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Theorem ntrclskb 37849
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 3785 . . . . 5 (𝑠 = 𝑎 → (𝑠𝑡) = (𝑎𝑡))
21eqeq1d 2623 . . . 4 (𝑠 = 𝑎 → ((𝑠𝑡) = ∅ ↔ (𝑎𝑡) = ∅))
3 fveq2 6148 . . . . . 6 (𝑠 = 𝑎 → (𝐼𝑠) = (𝐼𝑎))
43ineq1d 3791 . . . . 5 (𝑠 = 𝑎 → ((𝐼𝑠) ∩ (𝐼𝑡)) = ((𝐼𝑎) ∩ (𝐼𝑡)))
54eqeq1d 2623 . . . 4 (𝑠 = 𝑎 → (((𝐼𝑠) ∩ (𝐼𝑡)) = ∅ ↔ ((𝐼𝑎) ∩ (𝐼𝑡)) = ∅))
62, 5imbi12d 334 . . 3 (𝑠 = 𝑎 → (((𝑠𝑡) = ∅ → ((𝐼𝑠) ∩ (𝐼𝑡)) = ∅) ↔ ((𝑎𝑡) = ∅ → ((𝐼𝑎) ∩ (𝐼𝑡)) = ∅)))
7 ineq2 3786 . . . . 5 (𝑡 = 𝑏 → (𝑎𝑡) = (𝑎𝑏))
87eqeq1d 2623 . . . 4 (𝑡 = 𝑏 → ((𝑎𝑡) = ∅ ↔ (𝑎𝑏) = ∅))
9 fveq2 6148 . . . . . 6 (𝑡 = 𝑏 → (𝐼𝑡) = (𝐼𝑏))
109ineq2d 3792 . . . . 5 (𝑡 = 𝑏 → ((𝐼𝑎) ∩ (𝐼𝑡)) = ((𝐼𝑎) ∩ (𝐼𝑏)))
1110eqeq1d 2623 . . . 4 (𝑡 = 𝑏 → (((𝐼𝑎) ∩ (𝐼𝑡)) = ∅ ↔ ((𝐼𝑎) ∩ (𝐼𝑏)) = ∅))
128, 11imbi12d 334 . . 3 (𝑡 = 𝑏 → (((𝑎𝑡) = ∅ → ((𝐼𝑎) ∩ (𝐼𝑡)) = ∅) ↔ ((𝑎𝑏) = ∅ → ((𝐼𝑎) ∩ (𝐼𝑏)) = ∅)))
136, 12cbvral2v 3167 . 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 37815 . . . 4 (𝜑 → (𝐵𝑠) ∈ 𝒫 𝐵)
1716adantr 481 . . 3 ((𝜑𝑠 ∈ 𝒫 𝐵) → (𝐵𝑠) ∈ 𝒫 𝐵)
1814, 15ntrclsrcomplex 37815 . . . . 5 (𝜑 → (𝐵𝑎) ∈ 𝒫 𝐵)
1918adantr 481 . . . 4 ((𝜑𝑎 ∈ 𝒫 𝐵) → (𝐵𝑎) ∈ 𝒫 𝐵)
20 difeq2 3700 . . . . . 6 (𝑠 = (𝐵𝑎) → (𝐵𝑠) = (𝐵 ∖ (𝐵𝑎)))
2120eqeq2d 2631 . . . . 5 (𝑠 = (𝐵𝑎) → (𝑎 = (𝐵𝑠) ↔ 𝑎 = (𝐵 ∖ (𝐵𝑎))))
2221adantl 482 . . . 4 (((𝜑𝑎 ∈ 𝒫 𝐵) ∧ 𝑠 = (𝐵𝑎)) → (𝑎 = (𝐵𝑠) ↔ 𝑎 = (𝐵 ∖ (𝐵𝑎))))
23 elpwi 4140 . . . . . . 7 (𝑎 ∈ 𝒫 𝐵𝑎𝐵)
24 dfss4 3836 . . . . . . 7 (𝑎𝐵 ↔ (𝐵 ∖ (𝐵𝑎)) = 𝑎)
2523, 24sylib 208 . . . . . 6 (𝑎 ∈ 𝒫 𝐵 → (𝐵 ∖ (𝐵𝑎)) = 𝑎)
2625eqcomd 2627 . . . . 5 (𝑎 ∈ 𝒫 𝐵𝑎 = (𝐵 ∖ (𝐵𝑎)))
2726adantl 482 . . . 4 ((𝜑𝑎 ∈ 𝒫 𝐵) → 𝑎 = (𝐵 ∖ (𝐵𝑎)))
2819, 22, 27rspcedvd 3302 . . 3 ((𝜑𝑎 ∈ 𝒫 𝐵) → ∃𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠))
29 simpl1 1062 . . . . 5 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵) → 𝜑)
3014, 15ntrclsrcomplex 37815 . . . . 5 (𝜑 → (𝐵𝑡) ∈ 𝒫 𝐵)
3129, 30syl 17 . . . 4 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵) → (𝐵𝑡) ∈ 𝒫 𝐵)
3214, 15ntrclsrcomplex 37815 . . . . . . 7 (𝜑 → (𝐵𝑏) ∈ 𝒫 𝐵)
3332adantr 481 . . . . . 6 ((𝜑𝑏 ∈ 𝒫 𝐵) → (𝐵𝑏) ∈ 𝒫 𝐵)
34 difeq2 3700 . . . . . . . 8 (𝑡 = (𝐵𝑏) → (𝐵𝑡) = (𝐵 ∖ (𝐵𝑏)))
3534eqeq2d 2631 . . . . . . 7 (𝑡 = (𝐵𝑏) → (𝑏 = (𝐵𝑡) ↔ 𝑏 = (𝐵 ∖ (𝐵𝑏))))
3635adantl 482 . . . . . 6 (((𝜑𝑏 ∈ 𝒫 𝐵) ∧ 𝑡 = (𝐵𝑏)) → (𝑏 = (𝐵𝑡) ↔ 𝑏 = (𝐵 ∖ (𝐵𝑏))))
37 elpwi 4140 . . . . . . . . 9 (𝑏 ∈ 𝒫 𝐵𝑏𝐵)
38 dfss4 3836 . . . . . . . . 9 (𝑏𝐵 ↔ (𝐵 ∖ (𝐵𝑏)) = 𝑏)
3937, 38sylib 208 . . . . . . . 8 (𝑏 ∈ 𝒫 𝐵 → (𝐵 ∖ (𝐵𝑏)) = 𝑏)
4039eqcomd 2627 . . . . . . 7 (𝑏 ∈ 𝒫 𝐵𝑏 = (𝐵 ∖ (𝐵𝑏)))
4140adantl 482 . . . . . 6 ((𝜑𝑏 ∈ 𝒫 𝐵) → 𝑏 = (𝐵 ∖ (𝐵𝑏)))
4233, 36, 41rspcedvd 3302 . . . . 5 ((𝜑𝑏 ∈ 𝒫 𝐵) → ∃𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡))
43423ad2antl1 1221 . . . 4 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑏 ∈ 𝒫 𝐵) → ∃𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡))
44 simp13 1091 . . . . . 6 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → 𝑎 = (𝐵𝑠))
45 ineq1 3785 . . . . . . . 8 (𝑎 = (𝐵𝑠) → (𝑎𝑏) = ((𝐵𝑠) ∩ 𝑏))
4645eqeq1d 2623 . . . . . . 7 (𝑎 = (𝐵𝑠) → ((𝑎𝑏) = ∅ ↔ ((𝐵𝑠) ∩ 𝑏) = ∅))
47 fveq2 6148 . . . . . . . . 9 (𝑎 = (𝐵𝑠) → (𝐼𝑎) = (𝐼‘(𝐵𝑠)))
4847ineq1d 3791 . . . . . . . 8 (𝑎 = (𝐵𝑠) → ((𝐼𝑎) ∩ (𝐼𝑏)) = ((𝐼‘(𝐵𝑠)) ∩ (𝐼𝑏)))
4948eqeq1d 2623 . . . . . . 7 (𝑎 = (𝐵𝑠) → (((𝐼𝑎) ∩ (𝐼𝑏)) = ∅ ↔ ((𝐼‘(𝐵𝑠)) ∩ (𝐼𝑏)) = ∅))
5046, 49imbi12d 334 . . . . . 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 1061 . . . . . 6 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → 𝑏 = (𝐵𝑡))
53 ineq2 3786 . . . . . . . 8 (𝑏 = (𝐵𝑡) → ((𝐵𝑠) ∩ 𝑏) = ((𝐵𝑠) ∩ (𝐵𝑡)))
5453eqeq1d 2623 . . . . . . 7 (𝑏 = (𝐵𝑡) → (((𝐵𝑠) ∩ 𝑏) = ∅ ↔ ((𝐵𝑠) ∩ (𝐵𝑡)) = ∅))
55 fveq2 6148 . . . . . . . . 9 (𝑏 = (𝐵𝑡) → (𝐼𝑏) = (𝐼‘(𝐵𝑡)))
5655ineq2d 3792 . . . . . . . 8 (𝑏 = (𝐵𝑡) → ((𝐼‘(𝐵𝑠)) ∩ (𝐼𝑏)) = ((𝐼‘(𝐵𝑠)) ∩ (𝐼‘(𝐵𝑡))))
5756eqeq1d 2623 . . . . . . 7 (𝑏 = (𝐵𝑡) → (((𝐼‘(𝐵𝑠)) ∩ (𝐼𝑏)) = ∅ ↔ ((𝐼‘(𝐵𝑠)) ∩ (𝐼‘(𝐵𝑡))) = ∅))
5854, 57imbi12d 334 . . . . . 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 1089 . . . . . 6 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → 𝜑)
61 simp12 1090 . . . . . 6 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → 𝑠 ∈ 𝒫 𝐵)
62 simp2 1060 . . . . . 6 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → 𝑡 ∈ 𝒫 𝐵)
63 simp2 1060 . . . . . . . . . . . 12 ((𝜑𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵) → 𝑠 ∈ 𝒫 𝐵)
6463elpwid 4141 . . . . . . . . . . 11 ((𝜑𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵) → 𝑠𝐵)
65 simp3 1061 . . . . . . . . . . . 12 ((𝜑𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵) → 𝑡 ∈ 𝒫 𝐵)
6665elpwid 4141 . . . . . . . . . . 11 ((𝜑𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵) → 𝑡𝐵)
6764, 66unssd 3767 . . . . . . . . . 10 ((𝜑𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵) → (𝑠𝑡) ⊆ 𝐵)
68 ssid 3603 . . . . . . . . . 10 𝐵𝐵
69 rcompleq 37800 . . . . . . . . . 10 (((𝑠𝑡) ⊆ 𝐵𝐵𝐵) → ((𝑠𝑡) = 𝐵 ↔ (𝐵 ∖ (𝑠𝑡)) = (𝐵𝐵)))
7067, 68, 69sylancl 693 . . . . . . . . 9 ((𝜑𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵) → ((𝑠𝑡) = 𝐵 ↔ (𝐵 ∖ (𝑠𝑡)) = (𝐵𝐵)))
71 difundi 3855 . . . . . . . . . 10 (𝐵 ∖ (𝑠𝑡)) = ((𝐵𝑠) ∩ (𝐵𝑡))
72 difid 3922 . . . . . . . . . 10 (𝐵𝐵) = ∅
7371, 72eqeq12i 2635 . . . . . . . . 9 ((𝐵 ∖ (𝑠𝑡)) = (𝐵𝐵) ↔ ((𝐵𝑠) ∩ (𝐵𝑡)) = ∅)
7470, 73syl6rbb 277 . . . . . . . 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 37833 . . . . . . . . . . . . . . 15 (𝜑𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵))
77763ad2ant1 1080 . . . . . . . . . . . . . 14 ((𝜑𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵) → 𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵))
78 elmapi 7823 . . . . . . . . . . . . . 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 37814 . . . . . . . . . . . . . . 15 (𝜑𝐵 ∈ V)
81803ad2ant1 1080 . . . . . . . . . . . . . 14 ((𝜑𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵) → 𝐵 ∈ V)
82 difssd 3716 . . . . . . . . . . . . . 14 ((𝜑𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵) → (𝐵𝑠) ⊆ 𝐵)
8381, 82sselpwd 4767 . . . . . . . . . . . . 13 ((𝜑𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵) → (𝐵𝑠) ∈ 𝒫 𝐵)
8479, 83ffvelrnd 6316 . . . . . . . . . . . 12 ((𝜑𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵) → (𝐼‘(𝐵𝑠)) ∈ 𝒫 𝐵)
8584elpwid 4141 . . . . . . . . . . 11 ((𝜑𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵) → (𝐼‘(𝐵𝑠)) ⊆ 𝐵)
86 ssinss1 3819 . . . . . . . . . . 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 3944 . . . . . . . . . 10 ∅ ⊆ 𝐵
89 rcompleq 37800 . . . . . . . . . 10 ((((𝐼‘(𝐵𝑠)) ∩ (𝐼‘(𝐵𝑡))) ⊆ 𝐵 ∧ ∅ ⊆ 𝐵) → (((𝐼‘(𝐵𝑠)) ∩ (𝐼‘(𝐵𝑡))) = ∅ ↔ (𝐵 ∖ ((𝐼‘(𝐵𝑠)) ∩ (𝐼‘(𝐵𝑡)))) = (𝐵 ∖ ∅)))
9087, 88, 89sylancl 693 . . . . . . . . 9 ((𝜑𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵) → (((𝐼‘(𝐵𝑠)) ∩ (𝐼‘(𝐵𝑡))) = ∅ ↔ (𝐵 ∖ ((𝐼‘(𝐵𝑠)) ∩ (𝐼‘(𝐵𝑡)))) = (𝐵 ∖ ∅)))
91 difindi 3857 . . . . . . . . . 10 (𝐵 ∖ ((𝐼‘(𝐵𝑠)) ∩ (𝐼‘(𝐵𝑡)))) = ((𝐵 ∖ (𝐼‘(𝐵𝑠))) ∪ (𝐵 ∖ (𝐼‘(𝐵𝑡))))
92 dif0 3924 . . . . . . . . . 10 (𝐵 ∖ ∅) = 𝐵
9391, 92eqeq12i 2635 . . . . . . . . 9 ((𝐵 ∖ ((𝐼‘(𝐵𝑠)) ∩ (𝐼‘(𝐵𝑡)))) = (𝐵 ∖ ∅) ↔ ((𝐵 ∖ (𝐼‘(𝐵𝑠))) ∪ (𝐵 ∖ (𝐼‘(𝐵𝑡)))) = 𝐵)
9490, 93syl6bb 276 . . . . . . . 8 ((𝜑𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵) → (((𝐼‘(𝐵𝑠)) ∩ (𝐼‘(𝐵𝑡))) = ∅ ↔ ((𝐵 ∖ (𝐼‘(𝐵𝑠))) ∪ (𝐵 ∖ (𝐼‘(𝐵𝑡)))) = 𝐵))
9574, 94imbi12d 334 . . . . . . 7 ((𝜑𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵) → ((((𝐵𝑠) ∩ (𝐵𝑡)) = ∅ → ((𝐼‘(𝐵𝑠)) ∩ (𝐼‘(𝐵𝑡))) = ∅) ↔ ((𝑠𝑡) = 𝐵 → ((𝐵 ∖ (𝐼‘(𝐵𝑠))) ∪ (𝐵 ∖ (𝐼‘(𝐵𝑡)))) = 𝐵)))
96 eqid 2621 . . . . . . . . . . . 12 (𝐷𝐼) = (𝐷𝐼)
97 eqid 2621 . . . . . . . . . . . 12 ((𝐷𝐼)‘𝑠) = ((𝐷𝐼)‘𝑠)
9875, 14, 81, 77, 96, 63, 97dssmapfv3d 37795 . . . . . . . . . . 11 ((𝜑𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵) → ((𝐷𝐼)‘𝑠) = (𝐵 ∖ (𝐼‘(𝐵𝑠))))
99 eqid 2621 . . . . . . . . . . . 12 ((𝐷𝐼)‘𝑡) = ((𝐷𝐼)‘𝑡)
10075, 14, 81, 77, 96, 65, 99dssmapfv3d 37795 . . . . . . . . . . 11 ((𝜑𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵) → ((𝐷𝐼)‘𝑡) = (𝐵 ∖ (𝐼‘(𝐵𝑡))))
10198, 100uneq12d 3746 . . . . . . . . . 10 ((𝜑𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵) → (((𝐷𝐼)‘𝑠) ∪ ((𝐷𝐼)‘𝑡)) = ((𝐵 ∖ (𝐼‘(𝐵𝑠))) ∪ (𝐵 ∖ (𝐼‘(𝐵𝑡)))))
10275, 14, 15ntrclsfv1 37835 . . . . . . . . . . . 12 (𝜑 → (𝐷𝐼) = 𝐾)
1031023ad2ant1 1080 . . . . . . . . . . 11 ((𝜑𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵) → (𝐷𝐼) = 𝐾)
104 fveq1 6147 . . . . . . . . . . . 12 ((𝐷𝐼) = 𝐾 → ((𝐷𝐼)‘𝑠) = (𝐾𝑠))
105 fveq1 6147 . . . . . . . . . . . 12 ((𝐷𝐼) = 𝐾 → ((𝐷𝐼)‘𝑡) = (𝐾𝑡))
106104, 105uneq12d 3746 . . . . . . . . . . 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 2657 . . . . . . . . 9 ((𝜑𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵) → ((𝐵 ∖ (𝐼‘(𝐵𝑠))) ∪ (𝐵 ∖ (𝐼‘(𝐵𝑡)))) = ((𝐾𝑠) ∪ (𝐾𝑡)))
109108eqeq1d 2623 . . . . . . . 8 ((𝜑𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵) → (((𝐵 ∖ (𝐼‘(𝐵𝑠))) ∪ (𝐵 ∖ (𝐼‘(𝐵𝑡)))) = 𝐵 ↔ ((𝐾𝑠) ∪ (𝐾𝑡)) = 𝐵))
110109imbi2d 330 . . . . . . 7 ((𝜑𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵) → (((𝑠𝑡) = 𝐵 → ((𝐵 ∖ (𝐼‘(𝐵𝑠))) ∪ (𝐵 ∖ (𝐼‘(𝐵𝑡)))) = 𝐵) ↔ ((𝑠𝑡) = 𝐵 → ((𝐾𝑠) ∪ (𝐾𝑡)) = 𝐵)))
11195, 110bitrd 268 . . . . . 6 ((𝜑𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵) → ((((𝐵𝑠) ∩ (𝐵𝑡)) = ∅ → ((𝐼‘(𝐵𝑠)) ∩ (𝐼‘(𝐵𝑡))) = ∅) ↔ ((𝑠𝑡) = 𝐵 → ((𝐾𝑠) ∪ (𝐾𝑡)) = 𝐵)))
11260, 61, 62, 111syl3anc 1323 . . . . 5 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → ((((𝐵𝑠) ∩ (𝐵𝑡)) = ∅ → ((𝐼‘(𝐵𝑠)) ∩ (𝐼‘(𝐵𝑡))) = ∅) ↔ ((𝑠𝑡) = 𝐵 → ((𝐾𝑠) ∪ (𝐾𝑡)) = 𝐵)))
11351, 59, 1123bitrd 294 . . . 4 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → (((𝑎𝑏) = ∅ → ((𝐼𝑎) ∩ (𝐼𝑏)) = ∅) ↔ ((𝑠𝑡) = 𝐵 → ((𝐾𝑠) ∪ (𝐾𝑡)) = 𝐵)))
11431, 43, 113ralxfrd2 4844 . . 3 ((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) → (∀𝑏 ∈ 𝒫 𝐵((𝑎𝑏) = ∅ → ((𝐼𝑎) ∩ (𝐼𝑏)) = ∅) ↔ ∀𝑡 ∈ 𝒫 𝐵((𝑠𝑡) = 𝐵 → ((𝐾𝑠) ∪ (𝐾𝑡)) = 𝐵)))
11517, 28, 114ralxfrd2 4844 . 2 (𝜑 → (∀𝑎 ∈ 𝒫 𝐵𝑏 ∈ 𝒫 𝐵((𝑎𝑏) = ∅ → ((𝐼𝑎) ∩ (𝐼𝑏)) = ∅) ↔ ∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝑠𝑡) = 𝐵 → ((𝐾𝑠) ∪ (𝐾𝑡)) = 𝐵)))
11613, 115syl5bb 272 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 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907  wrex 2908  Vcvv 3186  cdif 3552  cun 3553  cin 3554  wss 3555  c0 3891  𝒫 cpw 4130   class class class wbr 4613  cmpt 4673  wf 5843  cfv 5847  (class class class)co 6604  𝑚 cmap 7802
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-1st 7113  df-2nd 7114  df-map 7804
This theorem is referenced by: (None)
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