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Theorem ntrclskb 41145
 Description: The interiors of disjoint sets are disjoint if and only if the closures of sets that span the base set also span the base set. (Contributed by RP, 10-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
ntrclskb (𝜑 → (∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝑠𝑡) = ∅ → ((𝐼𝑠) ∩ (𝐼𝑡)) = ∅) ↔ ∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝑠𝑡) = 𝐵 → ((𝐾𝑠) ∪ (𝐾𝑡)) = 𝐵)))
Distinct variable groups:   𝐵,𝑠,𝑡,𝑖,𝑗,𝑘   𝐼,𝑠,𝑡,𝑗,𝑘   𝜑,𝑠,𝑡,𝑖,𝑗,𝑘
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 ntrclskb
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ineq1 4109 . . . . 5 (𝑠 = 𝑎 → (𝑠𝑡) = (𝑎𝑡))
21eqeq1d 2760 . . . 4 (𝑠 = 𝑎 → ((𝑠𝑡) = ∅ ↔ (𝑎𝑡) = ∅))
3 fveq2 6658 . . . . . 6 (𝑠 = 𝑎 → (𝐼𝑠) = (𝐼𝑎))
43ineq1d 4116 . . . . 5 (𝑠 = 𝑎 → ((𝐼𝑠) ∩ (𝐼𝑡)) = ((𝐼𝑎) ∩ (𝐼𝑡)))
54eqeq1d 2760 . . . 4 (𝑠 = 𝑎 → (((𝐼𝑠) ∩ (𝐼𝑡)) = ∅ ↔ ((𝐼𝑎) ∩ (𝐼𝑡)) = ∅))
62, 5imbi12d 348 . . 3 (𝑠 = 𝑎 → (((𝑠𝑡) = ∅ → ((𝐼𝑠) ∩ (𝐼𝑡)) = ∅) ↔ ((𝑎𝑡) = ∅ → ((𝐼𝑎) ∩ (𝐼𝑡)) = ∅)))
7 ineq2 4111 . . . . 5 (𝑡 = 𝑏 → (𝑎𝑡) = (𝑎𝑏))
87eqeq1d 2760 . . . 4 (𝑡 = 𝑏 → ((𝑎𝑡) = ∅ ↔ (𝑎𝑏) = ∅))
9 fveq2 6658 . . . . . 6 (𝑡 = 𝑏 → (𝐼𝑡) = (𝐼𝑏))
109ineq2d 4117 . . . . 5 (𝑡 = 𝑏 → ((𝐼𝑎) ∩ (𝐼𝑡)) = ((𝐼𝑎) ∩ (𝐼𝑏)))
1110eqeq1d 2760 . . . 4 (𝑡 = 𝑏 → (((𝐼𝑎) ∩ (𝐼𝑡)) = ∅ ↔ ((𝐼𝑎) ∩ (𝐼𝑏)) = ∅))
128, 11imbi12d 348 . . 3 (𝑡 = 𝑏 → (((𝑎𝑡) = ∅ → ((𝐼𝑎) ∩ (𝐼𝑡)) = ∅) ↔ ((𝑎𝑏) = ∅ → ((𝐼𝑎) ∩ (𝐼𝑏)) = ∅)))
136, 12cbvral2vw 3373 . 2 (∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝑠𝑡) = ∅ → ((𝐼𝑠) ∩ (𝐼𝑡)) = ∅) ↔ ∀𝑎 ∈ 𝒫 𝐵𝑏 ∈ 𝒫 𝐵((𝑎𝑏) = ∅ → ((𝐼𝑎) ∩ (𝐼𝑏)) = ∅))
14 ntrcls.d . . . . 5 𝐷 = (𝑂𝐵)
15 ntrcls.r . . . . 5 (𝜑𝐼𝐷𝐾)
1614, 15ntrclsrcomplex 41111 . . . 4 (𝜑 → (𝐵𝑠) ∈ 𝒫 𝐵)
1716adantr 484 . . 3 ((𝜑𝑠 ∈ 𝒫 𝐵) → (𝐵𝑠) ∈ 𝒫 𝐵)
1814, 15ntrclsrcomplex 41111 . . . . 5 (𝜑 → (𝐵𝑎) ∈ 𝒫 𝐵)
1918adantr 484 . . . 4 ((𝜑𝑎 ∈ 𝒫 𝐵) → (𝐵𝑎) ∈ 𝒫 𝐵)
20 difeq2 4022 . . . . . 6 (𝑠 = (𝐵𝑎) → (𝐵𝑠) = (𝐵 ∖ (𝐵𝑎)))
2120eqeq2d 2769 . . . . 5 (𝑠 = (𝐵𝑎) → (𝑎 = (𝐵𝑠) ↔ 𝑎 = (𝐵 ∖ (𝐵𝑎))))
2221adantl 485 . . . 4 (((𝜑𝑎 ∈ 𝒫 𝐵) ∧ 𝑠 = (𝐵𝑎)) → (𝑎 = (𝐵𝑠) ↔ 𝑎 = (𝐵 ∖ (𝐵𝑎))))
23 elpwi 4503 . . . . . . 7 (𝑎 ∈ 𝒫 𝐵𝑎𝐵)
24 dfss4 4163 . . . . . . 7 (𝑎𝐵 ↔ (𝐵 ∖ (𝐵𝑎)) = 𝑎)
2523, 24sylib 221 . . . . . 6 (𝑎 ∈ 𝒫 𝐵 → (𝐵 ∖ (𝐵𝑎)) = 𝑎)
2625eqcomd 2764 . . . . 5 (𝑎 ∈ 𝒫 𝐵𝑎 = (𝐵 ∖ (𝐵𝑎)))
2726adantl 485 . . . 4 ((𝜑𝑎 ∈ 𝒫 𝐵) → 𝑎 = (𝐵 ∖ (𝐵𝑎)))
2819, 22, 27rspcedvd 3544 . . 3 ((𝜑𝑎 ∈ 𝒫 𝐵) → ∃𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠))
29 simpl1 1188 . . . . 5 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵) → 𝜑)
3014, 15ntrclsrcomplex 41111 . . . . 5 (𝜑 → (𝐵𝑡) ∈ 𝒫 𝐵)
3129, 30syl 17 . . . 4 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵) → (𝐵𝑡) ∈ 𝒫 𝐵)
3214, 15ntrclsrcomplex 41111 . . . . . . 7 (𝜑 → (𝐵𝑏) ∈ 𝒫 𝐵)
3332adantr 484 . . . . . 6 ((𝜑𝑏 ∈ 𝒫 𝐵) → (𝐵𝑏) ∈ 𝒫 𝐵)
34 difeq2 4022 . . . . . . . 8 (𝑡 = (𝐵𝑏) → (𝐵𝑡) = (𝐵 ∖ (𝐵𝑏)))
3534eqeq2d 2769 . . . . . . 7 (𝑡 = (𝐵𝑏) → (𝑏 = (𝐵𝑡) ↔ 𝑏 = (𝐵 ∖ (𝐵𝑏))))
3635adantl 485 . . . . . 6 (((𝜑𝑏 ∈ 𝒫 𝐵) ∧ 𝑡 = (𝐵𝑏)) → (𝑏 = (𝐵𝑡) ↔ 𝑏 = (𝐵 ∖ (𝐵𝑏))))
37 elpwi 4503 . . . . . . . . 9 (𝑏 ∈ 𝒫 𝐵𝑏𝐵)
38 dfss4 4163 . . . . . . . . 9 (𝑏𝐵 ↔ (𝐵 ∖ (𝐵𝑏)) = 𝑏)
3937, 38sylib 221 . . . . . . . 8 (𝑏 ∈ 𝒫 𝐵 → (𝐵 ∖ (𝐵𝑏)) = 𝑏)
4039eqcomd 2764 . . . . . . 7 (𝑏 ∈ 𝒫 𝐵𝑏 = (𝐵 ∖ (𝐵𝑏)))
4140adantl 485 . . . . . 6 ((𝜑𝑏 ∈ 𝒫 𝐵) → 𝑏 = (𝐵 ∖ (𝐵𝑏)))
4233, 36, 41rspcedvd 3544 . . . . 5 ((𝜑𝑏 ∈ 𝒫 𝐵) → ∃𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡))
43423ad2antl1 1182 . . . 4 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑏 ∈ 𝒫 𝐵) → ∃𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡))
44 simp13 1202 . . . . . 6 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → 𝑎 = (𝐵𝑠))
45 ineq1 4109 . . . . . . . 8 (𝑎 = (𝐵𝑠) → (𝑎𝑏) = ((𝐵𝑠) ∩ 𝑏))
4645eqeq1d 2760 . . . . . . 7 (𝑎 = (𝐵𝑠) → ((𝑎𝑏) = ∅ ↔ ((𝐵𝑠) ∩ 𝑏) = ∅))
47 fveq2 6658 . . . . . . . . 9 (𝑎 = (𝐵𝑠) → (𝐼𝑎) = (𝐼‘(𝐵𝑠)))
4847ineq1d 4116 . . . . . . . 8 (𝑎 = (𝐵𝑠) → ((𝐼𝑎) ∩ (𝐼𝑏)) = ((𝐼‘(𝐵𝑠)) ∩ (𝐼𝑏)))
4948eqeq1d 2760 . . . . . . 7 (𝑎 = (𝐵𝑠) → (((𝐼𝑎) ∩ (𝐼𝑏)) = ∅ ↔ ((𝐼‘(𝐵𝑠)) ∩ (𝐼𝑏)) = ∅))
5046, 49imbi12d 348 . . . . . 6 (𝑎 = (𝐵𝑠) → (((𝑎𝑏) = ∅ → ((𝐼𝑎) ∩ (𝐼𝑏)) = ∅) ↔ (((𝐵𝑠) ∩ 𝑏) = ∅ → ((𝐼‘(𝐵𝑠)) ∩ (𝐼𝑏)) = ∅)))
5144, 50syl 17 . . . . 5 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → (((𝑎𝑏) = ∅ → ((𝐼𝑎) ∩ (𝐼𝑏)) = ∅) ↔ (((𝐵𝑠) ∩ 𝑏) = ∅ → ((𝐼‘(𝐵𝑠)) ∩ (𝐼𝑏)) = ∅)))
52 simp3 1135 . . . . . 6 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → 𝑏 = (𝐵𝑡))
53 ineq2 4111 . . . . . . . 8 (𝑏 = (𝐵𝑡) → ((𝐵𝑠) ∩ 𝑏) = ((𝐵𝑠) ∩ (𝐵𝑡)))
5453eqeq1d 2760 . . . . . . 7 (𝑏 = (𝐵𝑡) → (((𝐵𝑠) ∩ 𝑏) = ∅ ↔ ((𝐵𝑠) ∩ (𝐵𝑡)) = ∅))
55 fveq2 6658 . . . . . . . . 9 (𝑏 = (𝐵𝑡) → (𝐼𝑏) = (𝐼‘(𝐵𝑡)))
5655ineq2d 4117 . . . . . . . 8 (𝑏 = (𝐵𝑡) → ((𝐼‘(𝐵𝑠)) ∩ (𝐼𝑏)) = ((𝐼‘(𝐵𝑠)) ∩ (𝐼‘(𝐵𝑡))))
5756eqeq1d 2760 . . . . . . 7 (𝑏 = (𝐵𝑡) → (((𝐼‘(𝐵𝑠)) ∩ (𝐼𝑏)) = ∅ ↔ ((𝐼‘(𝐵𝑠)) ∩ (𝐼‘(𝐵𝑡))) = ∅))
5854, 57imbi12d 348 . . . . . 6 (𝑏 = (𝐵𝑡) → ((((𝐵𝑠) ∩ 𝑏) = ∅ → ((𝐼‘(𝐵𝑠)) ∩ (𝐼𝑏)) = ∅) ↔ (((𝐵𝑠) ∩ (𝐵𝑡)) = ∅ → ((𝐼‘(𝐵𝑠)) ∩ (𝐼‘(𝐵𝑡))) = ∅)))
5952, 58syl 17 . . . . 5 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → ((((𝐵𝑠) ∩ 𝑏) = ∅ → ((𝐼‘(𝐵𝑠)) ∩ (𝐼𝑏)) = ∅) ↔ (((𝐵𝑠) ∩ (𝐵𝑡)) = ∅ → ((𝐼‘(𝐵𝑠)) ∩ (𝐼‘(𝐵𝑡))) = ∅)))
60 simp11 1200 . . . . . 6 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → 𝜑)
61 simp12 1201 . . . . . 6 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → 𝑠 ∈ 𝒫 𝐵)
62 simp2 1134 . . . . . 6 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → 𝑡 ∈ 𝒫 𝐵)
63 simp2 1134 . . . . . . . . . . . 12 ((𝜑𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵) → 𝑠 ∈ 𝒫 𝐵)
6463elpwid 4505 . . . . . . . . . . 11 ((𝜑𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵) → 𝑠𝐵)
65 simp3 1135 . . . . . . . . . . . 12 ((𝜑𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵) → 𝑡 ∈ 𝒫 𝐵)
6665elpwid 4505 . . . . . . . . . . 11 ((𝜑𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵) → 𝑡𝐵)
6764, 66unssd 4091 . . . . . . . . . 10 ((𝜑𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵) → (𝑠𝑡) ⊆ 𝐵)
68 ssid 3914 . . . . . . . . . 10 𝐵𝐵
69 rcompleq 4200 . . . . . . . . . 10 (((𝑠𝑡) ⊆ 𝐵𝐵𝐵) → ((𝑠𝑡) = 𝐵 ↔ (𝐵 ∖ (𝑠𝑡)) = (𝐵𝐵)))
7067, 68, 69sylancl 589 . . . . . . . . 9 ((𝜑𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵) → ((𝑠𝑡) = 𝐵 ↔ (𝐵 ∖ (𝑠𝑡)) = (𝐵𝐵)))
71 difundi 4184 . . . . . . . . . 10 (𝐵 ∖ (𝑠𝑡)) = ((𝐵𝑠) ∩ (𝐵𝑡))
72 difid 4269 . . . . . . . . . 10 (𝐵𝐵) = ∅
7371, 72eqeq12i 2773 . . . . . . . . 9 ((𝐵 ∖ (𝑠𝑡)) = (𝐵𝐵) ↔ ((𝐵𝑠) ∩ (𝐵𝑡)) = ∅)
7470, 73bitr2di 291 . . . . . . . 8 ((𝜑𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵) → (((𝐵𝑠) ∩ (𝐵𝑡)) = ∅ ↔ (𝑠𝑡) = 𝐵))
75 ntrcls.o . . . . . . . . . . . . . . . 16 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖𝑗))))))
7675, 14, 15ntrclsiex 41129 . . . . . . . . . . . . . . 15 (𝜑𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵))
77763ad2ant1 1130 . . . . . . . . . . . . . 14 ((𝜑𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵) → 𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵))
78 elmapi 8438 . . . . . . . . . . . . . 14 (𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵) → 𝐼:𝒫 𝐵⟶𝒫 𝐵)
7977, 78syl 17 . . . . . . . . . . . . 13 ((𝜑𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵) → 𝐼:𝒫 𝐵⟶𝒫 𝐵)
8014, 15ntrclsbex 41110 . . . . . . . . . . . . . . 15 (𝜑𝐵 ∈ V)
81803ad2ant1 1130 . . . . . . . . . . . . . 14 ((𝜑𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵) → 𝐵 ∈ V)
82 difssd 4038 . . . . . . . . . . . . . 14 ((𝜑𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵) → (𝐵𝑠) ⊆ 𝐵)
8381, 82sselpwd 5196 . . . . . . . . . . . . 13 ((𝜑𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵) → (𝐵𝑠) ∈ 𝒫 𝐵)
8479, 83ffvelrnd 6843 . . . . . . . . . . . 12 ((𝜑𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵) → (𝐼‘(𝐵𝑠)) ∈ 𝒫 𝐵)
8584elpwid 4505 . . . . . . . . . . 11 ((𝜑𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵) → (𝐼‘(𝐵𝑠)) ⊆ 𝐵)
86 ssinss1 4142 . . . . . . . . . . 11 ((𝐼‘(𝐵𝑠)) ⊆ 𝐵 → ((𝐼‘(𝐵𝑠)) ∩ (𝐼‘(𝐵𝑡))) ⊆ 𝐵)
8785, 86syl 17 . . . . . . . . . 10 ((𝜑𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵) → ((𝐼‘(𝐵𝑠)) ∩ (𝐼‘(𝐵𝑡))) ⊆ 𝐵)
88 0ss 4292 . . . . . . . . . 10 ∅ ⊆ 𝐵
89 rcompleq 4200 . . . . . . . . . 10 ((((𝐼‘(𝐵𝑠)) ∩ (𝐼‘(𝐵𝑡))) ⊆ 𝐵 ∧ ∅ ⊆ 𝐵) → (((𝐼‘(𝐵𝑠)) ∩ (𝐼‘(𝐵𝑡))) = ∅ ↔ (𝐵 ∖ ((𝐼‘(𝐵𝑠)) ∩ (𝐼‘(𝐵𝑡)))) = (𝐵 ∖ ∅)))
9087, 88, 89sylancl 589 . . . . . . . . 9 ((𝜑𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵) → (((𝐼‘(𝐵𝑠)) ∩ (𝐼‘(𝐵𝑡))) = ∅ ↔ (𝐵 ∖ ((𝐼‘(𝐵𝑠)) ∩ (𝐼‘(𝐵𝑡)))) = (𝐵 ∖ ∅)))
91 difindi 4186 . . . . . . . . . 10 (𝐵 ∖ ((𝐼‘(𝐵𝑠)) ∩ (𝐼‘(𝐵𝑡)))) = ((𝐵 ∖ (𝐼‘(𝐵𝑠))) ∪ (𝐵 ∖ (𝐼‘(𝐵𝑡))))
92 dif0 4271 . . . . . . . . . 10 (𝐵 ∖ ∅) = 𝐵
9391, 92eqeq12i 2773 . . . . . . . . 9 ((𝐵 ∖ ((𝐼‘(𝐵𝑠)) ∩ (𝐼‘(𝐵𝑡)))) = (𝐵 ∖ ∅) ↔ ((𝐵 ∖ (𝐼‘(𝐵𝑠))) ∪ (𝐵 ∖ (𝐼‘(𝐵𝑡)))) = 𝐵)
9490, 93bitrdi 290 . . . . . . . 8 ((𝜑𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵) → (((𝐼‘(𝐵𝑠)) ∩ (𝐼‘(𝐵𝑡))) = ∅ ↔ ((𝐵 ∖ (𝐼‘(𝐵𝑠))) ∪ (𝐵 ∖ (𝐼‘(𝐵𝑡)))) = 𝐵))
9574, 94imbi12d 348 . . . . . . 7 ((𝜑𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵) → ((((𝐵𝑠) ∩ (𝐵𝑡)) = ∅ → ((𝐼‘(𝐵𝑠)) ∩ (𝐼‘(𝐵𝑡))) = ∅) ↔ ((𝑠𝑡) = 𝐵 → ((𝐵 ∖ (𝐼‘(𝐵𝑠))) ∪ (𝐵 ∖ (𝐼‘(𝐵𝑡)))) = 𝐵)))
96 eqid 2758 . . . . . . . . . . . 12 (𝐷𝐼) = (𝐷𝐼)
97 eqid 2758 . . . . . . . . . . . 12 ((𝐷𝐼)‘𝑠) = ((𝐷𝐼)‘𝑠)
9875, 14, 81, 77, 96, 63, 97dssmapfv3d 41093 . . . . . . . . . . 11 ((𝜑𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵) → ((𝐷𝐼)‘𝑠) = (𝐵 ∖ (𝐼‘(𝐵𝑠))))
99 eqid 2758 . . . . . . . . . . . 12 ((𝐷𝐼)‘𝑡) = ((𝐷𝐼)‘𝑡)
10075, 14, 81, 77, 96, 65, 99dssmapfv3d 41093 . . . . . . . . . . 11 ((𝜑𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵) → ((𝐷𝐼)‘𝑡) = (𝐵 ∖ (𝐼‘(𝐵𝑡))))
10198, 100uneq12d 4069 . . . . . . . . . 10 ((𝜑𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵) → (((𝐷𝐼)‘𝑠) ∪ ((𝐷𝐼)‘𝑡)) = ((𝐵 ∖ (𝐼‘(𝐵𝑠))) ∪ (𝐵 ∖ (𝐼‘(𝐵𝑡)))))
10275, 14, 15ntrclsfv1 41131 . . . . . . . . . . . 12 (𝜑 → (𝐷𝐼) = 𝐾)
1031023ad2ant1 1130 . . . . . . . . . . 11 ((𝜑𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵) → (𝐷𝐼) = 𝐾)
104 fveq1 6657 . . . . . . . . . . . 12 ((𝐷𝐼) = 𝐾 → ((𝐷𝐼)‘𝑠) = (𝐾𝑠))
105 fveq1 6657 . . . . . . . . . . . 12 ((𝐷𝐼) = 𝐾 → ((𝐷𝐼)‘𝑡) = (𝐾𝑡))
106104, 105uneq12d 4069 . . . . . . . . . . 11 ((𝐷𝐼) = 𝐾 → (((𝐷𝐼)‘𝑠) ∪ ((𝐷𝐼)‘𝑡)) = ((𝐾𝑠) ∪ (𝐾𝑡)))
107103, 106syl 17 . . . . . . . . . 10 ((𝜑𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵) → (((𝐷𝐼)‘𝑠) ∪ ((𝐷𝐼)‘𝑡)) = ((𝐾𝑠) ∪ (𝐾𝑡)))
108101, 107eqtr3d 2795 . . . . . . . . 9 ((𝜑𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵) → ((𝐵 ∖ (𝐼‘(𝐵𝑠))) ∪ (𝐵 ∖ (𝐼‘(𝐵𝑡)))) = ((𝐾𝑠) ∪ (𝐾𝑡)))
109108eqeq1d 2760 . . . . . . . 8 ((𝜑𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵) → (((𝐵 ∖ (𝐼‘(𝐵𝑠))) ∪ (𝐵 ∖ (𝐼‘(𝐵𝑡)))) = 𝐵 ↔ ((𝐾𝑠) ∪ (𝐾𝑡)) = 𝐵))
110109imbi2d 344 . . . . . . 7 ((𝜑𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵) → (((𝑠𝑡) = 𝐵 → ((𝐵 ∖ (𝐼‘(𝐵𝑠))) ∪ (𝐵 ∖ (𝐼‘(𝐵𝑡)))) = 𝐵) ↔ ((𝑠𝑡) = 𝐵 → ((𝐾𝑠) ∪ (𝐾𝑡)) = 𝐵)))
11195, 110bitrd 282 . . . . . 6 ((𝜑𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵) → ((((𝐵𝑠) ∩ (𝐵𝑡)) = ∅ → ((𝐼‘(𝐵𝑠)) ∩ (𝐼‘(𝐵𝑡))) = ∅) ↔ ((𝑠𝑡) = 𝐵 → ((𝐾𝑠) ∪ (𝐾𝑡)) = 𝐵)))
11260, 61, 62, 111syl3anc 1368 . . . . 5 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → ((((𝐵𝑠) ∩ (𝐵𝑡)) = ∅ → ((𝐼‘(𝐵𝑠)) ∩ (𝐼‘(𝐵𝑡))) = ∅) ↔ ((𝑠𝑡) = 𝐵 → ((𝐾𝑠) ∪ (𝐾𝑡)) = 𝐵)))
11351, 59, 1123bitrd 308 . . . 4 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → (((𝑎𝑏) = ∅ → ((𝐼𝑎) ∩ (𝐼𝑏)) = ∅) ↔ ((𝑠𝑡) = 𝐵 → ((𝐾𝑠) ∪ (𝐾𝑡)) = 𝐵)))
11431, 43, 113ralxfrd2 5281 . . 3 ((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) → (∀𝑏 ∈ 𝒫 𝐵((𝑎𝑏) = ∅ → ((𝐼𝑎) ∩ (𝐼𝑏)) = ∅) ↔ ∀𝑡 ∈ 𝒫 𝐵((𝑠𝑡) = 𝐵 → ((𝐾𝑠) ∪ (𝐾𝑡)) = 𝐵)))
11517, 28, 114ralxfrd2 5281 . 2 (𝜑 → (∀𝑎 ∈ 𝒫 𝐵𝑏 ∈ 𝒫 𝐵((𝑎𝑏) = ∅ → ((𝐼𝑎) ∩ (𝐼𝑏)) = ∅) ↔ ∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝑠𝑡) = 𝐵 → ((𝐾𝑠) ∪ (𝐾𝑡)) = 𝐵)))
11613, 115syl5bb 286 1 (𝜑 → (∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝑠𝑡) = ∅ → ((𝐼𝑠) ∩ (𝐼𝑡)) = ∅) ↔ ∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝑠𝑡) = 𝐵 → ((𝐾𝑠) ∪ (𝐾𝑡)) = 𝐵)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∀wral 3070  ∃wrex 3071  Vcvv 3409   ∖ cdif 3855   ∪ cun 3856   ∩ cin 3857   ⊆ wss 3858  ∅c0 4225  𝒫 cpw 4494   class class class wbr 5032   ↦ cmpt 5112  ⟶wf 6331  ‘cfv 6335  (class class class)co 7150   ↑m cmap 8416 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5156  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298  ax-un 7459 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-iun 4885  df-br 5033  df-opab 5095  df-mpt 5113  df-id 5430  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343  df-ov 7153  df-oprab 7154  df-mpo 7155  df-1st 7693  df-2nd 7694  df-map 8418 This theorem is referenced by: (None)
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