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Theorem ntrclsk3 44088
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 6905 . . . . 5 (𝑠 = 𝑎 → (𝐼𝑠) = (𝐼𝑎))
21ineq1d 4218 . . . 4 (𝑠 = 𝑎 → ((𝐼𝑠) ∩ (𝐼𝑡)) = ((𝐼𝑎) ∩ (𝐼𝑡)))
3 ineq1 4212 . . . . 5 (𝑠 = 𝑎 → (𝑠𝑡) = (𝑎𝑡))
43fveq2d 6909 . . . 4 (𝑠 = 𝑎 → (𝐼‘(𝑠𝑡)) = (𝐼‘(𝑎𝑡)))
52, 4sseq12d 4016 . . 3 (𝑠 = 𝑎 → (((𝐼𝑠) ∩ (𝐼𝑡)) ⊆ (𝐼‘(𝑠𝑡)) ↔ ((𝐼𝑎) ∩ (𝐼𝑡)) ⊆ (𝐼‘(𝑎𝑡))))
6 fveq2 6905 . . . . 5 (𝑡 = 𝑏 → (𝐼𝑡) = (𝐼𝑏))
76ineq2d 4219 . . . 4 (𝑡 = 𝑏 → ((𝐼𝑎) ∩ (𝐼𝑡)) = ((𝐼𝑎) ∩ (𝐼𝑏)))
8 ineq2 4213 . . . . 5 (𝑡 = 𝑏 → (𝑎𝑡) = (𝑎𝑏))
98fveq2d 6909 . . . 4 (𝑡 = 𝑏 → (𝐼‘(𝑎𝑡)) = (𝐼‘(𝑎𝑏)))
107, 9sseq12d 4016 . . 3 (𝑡 = 𝑏 → (((𝐼𝑎) ∩ (𝐼𝑡)) ⊆ (𝐼‘(𝑎𝑡)) ↔ ((𝐼𝑎) ∩ (𝐼𝑏)) ⊆ (𝐼‘(𝑎𝑏))))
115, 10cbvral2vw 3240 . 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 44052 . . . . 5 (𝜑𝐵 ∈ V)
15 difssd 4136 . . . . 5 (𝜑 → (𝐵𝑠) ⊆ 𝐵)
1614, 15sselpwd 5327 . . . 4 (𝜑 → (𝐵𝑠) ∈ 𝒫 𝐵)
1716adantr 480 . . 3 ((𝜑𝑠 ∈ 𝒫 𝐵) → (𝐵𝑠) ∈ 𝒫 𝐵)
18 elpwi 4606 . . . 4 (𝑎 ∈ 𝒫 𝐵𝑎𝐵)
19 simpl 482 . . . . . 6 ((𝐵 ∈ V ∧ 𝑎𝐵) → 𝐵 ∈ V)
20 difssd 4136 . . . . . 6 ((𝐵 ∈ V ∧ 𝑎𝐵) → (𝐵𝑎) ⊆ 𝐵)
2119, 20sselpwd 5327 . . . . 5 ((𝐵 ∈ V ∧ 𝑎𝐵) → (𝐵𝑎) ∈ 𝒫 𝐵)
22 simpr 484 . . . . . . . 8 (((𝐵 ∈ V ∧ 𝑎𝐵) ∧ 𝑠 = (𝐵𝑎)) → 𝑠 = (𝐵𝑎))
2322difeq2d 4125 . . . . . . 7 (((𝐵 ∈ V ∧ 𝑎𝐵) ∧ 𝑠 = (𝐵𝑎)) → (𝐵𝑠) = (𝐵 ∖ (𝐵𝑎)))
2423eqeq2d 2747 . . . . . 6 (((𝐵 ∈ V ∧ 𝑎𝐵) ∧ 𝑠 = (𝐵𝑎)) → (𝑎 = (𝐵𝑠) ↔ 𝑎 = (𝐵 ∖ (𝐵𝑎))))
25 eqcom 2743 . . . . . 6 (𝑎 = (𝐵 ∖ (𝐵𝑎)) ↔ (𝐵 ∖ (𝐵𝑎)) = 𝑎)
2624, 25bitrdi 287 . . . . 5 (((𝐵 ∈ V ∧ 𝑎𝐵) ∧ 𝑠 = (𝐵𝑎)) → (𝑎 = (𝐵𝑠) ↔ (𝐵 ∖ (𝐵𝑎)) = 𝑎))
27 dfss4 4268 . . . . . . 7 (𝑎𝐵 ↔ (𝐵 ∖ (𝐵𝑎)) = 𝑎)
2827biimpi 216 . . . . . 6 (𝑎𝐵 → (𝐵 ∖ (𝐵𝑎)) = 𝑎)
2928adantl 481 . . . . 5 ((𝐵 ∈ V ∧ 𝑎𝐵) → (𝐵 ∖ (𝐵𝑎)) = 𝑎)
3021, 26, 29rspcedvd 3623 . . . 4 ((𝐵 ∈ V ∧ 𝑎𝐵) → ∃𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠))
3114, 18, 30syl2an 596 . . 3 ((𝜑𝑎 ∈ 𝒫 𝐵) → ∃𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠))
32 simpl1 1191 . . . . 5 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵) → 𝜑)
33 difssd 4136 . . . . . 6 (𝜑 → (𝐵𝑡) ⊆ 𝐵)
3414, 33sselpwd 5327 . . . . 5 (𝜑 → (𝐵𝑡) ∈ 𝒫 𝐵)
3532, 34syl 17 . . . 4 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵) → (𝐵𝑡) ∈ 𝒫 𝐵)
36 elpwi 4606 . . . . . 6 (𝑏 ∈ 𝒫 𝐵𝑏𝐵)
37 simpl 482 . . . . . . . 8 ((𝐵 ∈ V ∧ 𝑏𝐵) → 𝐵 ∈ V)
38 difssd 4136 . . . . . . . 8 ((𝐵 ∈ V ∧ 𝑏𝐵) → (𝐵𝑏) ⊆ 𝐵)
3937, 38sselpwd 5327 . . . . . . 7 ((𝐵 ∈ V ∧ 𝑏𝐵) → (𝐵𝑏) ∈ 𝒫 𝐵)
40 simpr 484 . . . . . . . . . 10 (((𝐵 ∈ V ∧ 𝑏𝐵) ∧ 𝑡 = (𝐵𝑏)) → 𝑡 = (𝐵𝑏))
4140difeq2d 4125 . . . . . . . . 9 (((𝐵 ∈ V ∧ 𝑏𝐵) ∧ 𝑡 = (𝐵𝑏)) → (𝐵𝑡) = (𝐵 ∖ (𝐵𝑏)))
4241eqeq2d 2747 . . . . . . . 8 (((𝐵 ∈ V ∧ 𝑏𝐵) ∧ 𝑡 = (𝐵𝑏)) → (𝑏 = (𝐵𝑡) ↔ 𝑏 = (𝐵 ∖ (𝐵𝑏))))
43 eqcom 2743 . . . . . . . 8 (𝑏 = (𝐵 ∖ (𝐵𝑏)) ↔ (𝐵 ∖ (𝐵𝑏)) = 𝑏)
4442, 43bitrdi 287 . . . . . . 7 (((𝐵 ∈ V ∧ 𝑏𝐵) ∧ 𝑡 = (𝐵𝑏)) → (𝑏 = (𝐵𝑡) ↔ (𝐵 ∖ (𝐵𝑏)) = 𝑏))
45 dfss4 4268 . . . . . . . . 9 (𝑏𝐵 ↔ (𝐵 ∖ (𝐵𝑏)) = 𝑏)
4645biimpi 216 . . . . . . . 8 (𝑏𝐵 → (𝐵 ∖ (𝐵𝑏)) = 𝑏)
4746adantl 481 . . . . . . 7 ((𝐵 ∈ V ∧ 𝑏𝐵) → (𝐵 ∖ (𝐵𝑏)) = 𝑏)
4839, 44, 47rspcedvd 3623 . . . . . 6 ((𝐵 ∈ V ∧ 𝑏𝐵) → ∃𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡))
4914, 36, 48syl2an 596 . . . . 5 ((𝜑𝑏 ∈ 𝒫 𝐵) → ∃𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡))
50493ad2antl1 1185 . . . 4 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑏 ∈ 𝒫 𝐵) → ∃𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡))
51 simp13 1205 . . . . . 6 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → 𝑎 = (𝐵𝑠))
52 fveq2 6905 . . . . . . . 8 (𝑎 = (𝐵𝑠) → (𝐼𝑎) = (𝐼‘(𝐵𝑠)))
5352ineq1d 4218 . . . . . . 7 (𝑎 = (𝐵𝑠) → ((𝐼𝑎) ∩ (𝐼𝑏)) = ((𝐼‘(𝐵𝑠)) ∩ (𝐼𝑏)))
54 ineq1 4212 . . . . . . . 8 (𝑎 = (𝐵𝑠) → (𝑎𝑏) = ((𝐵𝑠) ∩ 𝑏))
5554fveq2d 6909 . . . . . . 7 (𝑎 = (𝐵𝑠) → (𝐼‘(𝑎𝑏)) = (𝐼‘((𝐵𝑠) ∩ 𝑏)))
5653, 55sseq12d 4016 . . . . . 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 6905 . . . . . . . 8 (𝑏 = (𝐵𝑡) → (𝐼𝑏) = (𝐼‘(𝐵𝑡)))
5958ineq2d 4219 . . . . . . 7 (𝑏 = (𝐵𝑡) → ((𝐼‘(𝐵𝑠)) ∩ (𝐼𝑏)) = ((𝐼‘(𝐵𝑠)) ∩ (𝐼‘(𝐵𝑡))))
60 ineq2 4213 . . . . . . . . 9 (𝑏 = (𝐵𝑡) → ((𝐵𝑠) ∩ 𝑏) = ((𝐵𝑠) ∩ (𝐵𝑡)))
61 difundi 4289 . . . . . . . . 9 (𝐵 ∖ (𝑠𝑡)) = ((𝐵𝑠) ∩ (𝐵𝑡))
6260, 61eqtr4di 2794 . . . . . . . 8 (𝑏 = (𝐵𝑡) → ((𝐵𝑠) ∩ 𝑏) = (𝐵 ∖ (𝑠𝑡)))
6362fveq2d 6909 . . . . . . 7 (𝑏 = (𝐵𝑡) → (𝐼‘((𝐵𝑠) ∩ 𝑏)) = (𝐼‘(𝐵 ∖ (𝑠𝑡))))
6459, 63sseq12d 4016 . . . . . 6 (𝑏 = (𝐵𝑡) → (((𝐼‘(𝐵𝑠)) ∩ (𝐼𝑏)) ⊆ (𝐼‘((𝐵𝑠) ∩ 𝑏)) ↔ ((𝐼‘(𝐵𝑠)) ∩ (𝐼‘(𝐵𝑡))) ⊆ (𝐼‘(𝐵 ∖ (𝑠𝑡)))))
65643ad2ant3 1135 . . . . 5 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → (((𝐼‘(𝐵𝑠)) ∩ (𝐼𝑏)) ⊆ (𝐼‘((𝐵𝑠) ∩ 𝑏)) ↔ ((𝐼‘(𝐵𝑠)) ∩ (𝐼‘(𝐵𝑡))) ⊆ (𝐼‘(𝐵 ∖ (𝑠𝑡)))))
66 simp11 1203 . . . . . . 7 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → 𝜑)
67 ntrcls.o . . . . . . . . 9 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖𝑗))))))
6867, 12, 13ntrclsiex 44071 . . . . . . . 8 (𝜑𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵))
6968, 14jca 511 . . . . . . 7 (𝜑 → (𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵) ∧ 𝐵 ∈ V))
70 elmapi 8890 . . . . . . . . . . . 12 (𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵) → 𝐼:𝒫 𝐵⟶𝒫 𝐵)
7170adantr 480 . . . . . . . . . . 11 ((𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵) ∧ 𝐵 ∈ V) → 𝐼:𝒫 𝐵⟶𝒫 𝐵)
72 simpr 484 . . . . . . . . . . . 12 ((𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵) ∧ 𝐵 ∈ V) → 𝐵 ∈ V)
73 difssd 4136 . . . . . . . . . . . 12 ((𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵) ∧ 𝐵 ∈ V) → (𝐵𝑠) ⊆ 𝐵)
7472, 73sselpwd 5327 . . . . . . . . . . 11 ((𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵) ∧ 𝐵 ∈ V) → (𝐵𝑠) ∈ 𝒫 𝐵)
7571, 74ffvelcdmd 7104 . . . . . . . . . 10 ((𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵) ∧ 𝐵 ∈ V) → (𝐼‘(𝐵𝑠)) ∈ 𝒫 𝐵)
7675elpwid 4608 . . . . . . . . 9 ((𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵) ∧ 𝐵 ∈ V) → (𝐼‘(𝐵𝑠)) ⊆ 𝐵)
77 orc 867 . . . . . . . . 9 ((𝐼‘(𝐵𝑠)) ⊆ 𝐵 → ((𝐼‘(𝐵𝑠)) ⊆ 𝐵 ∨ (𝐼‘(𝐵𝑡)) ⊆ 𝐵))
78 inss 4247 . . . . . . . . 9 (((𝐼‘(𝐵𝑠)) ⊆ 𝐵 ∨ (𝐼‘(𝐵𝑡)) ⊆ 𝐵) → ((𝐼‘(𝐵𝑠)) ∩ (𝐼‘(𝐵𝑡))) ⊆ 𝐵)
7976, 77, 783syl 18 . . . . . . . 8 ((𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵) ∧ 𝐵 ∈ V) → ((𝐼‘(𝐵𝑠)) ∩ (𝐼‘(𝐵𝑡))) ⊆ 𝐵)
80 difssd 4136 . . . . . . . . . . 11 ((𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵) ∧ 𝐵 ∈ V) → (𝐵 ∖ (𝑠𝑡)) ⊆ 𝐵)
8172, 80sselpwd 5327 . . . . . . . . . 10 ((𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵) ∧ 𝐵 ∈ V) → (𝐵 ∖ (𝑠𝑡)) ∈ 𝒫 𝐵)
8271, 81ffvelcdmd 7104 . . . . . . . . 9 ((𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵) ∧ 𝐵 ∈ V) → (𝐼‘(𝐵 ∖ (𝑠𝑡))) ∈ 𝒫 𝐵)
8382elpwid 4608 . . . . . . . 8 ((𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵) ∧ 𝐵 ∈ V) → (𝐼‘(𝐵 ∖ (𝑠𝑡))) ⊆ 𝐵)
8479, 83jca 511 . . . . . . 7 ((𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵) ∧ 𝐵 ∈ V) → (((𝐼‘(𝐵𝑠)) ∩ (𝐼‘(𝐵𝑡))) ⊆ 𝐵 ∧ (𝐼‘(𝐵 ∖ (𝑠𝑡))) ⊆ 𝐵))
85 sscon34b 4303 . . . . . . 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))))))
8666, 69, 84, 854syl 19 . . . . . 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))))))
87 difindi 4291 . . . . . . . 8 (𝐵 ∖ ((𝐼‘(𝐵𝑠)) ∩ (𝐼‘(𝐵𝑡)))) = ((𝐵 ∖ (𝐼‘(𝐵𝑠))) ∪ (𝐵 ∖ (𝐼‘(𝐵𝑡))))
8887sseq2i 4012 . . . . . . 7 ((𝐵 ∖ (𝐼‘(𝐵 ∖ (𝑠𝑡)))) ⊆ (𝐵 ∖ ((𝐼‘(𝐵𝑠)) ∩ (𝐼‘(𝐵𝑡)))) ↔ (𝐵 ∖ (𝐼‘(𝐵 ∖ (𝑠𝑡)))) ⊆ ((𝐵 ∖ (𝐼‘(𝐵𝑠))) ∪ (𝐵 ∖ (𝐼‘(𝐵𝑡)))))
8988a1i 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))))))
9066, 14syl 17 . . . . . . 7 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → 𝐵 ∈ V)
9166, 68syl 17 . . . . . . 7 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → 𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵))
92 simp12 1204 . . . . . . 7 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → 𝑠 ∈ 𝒫 𝐵)
93 rp-simp2 43811 . . . . . . 7 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → 𝑡 ∈ 𝒫 𝐵)
94 simpl2 1192 . . . . . . . . . 10 (((𝜑𝐵 ∈ V ∧ 𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵)) ∧ (𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵)) → 𝐵 ∈ V)
95 simpl3 1193 . . . . . . . . . 10 (((𝜑𝐵 ∈ V ∧ 𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵)) ∧ (𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵)) → 𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵))
96 eqid 2736 . . . . . . . . . 10 (𝐷𝐼) = (𝐷𝐼)
97 simpl 482 . . . . . . . . . . . 12 ((𝐵 ∈ V ∧ (𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵)) → 𝐵 ∈ V)
98 simprl 770 . . . . . . . . . . . . . 14 ((𝐵 ∈ V ∧ (𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵)) → 𝑠 ∈ 𝒫 𝐵)
9998elpwid 4608 . . . . . . . . . . . . 13 ((𝐵 ∈ V ∧ (𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵)) → 𝑠𝐵)
100 simprr 772 . . . . . . . . . . . . . 14 ((𝐵 ∈ V ∧ (𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵)) → 𝑡 ∈ 𝒫 𝐵)
101100elpwid 4608 . . . . . . . . . . . . 13 ((𝐵 ∈ V ∧ (𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵)) → 𝑡𝐵)
10299, 101unssd 4191 . . . . . . . . . . . 12 ((𝐵 ∈ V ∧ (𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵)) → (𝑠𝑡) ⊆ 𝐵)
10397, 102sselpwd 5327 . . . . . . . . . . 11 ((𝐵 ∈ V ∧ (𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵)) → (𝑠𝑡) ∈ 𝒫 𝐵)
1041033ad2antl2 1186 . . . . . . . . . 10 (((𝜑𝐵 ∈ V ∧ 𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵)) ∧ (𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵)) → (𝑠𝑡) ∈ 𝒫 𝐵)
105 eqid 2736 . . . . . . . . . 10 ((𝐷𝐼)‘(𝑠𝑡)) = ((𝐷𝐼)‘(𝑠𝑡))
10667, 12, 94, 95, 96, 104, 105dssmapfv3d 44037 . . . . . . . . 9 (((𝜑𝐵 ∈ V ∧ 𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵)) ∧ (𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵)) → ((𝐷𝐼)‘(𝑠𝑡)) = (𝐵 ∖ (𝐼‘(𝐵 ∖ (𝑠𝑡)))))
107 simpl1 1191 . . . . . . . . . 10 (((𝜑𝐵 ∈ V ∧ 𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵)) ∧ (𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵)) → 𝜑)
10867, 12, 13ntrclsfv1 44073 . . . . . . . . . . 11 (𝜑 → (𝐷𝐼) = 𝐾)
109108fveq1d 6907 . . . . . . . . . 10 (𝜑 → ((𝐷𝐼)‘(𝑠𝑡)) = (𝐾‘(𝑠𝑡)))
110107, 109syl 17 . . . . . . . . 9 (((𝜑𝐵 ∈ V ∧ 𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵)) ∧ (𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵)) → ((𝐷𝐼)‘(𝑠𝑡)) = (𝐾‘(𝑠𝑡)))
111106, 110eqtr3d 2778 . . . . . . . 8 (((𝜑𝐵 ∈ V ∧ 𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵)) ∧ (𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵)) → (𝐵 ∖ (𝐼‘(𝐵 ∖ (𝑠𝑡)))) = (𝐾‘(𝑠𝑡)))
112 simprl 770 . . . . . . . . . . 11 (((𝜑𝐵 ∈ V ∧ 𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵)) ∧ (𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵)) → 𝑠 ∈ 𝒫 𝐵)
113 eqid 2736 . . . . . . . . . . 11 ((𝐷𝐼)‘𝑠) = ((𝐷𝐼)‘𝑠)
11467, 12, 94, 95, 96, 112, 113dssmapfv3d 44037 . . . . . . . . . 10 (((𝜑𝐵 ∈ V ∧ 𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵)) ∧ (𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵)) → ((𝐷𝐼)‘𝑠) = (𝐵 ∖ (𝐼‘(𝐵𝑠))))
115108fveq1d 6907 . . . . . . . . . . 11 (𝜑 → ((𝐷𝐼)‘𝑠) = (𝐾𝑠))
116107, 115syl 17 . . . . . . . . . 10 (((𝜑𝐵 ∈ V ∧ 𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵)) ∧ (𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵)) → ((𝐷𝐼)‘𝑠) = (𝐾𝑠))
117114, 116eqtr3d 2778 . . . . . . . . 9 (((𝜑𝐵 ∈ V ∧ 𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵)) ∧ (𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵)) → (𝐵 ∖ (𝐼‘(𝐵𝑠))) = (𝐾𝑠))
118 simprr 772 . . . . . . . . . . 11 (((𝜑𝐵 ∈ V ∧ 𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵)) ∧ (𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵)) → 𝑡 ∈ 𝒫 𝐵)
119 eqid 2736 . . . . . . . . . . 11 ((𝐷𝐼)‘𝑡) = ((𝐷𝐼)‘𝑡)
12067, 12, 94, 95, 96, 118, 119dssmapfv3d 44037 . . . . . . . . . 10 (((𝜑𝐵 ∈ V ∧ 𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵)) ∧ (𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵)) → ((𝐷𝐼)‘𝑡) = (𝐵 ∖ (𝐼‘(𝐵𝑡))))
121108fveq1d 6907 . . . . . . . . . . 11 (𝜑 → ((𝐷𝐼)‘𝑡) = (𝐾𝑡))
122107, 121syl 17 . . . . . . . . . 10 (((𝜑𝐵 ∈ V ∧ 𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵)) ∧ (𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵)) → ((𝐷𝐼)‘𝑡) = (𝐾𝑡))
123120, 122eqtr3d 2778 . . . . . . . . 9 (((𝜑𝐵 ∈ V ∧ 𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵)) ∧ (𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵)) → (𝐵 ∖ (𝐼‘(𝐵𝑡))) = (𝐾𝑡))
124117, 123uneq12d 4168 . . . . . . . 8 (((𝜑𝐵 ∈ V ∧ 𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵)) ∧ (𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵)) → ((𝐵 ∖ (𝐼‘(𝐵𝑠))) ∪ (𝐵 ∖ (𝐼‘(𝐵𝑡)))) = ((𝐾𝑠) ∪ (𝐾𝑡)))
125111, 124sseq12d 4016 . . . . . . 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))))
12666, 90, 91, 92, 93, 125syl32anc 1379 . . . . . 6 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → ((𝐵 ∖ (𝐼‘(𝐵 ∖ (𝑠𝑡)))) ⊆ ((𝐵 ∖ (𝐼‘(𝐵𝑠))) ∪ (𝐵 ∖ (𝐼‘(𝐵𝑡)))) ↔ (𝐾‘(𝑠𝑡)) ⊆ ((𝐾𝑠) ∪ (𝐾𝑡))))
12786, 89, 1263bitrd 305 . . . . 5 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → (((𝐼‘(𝐵𝑠)) ∩ (𝐼‘(𝐵𝑡))) ⊆ (𝐼‘(𝐵 ∖ (𝑠𝑡))) ↔ (𝐾‘(𝑠𝑡)) ⊆ ((𝐾𝑠) ∪ (𝐾𝑡))))
12857, 65, 1273bitrd 305 . . . 4 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → (((𝐼𝑎) ∩ (𝐼𝑏)) ⊆ (𝐼‘(𝑎𝑏)) ↔ (𝐾‘(𝑠𝑡)) ⊆ ((𝐾𝑠) ∪ (𝐾𝑡))))
12935, 50, 128ralxfrd2 5411 . . 3 ((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) → (∀𝑏 ∈ 𝒫 𝐵((𝐼𝑎) ∩ (𝐼𝑏)) ⊆ (𝐼‘(𝑎𝑏)) ↔ ∀𝑡 ∈ 𝒫 𝐵(𝐾‘(𝑠𝑡)) ⊆ ((𝐾𝑠) ∪ (𝐾𝑡))))
13017, 31, 129ralxfrd2 5411 . 2 (𝜑 → (∀𝑎 ∈ 𝒫 𝐵𝑏 ∈ 𝒫 𝐵((𝐼𝑎) ∩ (𝐼𝑏)) ⊆ (𝐼‘(𝑎𝑏)) ↔ ∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵(𝐾‘(𝑠𝑡)) ⊆ ((𝐾𝑠) ∪ (𝐾𝑡))))
13111, 130bitrid 283 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 206  wa 395  wo 847  w3a 1086   = wceq 1539  wcel 2107  wral 3060  wrex 3069  Vcvv 3479  cdif 3947  cun 3948  cin 3949  wss 3950  𝒫 cpw 4599   class class class wbr 5142  cmpt 5224  wf 6556  cfv 6560  (class class class)co 7432  m cmap 8867
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-frege1 43808
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-1st 8015  df-2nd 8016  df-map 8869
This theorem is referenced by: (None)
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