Metamath Proof Explorer

Theorem ntrclsk13

Description: The interior of the intersection of any pair is equal to the intersection of the interiors if and only if the closure of the unions of any pair is equal to the union of closures. (Contributed by RP, 19-Jun-2021)

Ref Expression
Hypotheses ntrcls.o ⊢ 𝑂 = ( 𝑖 ∈ V ↦ ( 𝑘 ∈ ( 𝒫 𝑖 ↑m 𝒫 𝑖 ) ↦ ( 𝑗 ∈ 𝒫 𝑖 ↦ ( 𝑖 ∖ ( 𝑘 ‘ ( 𝑖 ∖ 𝑗 ) ) ) ) ) )
ntrcls.d ⊢ 𝐷 = ( 𝑂 ‘ 𝐵 )
ntrcls.r ⊢ ( 𝜑 → 𝐼 𝐷 𝐾 )
Assertion ntrclsk13 ( 𝜑 → ( ∀ 𝑠 ∈ 𝒫 𝐵 ∀ 𝑡 ∈ 𝒫 𝐵 ( 𝐼 ‘ ( 𝑠 ∩ 𝑡 ) ) = ( ( 𝐼 ‘ 𝑠 ) ∩ ( 𝐼 ‘ 𝑡 ) ) ↔ ∀ 𝑠 ∈ 𝒫 𝐵 ∀ 𝑡 ∈ 𝒫 𝐵 ( 𝐾 ‘ ( 𝑠 ∪ 𝑡 ) ) = ( ( 𝐾 ‘ 𝑠 ) ∪ ( 𝐾 ‘ 𝑡 ) ) ) )

Proof

Step Hyp Ref Expression
1 ntrcls.o ⊢ 𝑂 = ( 𝑖 ∈ V ↦ ( 𝑘 ∈ ( 𝒫 𝑖 ↑m 𝒫 𝑖 ) ↦ ( 𝑗 ∈ 𝒫 𝑖 ↦ ( 𝑖 ∖ ( 𝑘 ‘ ( 𝑖 ∖ 𝑗 ) ) ) ) ) )
2 ntrcls.d ⊢ 𝐷 = ( 𝑂 ‘ 𝐵 )
3 ntrcls.r ⊢ ( 𝜑 → 𝐼 𝐷 𝐾 )
4 ineq1 ⊢ ( 𝑠 = 𝑎 → ( 𝑠 ∩ 𝑡 ) = ( 𝑎 ∩ 𝑡 ) )
5 4 fveq2d ⊢ ( 𝑠 = 𝑎 → ( 𝐼 ‘ ( 𝑠 ∩ 𝑡 ) ) = ( 𝐼 ‘ ( 𝑎 ∩ 𝑡 ) ) )
6 fveq2 ⊢ ( 𝑠 = 𝑎 → ( 𝐼 ‘ 𝑠 ) = ( 𝐼 ‘ 𝑎 ) )
7 6 ineq1d ⊢ ( 𝑠 = 𝑎 → ( ( 𝐼 ‘ 𝑠 ) ∩ ( 𝐼 ‘ 𝑡 ) ) = ( ( 𝐼 ‘ 𝑎 ) ∩ ( 𝐼 ‘ 𝑡 ) ) )
8 5 7 eqeq12d ⊢ ( 𝑠 = 𝑎 → ( ( 𝐼 ‘ ( 𝑠 ∩ 𝑡 ) ) = ( ( 𝐼 ‘ 𝑠 ) ∩ ( 𝐼 ‘ 𝑡 ) ) ↔ ( 𝐼 ‘ ( 𝑎 ∩ 𝑡 ) ) = ( ( 𝐼 ‘ 𝑎 ) ∩ ( 𝐼 ‘ 𝑡 ) ) ) )
9 ineq2 ⊢ ( 𝑡 = 𝑏 → ( 𝑎 ∩ 𝑡 ) = ( 𝑎 ∩ 𝑏 ) )
10 9 fveq2d ⊢ ( 𝑡 = 𝑏 → ( 𝐼 ‘ ( 𝑎 ∩ 𝑡 ) ) = ( 𝐼 ‘ ( 𝑎 ∩ 𝑏 ) ) )
11 fveq2 ⊢ ( 𝑡 = 𝑏 → ( 𝐼 ‘ 𝑡 ) = ( 𝐼 ‘ 𝑏 ) )
12 11 ineq2d ⊢ ( 𝑡 = 𝑏 → ( ( 𝐼 ‘ 𝑎 ) ∩ ( 𝐼 ‘ 𝑡 ) ) = ( ( 𝐼 ‘ 𝑎 ) ∩ ( 𝐼 ‘ 𝑏 ) ) )
13 10 12 eqeq12d ⊢ ( 𝑡 = 𝑏 → ( ( 𝐼 ‘ ( 𝑎 ∩ 𝑡 ) ) = ( ( 𝐼 ‘ 𝑎 ) ∩ ( 𝐼 ‘ 𝑡 ) ) ↔ ( 𝐼 ‘ ( 𝑎 ∩ 𝑏 ) ) = ( ( 𝐼 ‘ 𝑎 ) ∩ ( 𝐼 ‘ 𝑏 ) ) ) )
14 8 13 cbvral2vw ⊢ ( ∀ 𝑠 ∈ 𝒫 𝐵 ∀ 𝑡 ∈ 𝒫 𝐵 ( 𝐼 ‘ ( 𝑠 ∩ 𝑡 ) ) = ( ( 𝐼 ‘ 𝑠 ) ∩ ( 𝐼 ‘ 𝑡 ) ) ↔ ∀ 𝑎 ∈ 𝒫 𝐵 ∀ 𝑏 ∈ 𝒫 𝐵 ( 𝐼 ‘ ( 𝑎 ∩ 𝑏 ) ) = ( ( 𝐼 ‘ 𝑎 ) ∩ ( 𝐼 ‘ 𝑏 ) ) )
15 2 3 ntrclsbex ⊢ ( 𝜑 → 𝐵 ∈ V )
16 difssd ⊢ ( 𝜑 → ( 𝐵 ∖ 𝑠 ) ⊆ 𝐵 )
17 15 16 sselpwd ⊢ ( 𝜑 → ( 𝐵 ∖ 𝑠 ) ∈ 𝒫 𝐵 )
18 17 adantr ⊢ ( ( 𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ) → ( 𝐵 ∖ 𝑠 ) ∈ 𝒫 𝐵 )
19 elpwi ⊢ ( 𝑎 ∈ 𝒫 𝐵 → 𝑎 ⊆ 𝐵 )
20 15 adantr ⊢ ( ( 𝜑 ∧ 𝑎 ⊆ 𝐵 ) → 𝐵 ∈ V )
21 difssd ⊢ ( ( 𝜑 ∧ 𝑎 ⊆ 𝐵 ) → ( 𝐵 ∖ 𝑎 ) ⊆ 𝐵 )
22 20 21 sselpwd ⊢ ( ( 𝜑 ∧ 𝑎 ⊆ 𝐵 ) → ( 𝐵 ∖ 𝑎 ) ∈ 𝒫 𝐵 )
23 difeq2 ⊢ ( 𝑠 = ( 𝐵 ∖ 𝑎 ) → ( 𝐵 ∖ 𝑠 ) = ( 𝐵 ∖ ( 𝐵 ∖ 𝑎 ) ) )
24 23 eqeq2d ⊢ ( 𝑠 = ( 𝐵 ∖ 𝑎 ) → ( 𝑎 = ( 𝐵 ∖ 𝑠 ) ↔ 𝑎 = ( 𝐵 ∖ ( 𝐵 ∖ 𝑎 ) ) ) )
25 eqcom ⊢ ( 𝑎 = ( 𝐵 ∖ ( 𝐵 ∖ 𝑎 ) ) ↔ ( 𝐵 ∖ ( 𝐵 ∖ 𝑎 ) ) = 𝑎 )
26 24 25 bitrdi ⊢ ( 𝑠 = ( 𝐵 ∖ 𝑎 ) → ( 𝑎 = ( 𝐵 ∖ 𝑠 ) ↔ ( 𝐵 ∖ ( 𝐵 ∖ 𝑎 ) ) = 𝑎 ) )
27 26 adantl ⊢ ( ( ( 𝜑 ∧ 𝑎 ⊆ 𝐵 ) ∧ 𝑠 = ( 𝐵 ∖ 𝑎 ) ) → ( 𝑎 = ( 𝐵 ∖ 𝑠 ) ↔ ( 𝐵 ∖ ( 𝐵 ∖ 𝑎 ) ) = 𝑎 ) )
28 dfss4 ⊢ ( 𝑎 ⊆ 𝐵 ↔ ( 𝐵 ∖ ( 𝐵 ∖ 𝑎 ) ) = 𝑎 )
29 28 bilani ⊢ ( ( 𝜑 ∧ 𝑎 ⊆ 𝐵 ) → ( 𝐵 ∖ ( 𝐵 ∖ 𝑎 ) ) = 𝑎 )
30 22 27 29 rspcedvd ⊢ ( ( 𝜑 ∧ 𝑎 ⊆ 𝐵 ) → ∃ 𝑠 ∈ 𝒫 𝐵 𝑎 = ( 𝐵 ∖ 𝑠 ) )
31 19 30 sylan2 ⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝒫 𝐵 ) → ∃ 𝑠 ∈ 𝒫 𝐵 𝑎 = ( 𝐵 ∖ 𝑠 ) )
32 ineq1 ⊢ ( 𝑎 = ( 𝐵 ∖ 𝑠 ) → ( 𝑎 ∩ 𝑏 ) = ( ( 𝐵 ∖ 𝑠 ) ∩ 𝑏 ) )
33 32 fveq2d ⊢ ( 𝑎 = ( 𝐵 ∖ 𝑠 ) → ( 𝐼 ‘ ( 𝑎 ∩ 𝑏 ) ) = ( 𝐼 ‘ ( ( 𝐵 ∖ 𝑠 ) ∩ 𝑏 ) ) )
34 fveq2 ⊢ ( 𝑎 = ( 𝐵 ∖ 𝑠 ) → ( 𝐼 ‘ 𝑎 ) = ( 𝐼 ‘ ( 𝐵 ∖ 𝑠 ) ) )
35 34 ineq1d ⊢ ( 𝑎 = ( 𝐵 ∖ 𝑠 ) → ( ( 𝐼 ‘ 𝑎 ) ∩ ( 𝐼 ‘ 𝑏 ) ) = ( ( 𝐼 ‘ ( 𝐵 ∖ 𝑠 ) ) ∩ ( 𝐼 ‘ 𝑏 ) ) )
36 33 35 eqeq12d ⊢ ( 𝑎 = ( 𝐵 ∖ 𝑠 ) → ( ( 𝐼 ‘ ( 𝑎 ∩ 𝑏 ) ) = ( ( 𝐼 ‘ 𝑎 ) ∩ ( 𝐼 ‘ 𝑏 ) ) ↔ ( 𝐼 ‘ ( ( 𝐵 ∖ 𝑠 ) ∩ 𝑏 ) ) = ( ( 𝐼 ‘ ( 𝐵 ∖ 𝑠 ) ) ∩ ( 𝐼 ‘ 𝑏 ) ) ) )
37 36 ralbidv ⊢ ( 𝑎 = ( 𝐵 ∖ 𝑠 ) → ( ∀ 𝑏 ∈ 𝒫 𝐵 ( 𝐼 ‘ ( 𝑎 ∩ 𝑏 ) ) = ( ( 𝐼 ‘ 𝑎 ) ∩ ( 𝐼 ‘ 𝑏 ) ) ↔ ∀ 𝑏 ∈ 𝒫 𝐵 ( 𝐼 ‘ ( ( 𝐵 ∖ 𝑠 ) ∩ 𝑏 ) ) = ( ( 𝐼 ‘ ( 𝐵 ∖ 𝑠 ) ) ∩ ( 𝐼 ‘ 𝑏 ) ) ) )
38 37 3ad2ant3 ⊢ ( ( 𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = ( 𝐵 ∖ 𝑠 ) ) → ( ∀ 𝑏 ∈ 𝒫 𝐵 ( 𝐼 ‘ ( 𝑎 ∩ 𝑏 ) ) = ( ( 𝐼 ‘ 𝑎 ) ∩ ( 𝐼 ‘ 𝑏 ) ) ↔ ∀ 𝑏 ∈ 𝒫 𝐵 ( 𝐼 ‘ ( ( 𝐵 ∖ 𝑠 ) ∩ 𝑏 ) ) = ( ( 𝐼 ‘ ( 𝐵 ∖ 𝑠 ) ) ∩ ( 𝐼 ‘ 𝑏 ) ) ) )
39 difssd ⊢ ( 𝜑 → ( 𝐵 ∖ 𝑡 ) ⊆ 𝐵 )
40 15 39 sselpwd ⊢ ( 𝜑 → ( 𝐵 ∖ 𝑡 ) ∈ 𝒫 𝐵 )
41 40 ad2antrr ⊢ ( ( ( 𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ) ∧ 𝑡 ∈ 𝒫 𝐵 ) → ( 𝐵 ∖ 𝑡 ) ∈ 𝒫 𝐵 )
42 simpll ⊢ ( ( ( 𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ) ∧ 𝑏 ∈ 𝒫 𝐵 ) → 𝜑 )
43 elpwi ⊢ ( 𝑏 ∈ 𝒫 𝐵 → 𝑏 ⊆ 𝐵 )
44 43 adantl ⊢ ( ( ( 𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ) ∧ 𝑏 ∈ 𝒫 𝐵 ) → 𝑏 ⊆ 𝐵 )
45 difssd ⊢ ( 𝜑 → ( 𝐵 ∖ 𝑏 ) ⊆ 𝐵 )
46 15 45 sselpwd ⊢ ( 𝜑 → ( 𝐵 ∖ 𝑏 ) ∈ 𝒫 𝐵 )
47 46 adantr ⊢ ( ( 𝜑 ∧ 𝑏 ⊆ 𝐵 ) → ( 𝐵 ∖ 𝑏 ) ∈ 𝒫 𝐵 )
48 difeq2 ⊢ ( 𝑡 = ( 𝐵 ∖ 𝑏 ) → ( 𝐵 ∖ 𝑡 ) = ( 𝐵 ∖ ( 𝐵 ∖ 𝑏 ) ) )
49 48 eqeq2d ⊢ ( 𝑡 = ( 𝐵 ∖ 𝑏 ) → ( 𝑏 = ( 𝐵 ∖ 𝑡 ) ↔ 𝑏 = ( 𝐵 ∖ ( 𝐵 ∖ 𝑏 ) ) ) )
50 eqcom ⊢ ( 𝑏 = ( 𝐵 ∖ ( 𝐵 ∖ 𝑏 ) ) ↔ ( 𝐵 ∖ ( 𝐵 ∖ 𝑏 ) ) = 𝑏 )
51 49 50 bitrdi ⊢ ( 𝑡 = ( 𝐵 ∖ 𝑏 ) → ( 𝑏 = ( 𝐵 ∖ 𝑡 ) ↔ ( 𝐵 ∖ ( 𝐵 ∖ 𝑏 ) ) = 𝑏 ) )
52 51 adantl ⊢ ( ( ( 𝜑 ∧ 𝑏 ⊆ 𝐵 ) ∧ 𝑡 = ( 𝐵 ∖ 𝑏 ) ) → ( 𝑏 = ( 𝐵 ∖ 𝑡 ) ↔ ( 𝐵 ∖ ( 𝐵 ∖ 𝑏 ) ) = 𝑏 ) )
53 dfss4 ⊢ ( 𝑏 ⊆ 𝐵 ↔ ( 𝐵 ∖ ( 𝐵 ∖ 𝑏 ) ) = 𝑏 )
54 53 bilani ⊢ ( ( 𝜑 ∧ 𝑏 ⊆ 𝐵 ) → ( 𝐵 ∖ ( 𝐵 ∖ 𝑏 ) ) = 𝑏 )
55 47 52 54 rspcedvd ⊢ ( ( 𝜑 ∧ 𝑏 ⊆ 𝐵 ) → ∃ 𝑡 ∈ 𝒫 𝐵 𝑏 = ( 𝐵 ∖ 𝑡 ) )
56 42 44 55 syl2anc ⊢ ( ( ( 𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ) ∧ 𝑏 ∈ 𝒫 𝐵 ) → ∃ 𝑡 ∈ 𝒫 𝐵 𝑏 = ( 𝐵 ∖ 𝑡 ) )
57 ineq2 ⊢ ( 𝑏 = ( 𝐵 ∖ 𝑡 ) → ( ( 𝐵 ∖ 𝑠 ) ∩ 𝑏 ) = ( ( 𝐵 ∖ 𝑠 ) ∩ ( 𝐵 ∖ 𝑡 ) ) )
58 difundi ⊢ ( 𝐵 ∖ ( 𝑠 ∪ 𝑡 ) ) = ( ( 𝐵 ∖ 𝑠 ) ∩ ( 𝐵 ∖ 𝑡 ) )
59 57 58 eqtr4di ⊢ ( 𝑏 = ( 𝐵 ∖ 𝑡 ) → ( ( 𝐵 ∖ 𝑠 ) ∩ 𝑏 ) = ( 𝐵 ∖ ( 𝑠 ∪ 𝑡 ) ) )
60 59 fveq2d ⊢ ( 𝑏 = ( 𝐵 ∖ 𝑡 ) → ( 𝐼 ‘ ( ( 𝐵 ∖ 𝑠 ) ∩ 𝑏 ) ) = ( 𝐼 ‘ ( 𝐵 ∖ ( 𝑠 ∪ 𝑡 ) ) ) )
61 fveq2 ⊢ ( 𝑏 = ( 𝐵 ∖ 𝑡 ) → ( 𝐼 ‘ 𝑏 ) = ( 𝐼 ‘ ( 𝐵 ∖ 𝑡 ) ) )
62 61 ineq2d ⊢ ( 𝑏 = ( 𝐵 ∖ 𝑡 ) → ( ( 𝐼 ‘ ( 𝐵 ∖ 𝑠 ) ) ∩ ( 𝐼 ‘ 𝑏 ) ) = ( ( 𝐼 ‘ ( 𝐵 ∖ 𝑠 ) ) ∩ ( 𝐼 ‘ ( 𝐵 ∖ 𝑡 ) ) ) )
63 60 62 eqeq12d ⊢ ( 𝑏 = ( 𝐵 ∖ 𝑡 ) → ( ( 𝐼 ‘ ( ( 𝐵 ∖ 𝑠 ) ∩ 𝑏 ) ) = ( ( 𝐼 ‘ ( 𝐵 ∖ 𝑠 ) ) ∩ ( 𝐼 ‘ 𝑏 ) ) ↔ ( 𝐼 ‘ ( 𝐵 ∖ ( 𝑠 ∪ 𝑡 ) ) ) = ( ( 𝐼 ‘ ( 𝐵 ∖ 𝑠 ) ) ∩ ( 𝐼 ‘ ( 𝐵 ∖ 𝑡 ) ) ) ) )
64 63 3ad2ant3 ⊢ ( ( ( 𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = ( 𝐵 ∖ 𝑡 ) ) → ( ( 𝐼 ‘ ( ( 𝐵 ∖ 𝑠 ) ∩ 𝑏 ) ) = ( ( 𝐼 ‘ ( 𝐵 ∖ 𝑠 ) ) ∩ ( 𝐼 ‘ 𝑏 ) ) ↔ ( 𝐼 ‘ ( 𝐵 ∖ ( 𝑠 ∪ 𝑡 ) ) ) = ( ( 𝐼 ‘ ( 𝐵 ∖ 𝑠 ) ) ∩ ( 𝐼 ‘ ( 𝐵 ∖ 𝑡 ) ) ) ) )
65 simp1l ⊢ ( ( ( 𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = ( 𝐵 ∖ 𝑡 ) ) → 𝜑 )
66 65 15 jccir ⊢ ( ( ( 𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = ( 𝐵 ∖ 𝑡 ) ) → ( 𝜑 ∧ 𝐵 ∈ V ) )
67 simp1r ⊢ ( ( ( 𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = ( 𝐵 ∖ 𝑡 ) ) → 𝑠 ∈ 𝒫 𝐵 )
68 simp2 ⊢ ( ( ( 𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = ( 𝐵 ∖ 𝑡 ) ) → 𝑡 ∈ 𝒫 𝐵 )
69 1 2 3 ntrclsiex ⊢ ( 𝜑 → 𝐼 ∈ ( 𝒫 𝐵 ↑m 𝒫 𝐵 ) )
70 elmapi ⊢ ( 𝐼 ∈ ( 𝒫 𝐵 ↑m 𝒫 𝐵 ) → 𝐼 : 𝒫 𝐵 ⟶ 𝒫 𝐵 )
71 69 70 syl ⊢ ( 𝜑 → 𝐼 : 𝒫 𝐵 ⟶ 𝒫 𝐵 )
72 71 anim1i ⊢ ( ( 𝜑 ∧ 𝐵 ∈ V ) → ( 𝐼 : 𝒫 𝐵 ⟶ 𝒫 𝐵 ∧ 𝐵 ∈ V ) )
73 72 adantr ⊢ ( ( ( 𝜑 ∧ 𝐵 ∈ V ) ∧ ( 𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 ∈ 𝒫 𝐵 ) ) → ( 𝐼 : 𝒫 𝐵 ⟶ 𝒫 𝐵 ∧ 𝐵 ∈ V ) )
74 simpl ⊢ ( ( 𝐼 : 𝒫 𝐵 ⟶ 𝒫 𝐵 ∧ 𝐵 ∈ V ) → 𝐼 : 𝒫 𝐵 ⟶ 𝒫 𝐵 )
75 simpr ⊢ ( ( 𝐼 : 𝒫 𝐵 ⟶ 𝒫 𝐵 ∧ 𝐵 ∈ V ) → 𝐵 ∈ V )
76 difssd ⊢ ( ( 𝐼 : 𝒫 𝐵 ⟶ 𝒫 𝐵 ∧ 𝐵 ∈ V ) → ( 𝐵 ∖ ( 𝑠 ∪ 𝑡 ) ) ⊆ 𝐵 )
77 75 76 sselpwd ⊢ ( ( 𝐼 : 𝒫 𝐵 ⟶ 𝒫 𝐵 ∧ 𝐵 ∈ V ) → ( 𝐵 ∖ ( 𝑠 ∪ 𝑡 ) ) ∈ 𝒫 𝐵 )
78 74 77 ffvelcdmd ⊢ ( ( 𝐼 : 𝒫 𝐵 ⟶ 𝒫 𝐵 ∧ 𝐵 ∈ V ) → ( 𝐼 ‘ ( 𝐵 ∖ ( 𝑠 ∪ 𝑡 ) ) ) ∈ 𝒫 𝐵 )
79 78 elpwid ⊢ ( ( 𝐼 : 𝒫 𝐵 ⟶ 𝒫 𝐵 ∧ 𝐵 ∈ V ) → ( 𝐼 ‘ ( 𝐵 ∖ ( 𝑠 ∪ 𝑡 ) ) ) ⊆ 𝐵 )
80 difssd ⊢ ( ( 𝐼 : 𝒫 𝐵 ⟶ 𝒫 𝐵 ∧ 𝐵 ∈ V ) → ( 𝐵 ∖ 𝑠 ) ⊆ 𝐵 )
81 75 80 sselpwd ⊢ ( ( 𝐼 : 𝒫 𝐵 ⟶ 𝒫 𝐵 ∧ 𝐵 ∈ V ) → ( 𝐵 ∖ 𝑠 ) ∈ 𝒫 𝐵 )
82 74 81 ffvelcdmd ⊢ ( ( 𝐼 : 𝒫 𝐵 ⟶ 𝒫 𝐵 ∧ 𝐵 ∈ V ) → ( 𝐼 ‘ ( 𝐵 ∖ 𝑠 ) ) ∈ 𝒫 𝐵 )
83 82 elpwid ⊢ ( ( 𝐼 : 𝒫 𝐵 ⟶ 𝒫 𝐵 ∧ 𝐵 ∈ V ) → ( 𝐼 ‘ ( 𝐵 ∖ 𝑠 ) ) ⊆ 𝐵 )
84 ssinss1 ⊢ ( ( 𝐼 ‘ ( 𝐵 ∖ 𝑠 ) ) ⊆ 𝐵 → ( ( 𝐼 ‘ ( 𝐵 ∖ 𝑠 ) ) ∩ ( 𝐼 ‘ ( 𝐵 ∖ 𝑡 ) ) ) ⊆ 𝐵 )
85 83 84 syl ⊢ ( ( 𝐼 : 𝒫 𝐵 ⟶ 𝒫 𝐵 ∧ 𝐵 ∈ V ) → ( ( 𝐼 ‘ ( 𝐵 ∖ 𝑠 ) ) ∩ ( 𝐼 ‘ ( 𝐵 ∖ 𝑡 ) ) ) ⊆ 𝐵 )
86 79 85 jca ⊢ ( ( 𝐼 : 𝒫 𝐵 ⟶ 𝒫 𝐵 ∧ 𝐵 ∈ V ) → ( ( 𝐼 ‘ ( 𝐵 ∖ ( 𝑠 ∪ 𝑡 ) ) ) ⊆ 𝐵 ∧ ( ( 𝐼 ‘ ( 𝐵 ∖ 𝑠 ) ) ∩ ( 𝐼 ‘ ( 𝐵 ∖ 𝑡 ) ) ) ⊆ 𝐵 ) )
87 rcompleq ⊢ ( ( ( 𝐼 ‘ ( 𝐵 ∖ ( 𝑠 ∪ 𝑡 ) ) ) ⊆ 𝐵 ∧ ( ( 𝐼 ‘ ( 𝐵 ∖ 𝑠 ) ) ∩ ( 𝐼 ‘ ( 𝐵 ∖ 𝑡 ) ) ) ⊆ 𝐵 ) → ( ( 𝐼 ‘ ( 𝐵 ∖ ( 𝑠 ∪ 𝑡 ) ) ) = ( ( 𝐼 ‘ ( 𝐵 ∖ 𝑠 ) ) ∩ ( 𝐼 ‘ ( 𝐵 ∖ 𝑡 ) ) ) ↔ ( 𝐵 ∖ ( 𝐼 ‘ ( 𝐵 ∖ ( 𝑠 ∪ 𝑡 ) ) ) ) = ( 𝐵 ∖ ( ( 𝐼 ‘ ( 𝐵 ∖ 𝑠 ) ) ∩ ( 𝐼 ‘ ( 𝐵 ∖ 𝑡 ) ) ) ) ) )
88 73 86 87 3syl ⊢ ( ( ( 𝜑 ∧ 𝐵 ∈ V ) ∧ ( 𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 ∈ 𝒫 𝐵 ) ) → ( ( 𝐼 ‘ ( 𝐵 ∖ ( 𝑠 ∪ 𝑡 ) ) ) = ( ( 𝐼 ‘ ( 𝐵 ∖ 𝑠 ) ) ∩ ( 𝐼 ‘ ( 𝐵 ∖ 𝑡 ) ) ) ↔ ( 𝐵 ∖ ( 𝐼 ‘ ( 𝐵 ∖ ( 𝑠 ∪ 𝑡 ) ) ) ) = ( 𝐵 ∖ ( ( 𝐼 ‘ ( 𝐵 ∖ 𝑠 ) ) ∩ ( 𝐼 ‘ ( 𝐵 ∖ 𝑡 ) ) ) ) ) )
89 simplr ⊢ ( ( ( 𝜑 ∧ 𝐵 ∈ V ) ∧ ( 𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 ∈ 𝒫 𝐵 ) ) → 𝐵 ∈ V )
90 69 ad2antrr ⊢ ( ( ( 𝜑 ∧ 𝐵 ∈ V ) ∧ ( 𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 ∈ 𝒫 𝐵 ) ) → 𝐼 ∈ ( 𝒫 𝐵 ↑m 𝒫 𝐵 ) )
91 eqid ⊢ ( 𝐷 ‘ 𝐼 ) = ( 𝐷 ‘ 𝐼 )
92 simprl ⊢ ( ( ( 𝜑 ∧ 𝐵 ∈ V ) ∧ ( 𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 ∈ 𝒫 𝐵 ) ) → 𝑠 ∈ 𝒫 𝐵 )
93 92 elpwid ⊢ ( ( ( 𝜑 ∧ 𝐵 ∈ V ) ∧ ( 𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 ∈ 𝒫 𝐵 ) ) → 𝑠 ⊆ 𝐵 )
94 simprr ⊢ ( ( ( 𝜑 ∧ 𝐵 ∈ V ) ∧ ( 𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 ∈ 𝒫 𝐵 ) ) → 𝑡 ∈ 𝒫 𝐵 )
95 94 elpwid ⊢ ( ( ( 𝜑 ∧ 𝐵 ∈ V ) ∧ ( 𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 ∈ 𝒫 𝐵 ) ) → 𝑡 ⊆ 𝐵 )
96 93 95 unssd ⊢ ( ( ( 𝜑 ∧ 𝐵 ∈ V ) ∧ ( 𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 ∈ 𝒫 𝐵 ) ) → ( 𝑠 ∪ 𝑡 ) ⊆ 𝐵 )
97 89 96 sselpwd ⊢ ( ( ( 𝜑 ∧ 𝐵 ∈ V ) ∧ ( 𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 ∈ 𝒫 𝐵 ) ) → ( 𝑠 ∪ 𝑡 ) ∈ 𝒫 𝐵 )
98 eqid ⊢ ( ( 𝐷 ‘ 𝐼 ) ‘ ( 𝑠 ∪ 𝑡 ) ) = ( ( 𝐷 ‘ 𝐼 ) ‘ ( 𝑠 ∪ 𝑡 ) )
99 1 2 89 90 91 97 98 dssmapfv3d ⊢ ( ( ( 𝜑 ∧ 𝐵 ∈ V ) ∧ ( 𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 ∈ 𝒫 𝐵 ) ) → ( ( 𝐷 ‘ 𝐼 ) ‘ ( 𝑠 ∪ 𝑡 ) ) = ( 𝐵 ∖ ( 𝐼 ‘ ( 𝐵 ∖ ( 𝑠 ∪ 𝑡 ) ) ) ) )
100 simpl ⊢ ( ( 𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 ∈ 𝒫 𝐵 ) → 𝑠 ∈ 𝒫 𝐵 )
101 simplr ⊢ ( ( ( 𝜑 ∧ 𝐵 ∈ V ) ∧ 𝑠 ∈ 𝒫 𝐵 ) → 𝐵 ∈ V )
102 69 ad2antrr ⊢ ( ( ( 𝜑 ∧ 𝐵 ∈ V ) ∧ 𝑠 ∈ 𝒫 𝐵 ) → 𝐼 ∈ ( 𝒫 𝐵 ↑m 𝒫 𝐵 ) )
103 simpr ⊢ ( ( ( 𝜑 ∧ 𝐵 ∈ V ) ∧ 𝑠 ∈ 𝒫 𝐵 ) → 𝑠 ∈ 𝒫 𝐵 )
104 eqid ⊢ ( ( 𝐷 ‘ 𝐼 ) ‘ 𝑠 ) = ( ( 𝐷 ‘ 𝐼 ) ‘ 𝑠 )
105 1 2 101 102 91 103 104 dssmapfv3d ⊢ ( ( ( 𝜑 ∧ 𝐵 ∈ V ) ∧ 𝑠 ∈ 𝒫 𝐵 ) → ( ( 𝐷 ‘ 𝐼 ) ‘ 𝑠 ) = ( 𝐵 ∖ ( 𝐼 ‘ ( 𝐵 ∖ 𝑠 ) ) ) )
106 100 105 sylan2 ⊢ ( ( ( 𝜑 ∧ 𝐵 ∈ V ) ∧ ( 𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 ∈ 𝒫 𝐵 ) ) → ( ( 𝐷 ‘ 𝐼 ) ‘ 𝑠 ) = ( 𝐵 ∖ ( 𝐼 ‘ ( 𝐵 ∖ 𝑠 ) ) ) )
107 simpr ⊢ ( ( 𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 ∈ 𝒫 𝐵 ) → 𝑡 ∈ 𝒫 𝐵 )
108 simplr ⊢ ( ( ( 𝜑 ∧ 𝐵 ∈ V ) ∧ 𝑡 ∈ 𝒫 𝐵 ) → 𝐵 ∈ V )
109 69 ad2antrr ⊢ ( ( ( 𝜑 ∧ 𝐵 ∈ V ) ∧ 𝑡 ∈ 𝒫 𝐵 ) → 𝐼 ∈ ( 𝒫 𝐵 ↑m 𝒫 𝐵 ) )
110 simpr ⊢ ( ( ( 𝜑 ∧ 𝐵 ∈ V ) ∧ 𝑡 ∈ 𝒫 𝐵 ) → 𝑡 ∈ 𝒫 𝐵 )
111 eqid ⊢ ( ( 𝐷 ‘ 𝐼 ) ‘ 𝑡 ) = ( ( 𝐷 ‘ 𝐼 ) ‘ 𝑡 )
112 1 2 108 109 91 110 111 dssmapfv3d ⊢ ( ( ( 𝜑 ∧ 𝐵 ∈ V ) ∧ 𝑡 ∈ 𝒫 𝐵 ) → ( ( 𝐷 ‘ 𝐼 ) ‘ 𝑡 ) = ( 𝐵 ∖ ( 𝐼 ‘ ( 𝐵 ∖ 𝑡 ) ) ) )
113 107 112 sylan2 ⊢ ( ( ( 𝜑 ∧ 𝐵 ∈ V ) ∧ ( 𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 ∈ 𝒫 𝐵 ) ) → ( ( 𝐷 ‘ 𝐼 ) ‘ 𝑡 ) = ( 𝐵 ∖ ( 𝐼 ‘ ( 𝐵 ∖ 𝑡 ) ) ) )
114 106 113 uneq12d ⊢ ( ( ( 𝜑 ∧ 𝐵 ∈ V ) ∧ ( 𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 ∈ 𝒫 𝐵 ) ) → ( ( ( 𝐷 ‘ 𝐼 ) ‘ 𝑠 ) ∪ ( ( 𝐷 ‘ 𝐼 ) ‘ 𝑡 ) ) = ( ( 𝐵 ∖ ( 𝐼 ‘ ( 𝐵 ∖ 𝑠 ) ) ) ∪ ( 𝐵 ∖ ( 𝐼 ‘ ( 𝐵 ∖ 𝑡 ) ) ) ) )
115 difindi ⊢ ( 𝐵 ∖ ( ( 𝐼 ‘ ( 𝐵 ∖ 𝑠 ) ) ∩ ( 𝐼 ‘ ( 𝐵 ∖ 𝑡 ) ) ) ) = ( ( 𝐵 ∖ ( 𝐼 ‘ ( 𝐵 ∖ 𝑠 ) ) ) ∪ ( 𝐵 ∖ ( 𝐼 ‘ ( 𝐵 ∖ 𝑡 ) ) ) )
116 114 115 eqtr4di ⊢ ( ( ( 𝜑 ∧ 𝐵 ∈ V ) ∧ ( 𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 ∈ 𝒫 𝐵 ) ) → ( ( ( 𝐷 ‘ 𝐼 ) ‘ 𝑠 ) ∪ ( ( 𝐷 ‘ 𝐼 ) ‘ 𝑡 ) ) = ( 𝐵 ∖ ( ( 𝐼 ‘ ( 𝐵 ∖ 𝑠 ) ) ∩ ( 𝐼 ‘ ( 𝐵 ∖ 𝑡 ) ) ) ) )
117 99 116 eqeq12d ⊢ ( ( ( 𝜑 ∧ 𝐵 ∈ V ) ∧ ( 𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 ∈ 𝒫 𝐵 ) ) → ( ( ( 𝐷 ‘ 𝐼 ) ‘ ( 𝑠 ∪ 𝑡 ) ) = ( ( ( 𝐷 ‘ 𝐼 ) ‘ 𝑠 ) ∪ ( ( 𝐷 ‘ 𝐼 ) ‘ 𝑡 ) ) ↔ ( 𝐵 ∖ ( 𝐼 ‘ ( 𝐵 ∖ ( 𝑠 ∪ 𝑡 ) ) ) ) = ( 𝐵 ∖ ( ( 𝐼 ‘ ( 𝐵 ∖ 𝑠 ) ) ∩ ( 𝐼 ‘ ( 𝐵 ∖ 𝑡 ) ) ) ) ) )
118 simpll ⊢ ( ( ( 𝜑 ∧ 𝐵 ∈ V ) ∧ ( 𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 ∈ 𝒫 𝐵 ) ) → 𝜑 )
119 1 2 3 ntrclsfv1 ⊢ ( 𝜑 → ( 𝐷 ‘ 𝐼 ) = 𝐾 )
120 fveq1 ⊢ ( ( 𝐷 ‘ 𝐼 ) = 𝐾 → ( ( 𝐷 ‘ 𝐼 ) ‘ ( 𝑠 ∪ 𝑡 ) ) = ( 𝐾 ‘ ( 𝑠 ∪ 𝑡 ) ) )
121 fveq1 ⊢ ( ( 𝐷 ‘ 𝐼 ) = 𝐾 → ( ( 𝐷 ‘ 𝐼 ) ‘ 𝑠 ) = ( 𝐾 ‘ 𝑠 ) )
122 fveq1 ⊢ ( ( 𝐷 ‘ 𝐼 ) = 𝐾 → ( ( 𝐷 ‘ 𝐼 ) ‘ 𝑡 ) = ( 𝐾 ‘ 𝑡 ) )
123 121 122 uneq12d ⊢ ( ( 𝐷 ‘ 𝐼 ) = 𝐾 → ( ( ( 𝐷 ‘ 𝐼 ) ‘ 𝑠 ) ∪ ( ( 𝐷 ‘ 𝐼 ) ‘ 𝑡 ) ) = ( ( 𝐾 ‘ 𝑠 ) ∪ ( 𝐾 ‘ 𝑡 ) ) )
124 120 123 eqeq12d ⊢ ( ( 𝐷 ‘ 𝐼 ) = 𝐾 → ( ( ( 𝐷 ‘ 𝐼 ) ‘ ( 𝑠 ∪ 𝑡 ) ) = ( ( ( 𝐷 ‘ 𝐼 ) ‘ 𝑠 ) ∪ ( ( 𝐷 ‘ 𝐼 ) ‘ 𝑡 ) ) ↔ ( 𝐾 ‘ ( 𝑠 ∪ 𝑡 ) ) = ( ( 𝐾 ‘ 𝑠 ) ∪ ( 𝐾 ‘ 𝑡 ) ) ) )
125 118 119 124 3syl ⊢ ( ( ( 𝜑 ∧ 𝐵 ∈ V ) ∧ ( 𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 ∈ 𝒫 𝐵 ) ) → ( ( ( 𝐷 ‘ 𝐼 ) ‘ ( 𝑠 ∪ 𝑡 ) ) = ( ( ( 𝐷 ‘ 𝐼 ) ‘ 𝑠 ) ∪ ( ( 𝐷 ‘ 𝐼 ) ‘ 𝑡 ) ) ↔ ( 𝐾 ‘ ( 𝑠 ∪ 𝑡 ) ) = ( ( 𝐾 ‘ 𝑠 ) ∪ ( 𝐾 ‘ 𝑡 ) ) ) )
126 88 117 125 3bitr2d ⊢ ( ( ( 𝜑 ∧ 𝐵 ∈ V ) ∧ ( 𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 ∈ 𝒫 𝐵 ) ) → ( ( 𝐼 ‘ ( 𝐵 ∖ ( 𝑠 ∪ 𝑡 ) ) ) = ( ( 𝐼 ‘ ( 𝐵 ∖ 𝑠 ) ) ∩ ( 𝐼 ‘ ( 𝐵 ∖ 𝑡 ) ) ) ↔ ( 𝐾 ‘ ( 𝑠 ∪ 𝑡 ) ) = ( ( 𝐾 ‘ 𝑠 ) ∪ ( 𝐾 ‘ 𝑡 ) ) ) )
127 66 67 68 126 syl12anc ⊢ ( ( ( 𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = ( 𝐵 ∖ 𝑡 ) ) → ( ( 𝐼 ‘ ( 𝐵 ∖ ( 𝑠 ∪ 𝑡 ) ) ) = ( ( 𝐼 ‘ ( 𝐵 ∖ 𝑠 ) ) ∩ ( 𝐼 ‘ ( 𝐵 ∖ 𝑡 ) ) ) ↔ ( 𝐾 ‘ ( 𝑠 ∪ 𝑡 ) ) = ( ( 𝐾 ‘ 𝑠 ) ∪ ( 𝐾 ‘ 𝑡 ) ) ) )
128 64 127 bitrd ⊢ ( ( ( 𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = ( 𝐵 ∖ 𝑡 ) ) → ( ( 𝐼 ‘ ( ( 𝐵 ∖ 𝑠 ) ∩ 𝑏 ) ) = ( ( 𝐼 ‘ ( 𝐵 ∖ 𝑠 ) ) ∩ ( 𝐼 ‘ 𝑏 ) ) ↔ ( 𝐾 ‘ ( 𝑠 ∪ 𝑡 ) ) = ( ( 𝐾 ‘ 𝑠 ) ∪ ( 𝐾 ‘ 𝑡 ) ) ) )
129 41 56 128 ralxfrd2 ⊢ ( ( 𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ) → ( ∀ 𝑏 ∈ 𝒫 𝐵 ( 𝐼 ‘ ( ( 𝐵 ∖ 𝑠 ) ∩ 𝑏 ) ) = ( ( 𝐼 ‘ ( 𝐵 ∖ 𝑠 ) ) ∩ ( 𝐼 ‘ 𝑏 ) ) ↔ ∀ 𝑡 ∈ 𝒫 𝐵 ( 𝐾 ‘ ( 𝑠 ∪ 𝑡 ) ) = ( ( 𝐾 ‘ 𝑠 ) ∪ ( 𝐾 ‘ 𝑡 ) ) ) )
130 129 3adant3 ⊢ ( ( 𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = ( 𝐵 ∖ 𝑠 ) ) → ( ∀ 𝑏 ∈ 𝒫 𝐵 ( 𝐼 ‘ ( ( 𝐵 ∖ 𝑠 ) ∩ 𝑏 ) ) = ( ( 𝐼 ‘ ( 𝐵 ∖ 𝑠 ) ) ∩ ( 𝐼 ‘ 𝑏 ) ) ↔ ∀ 𝑡 ∈ 𝒫 𝐵 ( 𝐾 ‘ ( 𝑠 ∪ 𝑡 ) ) = ( ( 𝐾 ‘ 𝑠 ) ∪ ( 𝐾 ‘ 𝑡 ) ) ) )
131 38 130 bitrd ⊢ ( ( 𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = ( 𝐵 ∖ 𝑠 ) ) → ( ∀ 𝑏 ∈ 𝒫 𝐵 ( 𝐼 ‘ ( 𝑎 ∩ 𝑏 ) ) = ( ( 𝐼 ‘ 𝑎 ) ∩ ( 𝐼 ‘ 𝑏 ) ) ↔ ∀ 𝑡 ∈ 𝒫 𝐵 ( 𝐾 ‘ ( 𝑠 ∪ 𝑡 ) ) = ( ( 𝐾 ‘ 𝑠 ) ∪ ( 𝐾 ‘ 𝑡 ) ) ) )
132 18 31 131 ralxfrd2 ⊢ ( 𝜑 → ( ∀ 𝑎 ∈ 𝒫 𝐵 ∀ 𝑏 ∈ 𝒫 𝐵 ( 𝐼 ‘ ( 𝑎 ∩ 𝑏 ) ) = ( ( 𝐼 ‘ 𝑎 ) ∩ ( 𝐼 ‘ 𝑏 ) ) ↔ ∀ 𝑠 ∈ 𝒫 𝐵 ∀ 𝑡 ∈ 𝒫 𝐵 ( 𝐾 ‘ ( 𝑠 ∪ 𝑡 ) ) = ( ( 𝐾 ‘ 𝑠 ) ∪ ( 𝐾 ‘ 𝑡 ) ) ) )
133 14 132 bitrid ⊢ ( 𝜑 → ( ∀ 𝑠 ∈ 𝒫 𝐵 ∀ 𝑡 ∈ 𝒫 𝐵 ( 𝐼 ‘ ( 𝑠 ∩ 𝑡 ) ) = ( ( 𝐼 ‘ 𝑠 ) ∩ ( 𝐼 ‘ 𝑡 ) ) ↔ ∀ 𝑠 ∈ 𝒫 𝐵 ∀ 𝑡 ∈ 𝒫 𝐵 ( 𝐾 ‘ ( 𝑠 ∪ 𝑡 ) ) = ( ( 𝐾 ‘ 𝑠 ) ∪ ( 𝐾 ‘ 𝑡 ) ) ) )