8 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → (𝐼‘(𝐵𝑠)) ∈ 𝒫 𝐵)
7372elpwid 4509 . . . . . . 7 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → (𝐼‘(𝐵𝑠)) ⊆ 𝐵)
74 sscon34b 4195 . . . . . . 7 (((𝐼‘(𝐵𝑡)) ⊆ 𝐵 ∧ (𝐼‘(𝐵𝑠)) ⊆ 𝐵) → ((𝐼‘(𝐵𝑡)) ⊆ (𝐼‘(𝐵𝑠)) ↔ (𝐵 ∖ (𝐼‘(𝐵𝑠))) ⊆ (𝐵 ∖ (𝐼‘(𝐵𝑡)))))
7569, 73, 74syl2anc 587 . . . . . 6 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → ((𝐼‘(𝐵𝑡)) ⊆ (𝐼‘(𝐵𝑠)) ↔ (𝐵 ∖ (𝐼‘(𝐵𝑠))) ⊆ (𝐵 ∖ (𝐼‘(𝐵𝑡)))))
7658, 75imbi12d 348 . . . . 5 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → (((𝐵𝑡) ⊆ (𝐵𝑠) → (𝐼‘(𝐵𝑡)) ⊆ (𝐼‘(𝐵𝑠))) ↔ (𝑠𝑡 → (𝐵 ∖ (𝐼‘(𝐵𝑠))) ⊆ (𝐵 ∖ (𝐼‘(𝐵𝑡))))))
77 simp3 1139 . . . . . . 7 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → 𝑏 = (𝐵𝑡))
78 simp13 1206 . . . . . . 7 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → 𝑎 = (𝐵𝑠))
7977, 78sseq12d 3920 . . . . . 6 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → (𝑏𝑎 ↔ (𝐵𝑡) ⊆ (𝐵𝑠)))
8077fveq2d 6690 . . . . . . 7 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → (𝐼𝑏) = (𝐼‘(𝐵𝑡)))
8178fveq2d 6690 . . . . . . 7 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → (𝐼𝑎) = (𝐼‘(𝐵𝑠)))
8280, 81sseq12d 3920 . . . . . 6 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → ((𝐼𝑏) ⊆ (𝐼𝑎) ↔ (𝐼‘(𝐵𝑡)) ⊆ (𝐼‘(𝐵𝑠))))
8379, 82imbi12d 348 . . . . 5 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → ((𝑏𝑎 → (𝐼𝑏) ⊆ (𝐼𝑎)) ↔ ((𝐵𝑡) ⊆ (𝐵𝑠) → (𝐼‘(𝐵𝑡)) ⊆ (𝐼‘(𝐵𝑠)))))
8460, 13, 14ntrclsfv1 41251 . . . . . . . . . 10 (𝜑 → (𝐷𝐼) = 𝐾)
8559, 84syl 17 . . . . . . . . 9 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → (𝐷𝐼) = 𝐾)
8685fveq1d 6688 . . . . . . . 8 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → ((𝐷𝐼)‘𝑠) = (𝐾𝑠))
87 eqid 2739 . . . . . . . . 9 (𝐷𝐼) = (𝐷𝐼)
88 eqid 2739 . . . . . . . . 9 ((𝐷𝐼)‘𝑠) = ((𝐷𝐼)‘𝑠)
8960, 13, 65, 62, 87, 52, 88dssmapfv3d 41213 . . . . . . . 8 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → ((𝐷𝐼)‘𝑠) = (𝐵 ∖ (𝐼‘(𝐵𝑠))))
9086, 89eqtr3d 2776 . . . . . . 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 41251 . . . . . . . . 9 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → (𝐷𝐼) = 𝐾)
9392fveq1d 6688 . . . . . . . 8 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → ((𝐷𝐼)‘𝑡) = (𝐾𝑡))
94 eqid 2739 . . . . . . . . 9 ((𝐷𝐼)‘𝑡) = ((𝐷𝐼)‘𝑡)
9560, 13, 65, 62, 87, 54, 94dssmapfv3d 41213 . . . . . . . 8 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → ((𝐷𝐼)‘𝑡) = (𝐵 ∖ (𝐼‘(𝐵𝑡))))
9693, 95eqtr3d 2776 . . . . . . 7 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → (𝐾𝑡) = (𝐵 ∖ (𝐼‘(𝐵𝑡))))
9790, 96sseq12d 3920 . . . . . 6 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → ((𝐾𝑠) ⊆ (𝐾𝑡) ↔ (𝐵 ∖ (𝐼‘(𝐵𝑠))) ⊆ (𝐵 ∖ (𝐼‘(𝐵𝑡)))))
9897imbi2d 344 . . . . 5 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → ((𝑠𝑡 → (𝐾𝑠) ⊆ (𝐾𝑡)) ↔ (𝑠𝑡 → (𝐵 ∖ (𝐼‘(𝐵𝑠))) ⊆ (𝐵 ∖ (𝐼‘(𝐵𝑡))))))
9976, 83, 983bitr4d 314 . . . 4 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → ((𝑏𝑎 → (𝐼𝑏) ⊆ (𝐼𝑎)) ↔ (𝑠𝑡 → (𝐾𝑠) ⊆ (𝐾𝑡))))
10036, 51, 99ralxfrd2 5289 . . 3 ((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) → (∀𝑏 ∈ 𝒫 𝐵(𝑏𝑎 → (𝐼𝑏) ⊆ (𝐼𝑎)) ↔ ∀𝑡 ∈ 𝒫 𝐵(𝑠𝑡 → (𝐾𝑠) ⊆ (𝐾𝑡))))
10118, 32, 100ralxfrd2 5289 . 2 (𝜑 → (∀𝑎 ∈ 𝒫 𝐵𝑏 ∈ 𝒫 𝐵(𝑏𝑎 → (𝐼𝑏) ⊆ (𝐼𝑎)) ↔ ∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵(𝑠𝑡 → (𝐾𝑠) ⊆ (𝐾𝑡))))
10211, 101syl5bb 286 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 209  wa 399  w3a 1088   = wceq 1542  wcel 2114  wral 3054  wrex 3055  Vcvv 3400  cdif 3850  wss 3853  𝒫 cpw 4498   class class class wbr 5040  cmpt 5120  wf 6345  cfv 6349  (class class class)co 7182  m cmap 8449
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-rep 5164  ax-sep 5177  ax-nul 5184  ax-pow 5242  ax-pr 5306  ax-un 7491
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ne 2936  df-ral 3059  df-rex 3060  df-reu 3061  df-rab 3063  df-v 3402  df-sbc 3686  df-csb 3801  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4222  df-if 4425  df-pw 4500  df-sn 4527  df-pr 4529  df-op 4533  df-uni 4807  df-iun 4893  df-br 5041  df-opab 5103  df-mpt 5121  df-id 5439  df-xp 5541  df-rel 5542  df-cnv 5543  df-co 5544  df-dm 5545  df-rn 5546  df-res 5547  df-ima 5548  df-iota 6307  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-ov 7185  df-oprab 7186  df-mpo 7187  df-1st 7726  df-2nd 7727  df-map 8451
This theorem is referenced by: (None)
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