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Theorem ntrclsk3 38685
Description: The intersection of interiors of a every pair is a subset of the interior of the intersection of the pair if an only if the closure of the union of every pair is a subset of the union of closures of the pair. (Contributed by RP, 19-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
ntrclsk3 (𝜑 → (∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝐼𝑠) ∩ (𝐼𝑡)) ⊆ (𝐼‘(𝑠𝑡)) ↔ ∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵(𝐾‘(𝑠𝑡)) ⊆ ((𝐾𝑠) ∪ (𝐾𝑡))))
Distinct variable groups:   𝐵,𝑠,𝑡,𝑖,𝑗,𝑘   𝐼,𝑠,𝑡,𝑖,𝑗,𝑘   𝜑,𝑠,𝑡,𝑖,𝑗,𝑘
Allowed substitution hints:   𝐷(𝑡,𝑖,𝑗,𝑘,𝑠)   𝐾(𝑡,𝑖,𝑗,𝑘,𝑠)   𝑂(𝑡,𝑖,𝑗,𝑘,𝑠)

Proof of Theorem ntrclsk3
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 6229 . . . . 5 (𝑠 = 𝑎 → (𝐼𝑠) = (𝐼𝑎))
21ineq1d 3846 . . . 4 (𝑠 = 𝑎 → ((𝐼𝑠) ∩ (𝐼𝑡)) = ((𝐼𝑎) ∩ (𝐼𝑡)))
3 ineq1 3840 . . . . 5 (𝑠 = 𝑎 → (𝑠𝑡) = (𝑎𝑡))
43fveq2d 6233 . . . 4 (𝑠 = 𝑎 → (𝐼‘(𝑠𝑡)) = (𝐼‘(𝑎𝑡)))
52, 4sseq12d 3667 . . 3 (𝑠 = 𝑎 → (((𝐼𝑠) ∩ (𝐼𝑡)) ⊆ (𝐼‘(𝑠𝑡)) ↔ ((𝐼𝑎) ∩ (𝐼𝑡)) ⊆ (𝐼‘(𝑎𝑡))))
6 fveq2 6229 . . . . 5 (𝑡 = 𝑏 → (𝐼𝑡) = (𝐼𝑏))
76ineq2d 3847 . . . 4 (𝑡 = 𝑏 → ((𝐼𝑎) ∩ (𝐼𝑡)) = ((𝐼𝑎) ∩ (𝐼𝑏)))
8 ineq2 3841 . . . . 5 (𝑡 = 𝑏 → (𝑎𝑡) = (𝑎𝑏))
98fveq2d 6233 . . . 4 (𝑡 = 𝑏 → (𝐼‘(𝑎𝑡)) = (𝐼‘(𝑎𝑏)))
107, 9sseq12d 3667 . . 3 (𝑡 = 𝑏 → (((𝐼𝑎) ∩ (𝐼𝑡)) ⊆ (𝐼‘(𝑎𝑡)) ↔ ((𝐼𝑎) ∩ (𝐼𝑏)) ⊆ (𝐼‘(𝑎𝑏))))
115, 10cbvral2v 3209 . 2 (∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝐼𝑠) ∩ (𝐼𝑡)) ⊆ (𝐼‘(𝑠𝑡)) ↔ ∀𝑎 ∈ 𝒫 𝐵𝑏 ∈ 𝒫 𝐵((𝐼𝑎) ∩ (𝐼𝑏)) ⊆ (𝐼‘(𝑎𝑏)))
12 ntrcls.d . . . . . 6 𝐷 = (𝑂𝐵)
13 ntrcls.r . . . . . 6 (𝜑𝐼𝐷𝐾)
1412, 13ntrclsbex 38649 . . . . 5 (𝜑𝐵 ∈ V)
15 difssd 3771 . . . . 5 (𝜑 → (𝐵𝑠) ⊆ 𝐵)
1614, 15sselpwd 4840 . . . 4 (𝜑 → (𝐵𝑠) ∈ 𝒫 𝐵)
1716adantr 480 . . 3 ((𝜑𝑠 ∈ 𝒫 𝐵) → (𝐵𝑠) ∈ 𝒫 𝐵)
18 elpwi 4201 . . . 4 (𝑎 ∈ 𝒫 𝐵𝑎𝐵)
19 simpl 472 . . . . . 6 ((𝐵 ∈ V ∧ 𝑎𝐵) → 𝐵 ∈ V)
20 difssd 3771 . . . . . 6 ((𝐵 ∈ V ∧ 𝑎𝐵) → (𝐵𝑎) ⊆ 𝐵)
2119, 20sselpwd 4840 . . . . 5 ((𝐵 ∈ V ∧ 𝑎𝐵) → (𝐵𝑎) ∈ 𝒫 𝐵)
22 simpr 476 . . . . . . . 8 (((𝐵 ∈ V ∧ 𝑎𝐵) ∧ 𝑠 = (𝐵𝑎)) → 𝑠 = (𝐵𝑎))
2322difeq2d 3761 . . . . . . 7 (((𝐵 ∈ V ∧ 𝑎𝐵) ∧ 𝑠 = (𝐵𝑎)) → (𝐵𝑠) = (𝐵 ∖ (𝐵𝑎)))
2423eqeq2d 2661 . . . . . 6 (((𝐵 ∈ V ∧ 𝑎𝐵) ∧ 𝑠 = (𝐵𝑎)) → (𝑎 = (𝐵𝑠) ↔ 𝑎 = (𝐵 ∖ (𝐵𝑎))))
25 eqcom 2658 . . . . . 6 (𝑎 = (𝐵 ∖ (𝐵𝑎)) ↔ (𝐵 ∖ (𝐵𝑎)) = 𝑎)
2624, 25syl6bb 276 . . . . 5 (((𝐵 ∈ V ∧ 𝑎𝐵) ∧ 𝑠 = (𝐵𝑎)) → (𝑎 = (𝐵𝑠) ↔ (𝐵 ∖ (𝐵𝑎)) = 𝑎))
27 dfss4 3891 . . . . . . 7 (𝑎𝐵 ↔ (𝐵 ∖ (𝐵𝑎)) = 𝑎)
2827biimpi 206 . . . . . 6 (𝑎𝐵 → (𝐵 ∖ (𝐵𝑎)) = 𝑎)
2928adantl 481 . . . . 5 ((𝐵 ∈ V ∧ 𝑎𝐵) → (𝐵 ∖ (𝐵𝑎)) = 𝑎)
3021, 26, 29rspcedvd 3348 . . . 4 ((𝐵 ∈ V ∧ 𝑎𝐵) → ∃𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠))
3114, 18, 30syl2an 493 . . 3 ((𝜑𝑎 ∈ 𝒫 𝐵) → ∃𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠))
32 simpl1 1084 . . . . 5 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵) → 𝜑)
33 difssd 3771 . . . . . 6 (𝜑 → (𝐵𝑡) ⊆ 𝐵)
3414, 33sselpwd 4840 . . . . 5 (𝜑 → (𝐵𝑡) ∈ 𝒫 𝐵)
3532, 34syl 17 . . . 4 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵) → (𝐵𝑡) ∈ 𝒫 𝐵)
36 elpwi 4201 . . . . . 6 (𝑏 ∈ 𝒫 𝐵𝑏𝐵)
37 simpl 472 . . . . . . . 8 ((𝐵 ∈ V ∧ 𝑏𝐵) → 𝐵 ∈ V)
38 difssd 3771 . . . . . . . 8 ((𝐵 ∈ V ∧ 𝑏𝐵) → (𝐵𝑏) ⊆ 𝐵)
3937, 38sselpwd 4840 . . . . . . 7 ((𝐵 ∈ V ∧ 𝑏𝐵) → (𝐵𝑏) ∈ 𝒫 𝐵)
40 simpr 476 . . . . . . . . . 10 (((𝐵 ∈ V ∧ 𝑏𝐵) ∧ 𝑡 = (𝐵𝑏)) → 𝑡 = (𝐵𝑏))
4140difeq2d 3761 . . . . . . . . 9 (((𝐵 ∈ V ∧ 𝑏𝐵) ∧ 𝑡 = (𝐵𝑏)) → (𝐵𝑡) = (𝐵 ∖ (𝐵𝑏)))
4241eqeq2d 2661 . . . . . . . 8 (((𝐵 ∈ V ∧ 𝑏𝐵) ∧ 𝑡 = (𝐵𝑏)) → (𝑏 = (𝐵𝑡) ↔ 𝑏 = (𝐵 ∖ (𝐵𝑏))))
43 eqcom 2658 . . . . . . . 8 (𝑏 = (𝐵 ∖ (𝐵𝑏)) ↔ (𝐵 ∖ (𝐵𝑏)) = 𝑏)
4442, 43syl6bb 276 . . . . . . 7 (((𝐵 ∈ V ∧ 𝑏𝐵) ∧ 𝑡 = (𝐵𝑏)) → (𝑏 = (𝐵𝑡) ↔ (𝐵 ∖ (𝐵𝑏)) = 𝑏))
45 dfss4 3891 . . . . . . . . 9 (𝑏𝐵 ↔ (𝐵 ∖ (𝐵𝑏)) = 𝑏)
4645biimpi 206 . . . . . . . 8 (𝑏𝐵 → (𝐵 ∖ (𝐵𝑏)) = 𝑏)
4746adantl 481 . . . . . . 7 ((𝐵 ∈ V ∧ 𝑏𝐵) → (𝐵 ∖ (𝐵𝑏)) = 𝑏)
4839, 44, 47rspcedvd 3348 . . . . . 6 ((𝐵 ∈ V ∧ 𝑏𝐵) → ∃𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡))
4914, 36, 48syl2an 493 . . . . 5 ((𝜑𝑏 ∈ 𝒫 𝐵) → ∃𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡))
50493ad2antl1 1243 . . . 4 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑏 ∈ 𝒫 𝐵) → ∃𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡))
51 simp13 1113 . . . . . 6 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → 𝑎 = (𝐵𝑠))
52 fveq2 6229 . . . . . . . 8 (𝑎 = (𝐵𝑠) → (𝐼𝑎) = (𝐼‘(𝐵𝑠)))
5352ineq1d 3846 . . . . . . 7 (𝑎 = (𝐵𝑠) → ((𝐼𝑎) ∩ (𝐼𝑏)) = ((𝐼‘(𝐵𝑠)) ∩ (𝐼𝑏)))
54 ineq1 3840 . . . . . . . 8 (𝑎 = (𝐵𝑠) → (𝑎𝑏) = ((𝐵𝑠) ∩ 𝑏))
5554fveq2d 6233 . . . . . . 7 (𝑎 = (𝐵𝑠) → (𝐼‘(𝑎𝑏)) = (𝐼‘((𝐵𝑠) ∩ 𝑏)))
5653, 55sseq12d 3667 . . . . . 6 (𝑎 = (𝐵𝑠) → (((𝐼𝑎) ∩ (𝐼𝑏)) ⊆ (𝐼‘(𝑎𝑏)) ↔ ((𝐼‘(𝐵𝑠)) ∩ (𝐼𝑏)) ⊆ (𝐼‘((𝐵𝑠) ∩ 𝑏))))
5751, 56syl 17 . . . . 5 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → (((𝐼𝑎) ∩ (𝐼𝑏)) ⊆ (𝐼‘(𝑎𝑏)) ↔ ((𝐼‘(𝐵𝑠)) ∩ (𝐼𝑏)) ⊆ (𝐼‘((𝐵𝑠) ∩ 𝑏))))
58 fveq2 6229 . . . . . . . 8 (𝑏 = (𝐵𝑡) → (𝐼𝑏) = (𝐼‘(𝐵𝑡)))
5958ineq2d 3847 . . . . . . 7 (𝑏 = (𝐵𝑡) → ((𝐼‘(𝐵𝑠)) ∩ (𝐼𝑏)) = ((𝐼‘(𝐵𝑠)) ∩ (𝐼‘(𝐵𝑡))))
60 ineq2 3841 . . . . . . . . 9 (𝑏 = (𝐵𝑡) → ((𝐵𝑠) ∩ 𝑏) = ((𝐵𝑠) ∩ (𝐵𝑡)))
61 difundi 3912 . . . . . . . . 9 (𝐵 ∖ (𝑠𝑡)) = ((𝐵𝑠) ∩ (𝐵𝑡))
6260, 61syl6eqr 2703 . . . . . . . 8 (𝑏 = (𝐵𝑡) → ((𝐵𝑠) ∩ 𝑏) = (𝐵 ∖ (𝑠𝑡)))
6362fveq2d 6233 . . . . . . 7 (𝑏 = (𝐵𝑡) → (𝐼‘((𝐵𝑠) ∩ 𝑏)) = (𝐼‘(𝐵 ∖ (𝑠𝑡))))
6459, 63sseq12d 3667 . . . . . 6 (𝑏 = (𝐵𝑡) → (((𝐼‘(𝐵𝑠)) ∩ (𝐼𝑏)) ⊆ (𝐼‘((𝐵𝑠) ∩ 𝑏)) ↔ ((𝐼‘(𝐵𝑠)) ∩ (𝐼‘(𝐵𝑡))) ⊆ (𝐼‘(𝐵 ∖ (𝑠𝑡)))))
65643ad2ant3 1104 . . . . 5 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → (((𝐼‘(𝐵𝑠)) ∩ (𝐼𝑏)) ⊆ (𝐼‘((𝐵𝑠) ∩ 𝑏)) ↔ ((𝐼‘(𝐵𝑠)) ∩ (𝐼‘(𝐵𝑡))) ⊆ (𝐼‘(𝐵 ∖ (𝑠𝑡)))))
66 simp11 1111 . . . . . . . 8 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → 𝜑)
67 ntrcls.o . . . . . . . . . 10 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖𝑚 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖𝑗))))))
6867, 12, 13ntrclsiex 38668 . . . . . . . . 9 (𝜑𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵))
6968, 14jca 553 . . . . . . . 8 (𝜑 → (𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵) ∧ 𝐵 ∈ V))
7066, 69syl 17 . . . . . . 7 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → (𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵) ∧ 𝐵 ∈ V))
71 elmapi 7921 . . . . . . . . . . . 12 (𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵) → 𝐼:𝒫 𝐵⟶𝒫 𝐵)
7271adantr 480 . . . . . . . . . . 11 ((𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵) ∧ 𝐵 ∈ V) → 𝐼:𝒫 𝐵⟶𝒫 𝐵)
73 simpr 476 . . . . . . . . . . . 12 ((𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵) ∧ 𝐵 ∈ V) → 𝐵 ∈ V)
74 difssd 3771 . . . . . . . . . . . 12 ((𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵) ∧ 𝐵 ∈ V) → (𝐵𝑠) ⊆ 𝐵)
7573, 74sselpwd 4840 . . . . . . . . . . 11 ((𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵) ∧ 𝐵 ∈ V) → (𝐵𝑠) ∈ 𝒫 𝐵)
7672, 75ffvelrnd 6400 . . . . . . . . . 10 ((𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵) ∧ 𝐵 ∈ V) → (𝐼‘(𝐵𝑠)) ∈ 𝒫 𝐵)
7776elpwid 4203 . . . . . . . . 9 ((𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵) ∧ 𝐵 ∈ V) → (𝐼‘(𝐵𝑠)) ⊆ 𝐵)
78 orc 399 . . . . . . . . 9 ((𝐼‘(𝐵𝑠)) ⊆ 𝐵 → ((𝐼‘(𝐵𝑠)) ⊆ 𝐵 ∨ (𝐼‘(𝐵𝑡)) ⊆ 𝐵))
79 inss 3875 . . . . . . . . 9 (((𝐼‘(𝐵𝑠)) ⊆ 𝐵 ∨ (𝐼‘(𝐵𝑡)) ⊆ 𝐵) → ((𝐼‘(𝐵𝑠)) ∩ (𝐼‘(𝐵𝑡))) ⊆ 𝐵)
8077, 78, 793syl 18 . . . . . . . 8 ((𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵) ∧ 𝐵 ∈ V) → ((𝐼‘(𝐵𝑠)) ∩ (𝐼‘(𝐵𝑡))) ⊆ 𝐵)
81 difssd 3771 . . . . . . . . . . 11 ((𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵) ∧ 𝐵 ∈ V) → (𝐵 ∖ (𝑠𝑡)) ⊆ 𝐵)
8273, 81sselpwd 4840 . . . . . . . . . 10 ((𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵) ∧ 𝐵 ∈ V) → (𝐵 ∖ (𝑠𝑡)) ∈ 𝒫 𝐵)
8372, 82ffvelrnd 6400 . . . . . . . . 9 ((𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵) ∧ 𝐵 ∈ V) → (𝐼‘(𝐵 ∖ (𝑠𝑡))) ∈ 𝒫 𝐵)
8483elpwid 4203 . . . . . . . 8 ((𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵) ∧ 𝐵 ∈ V) → (𝐼‘(𝐵 ∖ (𝑠𝑡))) ⊆ 𝐵)
8580, 84jca 553 . . . . . . 7 ((𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵) ∧ 𝐵 ∈ V) → (((𝐼‘(𝐵𝑠)) ∩ (𝐼‘(𝐵𝑡))) ⊆ 𝐵 ∧ (𝐼‘(𝐵 ∖ (𝑠𝑡))) ⊆ 𝐵))
86 sscon34b 38634 . . . . . . 7 ((((𝐼‘(𝐵𝑠)) ∩ (𝐼‘(𝐵𝑡))) ⊆ 𝐵 ∧ (𝐼‘(𝐵 ∖ (𝑠𝑡))) ⊆ 𝐵) → (((𝐼‘(𝐵𝑠)) ∩ (𝐼‘(𝐵𝑡))) ⊆ (𝐼‘(𝐵 ∖ (𝑠𝑡))) ↔ (𝐵 ∖ (𝐼‘(𝐵 ∖ (𝑠𝑡)))) ⊆ (𝐵 ∖ ((𝐼‘(𝐵𝑠)) ∩ (𝐼‘(𝐵𝑡))))))
8770, 85, 863syl 18 . . . . . 6 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → (((𝐼‘(𝐵𝑠)) ∩ (𝐼‘(𝐵𝑡))) ⊆ (𝐼‘(𝐵 ∖ (𝑠𝑡))) ↔ (𝐵 ∖ (𝐼‘(𝐵 ∖ (𝑠𝑡)))) ⊆ (𝐵 ∖ ((𝐼‘(𝐵𝑠)) ∩ (𝐼‘(𝐵𝑡))))))
88 difindi 3914 . . . . . . . 8 (𝐵 ∖ ((𝐼‘(𝐵𝑠)) ∩ (𝐼‘(𝐵𝑡)))) = ((𝐵 ∖ (𝐼‘(𝐵𝑠))) ∪ (𝐵 ∖ (𝐼‘(𝐵𝑡))))
8988sseq2i 3663 . . . . . . 7 ((𝐵 ∖ (𝐼‘(𝐵 ∖ (𝑠𝑡)))) ⊆ (𝐵 ∖ ((𝐼‘(𝐵𝑠)) ∩ (𝐼‘(𝐵𝑡)))) ↔ (𝐵 ∖ (𝐼‘(𝐵 ∖ (𝑠𝑡)))) ⊆ ((𝐵 ∖ (𝐼‘(𝐵𝑠))) ∪ (𝐵 ∖ (𝐼‘(𝐵𝑡)))))
9089a1i 11 . . . . . 6 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → ((𝐵 ∖ (𝐼‘(𝐵 ∖ (𝑠𝑡)))) ⊆ (𝐵 ∖ ((𝐼‘(𝐵𝑠)) ∩ (𝐼‘(𝐵𝑡)))) ↔ (𝐵 ∖ (𝐼‘(𝐵 ∖ (𝑠𝑡)))) ⊆ ((𝐵 ∖ (𝐼‘(𝐵𝑠))) ∪ (𝐵 ∖ (𝐼‘(𝐵𝑡))))))
9166, 14syl 17 . . . . . . 7 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → 𝐵 ∈ V)
9266, 68syl 17 . . . . . . 7 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → 𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵))
93 simp12 1112 . . . . . . 7 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → 𝑠 ∈ 𝒫 𝐵)
94 rp-simp2 38404 . . . . . . 7 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → 𝑡 ∈ 𝒫 𝐵)
95 simpl2 1085 . . . . . . . . . 10 (((𝜑𝐵 ∈ V ∧ 𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵)) ∧ (𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵)) → 𝐵 ∈ V)
96 simpl3 1086 . . . . . . . . . 10 (((𝜑𝐵 ∈ V ∧ 𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵)) ∧ (𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵)) → 𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵))
97 eqid 2651 . . . . . . . . . 10 (𝐷𝐼) = (𝐷𝐼)
98 simpl 472 . . . . . . . . . . . 12 ((𝐵 ∈ V ∧ (𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵)) → 𝐵 ∈ V)
99 simprl 809 . . . . . . . . . . . . . 14 ((𝐵 ∈ V ∧ (𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵)) → 𝑠 ∈ 𝒫 𝐵)
10099elpwid 4203 . . . . . . . . . . . . 13 ((𝐵 ∈ V ∧ (𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵)) → 𝑠𝐵)
101 simprr 811 . . . . . . . . . . . . . 14 ((𝐵 ∈ V ∧ (𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵)) → 𝑡 ∈ 𝒫 𝐵)
102101elpwid 4203 . . . . . . . . . . . . 13 ((𝐵 ∈ V ∧ (𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵)) → 𝑡𝐵)
103100, 102unssd 3822 . . . . . . . . . . . 12 ((𝐵 ∈ V ∧ (𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵)) → (𝑠𝑡) ⊆ 𝐵)
10498, 103sselpwd 4840 . . . . . . . . . . 11 ((𝐵 ∈ V ∧ (𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵)) → (𝑠𝑡) ∈ 𝒫 𝐵)
1051043ad2antl2 1244 . . . . . . . . . 10 (((𝜑𝐵 ∈ V ∧ 𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵)) ∧ (𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵)) → (𝑠𝑡) ∈ 𝒫 𝐵)
106 eqid 2651 . . . . . . . . . 10 ((𝐷𝐼)‘(𝑠𝑡)) = ((𝐷𝐼)‘(𝑠𝑡))
10767, 12, 95, 96, 97, 105, 106dssmapfv3d 38630 . . . . . . . . 9 (((𝜑𝐵 ∈ V ∧ 𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵)) ∧ (𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵)) → ((𝐷𝐼)‘(𝑠𝑡)) = (𝐵 ∖ (𝐼‘(𝐵 ∖ (𝑠𝑡)))))
108 simpl1 1084 . . . . . . . . . 10 (((𝜑𝐵 ∈ V ∧ 𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵)) ∧ (𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵)) → 𝜑)
10967, 12, 13ntrclsfv1 38670 . . . . . . . . . . 11 (𝜑 → (𝐷𝐼) = 𝐾)
110109fveq1d 6231 . . . . . . . . . 10 (𝜑 → ((𝐷𝐼)‘(𝑠𝑡)) = (𝐾‘(𝑠𝑡)))
111108, 110syl 17 . . . . . . . . 9 (((𝜑𝐵 ∈ V ∧ 𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵)) ∧ (𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵)) → ((𝐷𝐼)‘(𝑠𝑡)) = (𝐾‘(𝑠𝑡)))
112107, 111eqtr3d 2687 . . . . . . . 8 (((𝜑𝐵 ∈ V ∧ 𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵)) ∧ (𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵)) → (𝐵 ∖ (𝐼‘(𝐵 ∖ (𝑠𝑡)))) = (𝐾‘(𝑠𝑡)))
113 simprl 809 . . . . . . . . . . 11 (((𝜑𝐵 ∈ V ∧ 𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵)) ∧ (𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵)) → 𝑠 ∈ 𝒫 𝐵)
114 eqid 2651 . . . . . . . . . . 11 ((𝐷𝐼)‘𝑠) = ((𝐷𝐼)‘𝑠)
11567, 12, 95, 96, 97, 113, 114dssmapfv3d 38630 . . . . . . . . . 10 (((𝜑𝐵 ∈ V ∧ 𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵)) ∧ (𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵)) → ((𝐷𝐼)‘𝑠) = (𝐵 ∖ (𝐼‘(𝐵𝑠))))
116109fveq1d 6231 . . . . . . . . . . 11 (𝜑 → ((𝐷𝐼)‘𝑠) = (𝐾𝑠))
117108, 116syl 17 . . . . . . . . . 10 (((𝜑𝐵 ∈ V ∧ 𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵)) ∧ (𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵)) → ((𝐷𝐼)‘𝑠) = (𝐾𝑠))
118115, 117eqtr3d 2687 . . . . . . . . 9 (((𝜑𝐵 ∈ V ∧ 𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵)) ∧ (𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵)) → (𝐵 ∖ (𝐼‘(𝐵𝑠))) = (𝐾𝑠))
119 simprr 811 . . . . . . . . . . 11 (((𝜑𝐵 ∈ V ∧ 𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵)) ∧ (𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵)) → 𝑡 ∈ 𝒫 𝐵)
120 eqid 2651 . . . . . . . . . . 11 ((𝐷𝐼)‘𝑡) = ((𝐷𝐼)‘𝑡)
12167, 12, 95, 96, 97, 119, 120dssmapfv3d 38630 . . . . . . . . . 10 (((𝜑𝐵 ∈ V ∧ 𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵)) ∧ (𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵)) → ((𝐷𝐼)‘𝑡) = (𝐵 ∖ (𝐼‘(𝐵𝑡))))
122109fveq1d 6231 . . . . . . . . . . 11 (𝜑 → ((𝐷𝐼)‘𝑡) = (𝐾𝑡))
123108, 122syl 17 . . . . . . . . . 10 (((𝜑𝐵 ∈ V ∧ 𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵)) ∧ (𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵)) → ((𝐷𝐼)‘𝑡) = (𝐾𝑡))
124121, 123eqtr3d 2687 . . . . . . . . 9 (((𝜑𝐵 ∈ V ∧ 𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵)) ∧ (𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵)) → (𝐵 ∖ (𝐼‘(𝐵𝑡))) = (𝐾𝑡))
125118, 124uneq12d 3801 . . . . . . . 8 (((𝜑𝐵 ∈ V ∧ 𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵)) ∧ (𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵)) → ((𝐵 ∖ (𝐼‘(𝐵𝑠))) ∪ (𝐵 ∖ (𝐼‘(𝐵𝑡)))) = ((𝐾𝑠) ∪ (𝐾𝑡)))
126112, 125sseq12d 3667 . . . . . . 7 (((𝜑𝐵 ∈ V ∧ 𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵)) ∧ (𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵)) → ((𝐵 ∖ (𝐼‘(𝐵 ∖ (𝑠𝑡)))) ⊆ ((𝐵 ∖ (𝐼‘(𝐵𝑠))) ∪ (𝐵 ∖ (𝐼‘(𝐵𝑡)))) ↔ (𝐾‘(𝑠𝑡)) ⊆ ((𝐾𝑠) ∪ (𝐾𝑡))))
12766, 91, 92, 93, 94, 126syl32anc 1374 . . . . . 6 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → ((𝐵 ∖ (𝐼‘(𝐵 ∖ (𝑠𝑡)))) ⊆ ((𝐵 ∖ (𝐼‘(𝐵𝑠))) ∪ (𝐵 ∖ (𝐼‘(𝐵𝑡)))) ↔ (𝐾‘(𝑠𝑡)) ⊆ ((𝐾𝑠) ∪ (𝐾𝑡))))
12887, 90, 1273bitrd 294 . . . . 5 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → (((𝐼‘(𝐵𝑠)) ∩ (𝐼‘(𝐵𝑡))) ⊆ (𝐼‘(𝐵 ∖ (𝑠𝑡))) ↔ (𝐾‘(𝑠𝑡)) ⊆ ((𝐾𝑠) ∪ (𝐾𝑡))))
12957, 65, 1283bitrd 294 . . . 4 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → (((𝐼𝑎) ∩ (𝐼𝑏)) ⊆ (𝐼‘(𝑎𝑏)) ↔ (𝐾‘(𝑠𝑡)) ⊆ ((𝐾𝑠) ∪ (𝐾𝑡))))
13035, 50, 129ralxfrd2 4914 . . 3 ((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) → (∀𝑏 ∈ 𝒫 𝐵((𝐼𝑎) ∩ (𝐼𝑏)) ⊆ (𝐼‘(𝑎𝑏)) ↔ ∀𝑡 ∈ 𝒫 𝐵(𝐾‘(𝑠𝑡)) ⊆ ((𝐾𝑠) ∪ (𝐾𝑡))))
13117, 31, 130ralxfrd2 4914 . 2 (𝜑 → (∀𝑎 ∈ 𝒫 𝐵𝑏 ∈ 𝒫 𝐵((𝐼𝑎) ∩ (𝐼𝑏)) ⊆ (𝐼‘(𝑎𝑏)) ↔ ∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵(𝐾‘(𝑠𝑡)) ⊆ ((𝐾𝑠) ∪ (𝐾𝑡))))
13211, 131syl5bb 272 1 (𝜑 → (∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝐼𝑠) ∩ (𝐼𝑡)) ⊆ (𝐼‘(𝑠𝑡)) ↔ ∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵(𝐾‘(𝑠𝑡)) ⊆ ((𝐾𝑠) ∪ (𝐾𝑡))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wo 382  wa 383  w3a 1054   = wceq 1523  wcel 2030  wral 2941  wrex 2942  Vcvv 3231  cdif 3604  cun 3605  cin 3606  wss 3607  𝒫 cpw 4191   class class class wbr 4685  cmpt 4762  wf 5922  cfv 5926  (class class class)co 6690  𝑚 cmap 7899
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-frege1 38401
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-1st 7210  df-2nd 7211  df-map 7901
This theorem is referenced by: (None)
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