Metamath Proof Explorer

Theorem ntrclsiso

Description: If (pseudo-)interior and (pseudo-)closure functions are related by the duality operator then conditions equal to claiming that either is isotonic hold equally. (Contributed by RP, 3-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 ntrclsiso ( 𝜑 → ( ∀ 𝑠 ∈ 𝒫 𝐵 ∀ 𝑡 ∈ 𝒫 𝐵 ( 𝑠 ⊆ 𝑡 → ( 𝐼 ‘ 𝑠 ) ⊆ ( 𝐼 ‘ 𝑡 ) ) ↔ ∀ 𝑠 ∈ 𝒫 𝐵 ∀ 𝑡 ∈ 𝒫 𝐵 ( 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 sseq1 ⊢ ( 𝑠 = 𝑏 → ( 𝑠 ⊆ 𝑡 ↔ 𝑏 ⊆ 𝑡 ) )
5 fveq2 ⊢ ( 𝑠 = 𝑏 → ( 𝐼 ‘ 𝑠 ) = ( 𝐼 ‘ 𝑏 ) )
6 5 sseq1d ⊢ ( 𝑠 = 𝑏 → ( ( 𝐼 ‘ 𝑠 ) ⊆ ( 𝐼 ‘ 𝑡 ) ↔ ( 𝐼 ‘ 𝑏 ) ⊆ ( 𝐼 ‘ 𝑡 ) ) )
7 4 6 imbi12d ⊢ ( 𝑠 = 𝑏 → ( ( 𝑠 ⊆ 𝑡 → ( 𝐼 ‘ 𝑠 ) ⊆ ( 𝐼 ‘ 𝑡 ) ) ↔ ( 𝑏 ⊆ 𝑡 → ( 𝐼 ‘ 𝑏 ) ⊆ ( 𝐼 ‘ 𝑡 ) ) ) )
8 sseq2 ⊢ ( 𝑡 = 𝑎 → ( 𝑏 ⊆ 𝑡 ↔ 𝑏 ⊆ 𝑎 ) )
9 fveq2 ⊢ ( 𝑡 = 𝑎 → ( 𝐼 ‘ 𝑡 ) = ( 𝐼 ‘ 𝑎 ) )
10 9 sseq2d ⊢ ( 𝑡 = 𝑎 → ( ( 𝐼 ‘ 𝑏 ) ⊆ ( 𝐼 ‘ 𝑡 ) ↔ ( 𝐼 ‘ 𝑏 ) ⊆ ( 𝐼 ‘ 𝑎 ) ) )
11 8 10 imbi12d ⊢ ( 𝑡 = 𝑎 → ( ( 𝑏 ⊆ 𝑡 → ( 𝐼 ‘ 𝑏 ) ⊆ ( 𝐼 ‘ 𝑡 ) ) ↔ ( 𝑏 ⊆ 𝑎 → ( 𝐼 ‘ 𝑏 ) ⊆ ( 𝐼 ‘ 𝑎 ) ) ) )
12 7 11 cbvral2vw ⊢ ( ∀ 𝑠 ∈ 𝒫 𝐵 ∀ 𝑡 ∈ 𝒫 𝐵 ( 𝑠 ⊆ 𝑡 → ( 𝐼 ‘ 𝑠 ) ⊆ ( 𝐼 ‘ 𝑡 ) ) ↔ ∀ 𝑏 ∈ 𝒫 𝐵 ∀ 𝑎 ∈ 𝒫 𝐵 ( 𝑏 ⊆ 𝑎 → ( 𝐼 ‘ 𝑏 ) ⊆ ( 𝐼 ‘ 𝑎 ) ) )
13 ralcom ⊢ ( ∀ 𝑏 ∈ 𝒫 𝐵 ∀ 𝑎 ∈ 𝒫 𝐵 ( 𝑏 ⊆ 𝑎 → ( 𝐼 ‘ 𝑏 ) ⊆ ( 𝐼 ‘ 𝑎 ) ) ↔ ∀ 𝑎 ∈ 𝒫 𝐵 ∀ 𝑏 ∈ 𝒫 𝐵 ( 𝑏 ⊆ 𝑎 → ( 𝐼 ‘ 𝑏 ) ⊆ ( 𝐼 ‘ 𝑎 ) ) )
14 12 13 bitri ⊢ ( ∀ 𝑠 ∈ 𝒫 𝐵 ∀ 𝑡 ∈ 𝒫 𝐵 ( 𝑠 ⊆ 𝑡 → ( 𝐼 ‘ 𝑠 ) ⊆ ( 𝐼 ‘ 𝑡 ) ) ↔ ∀ 𝑎 ∈ 𝒫 𝐵 ∀ 𝑏 ∈ 𝒫 𝐵 ( 𝑏 ⊆ 𝑎 → ( 𝐼 ‘ 𝑏 ) ⊆ ( 𝐼 ‘ 𝑎 ) ) )
15 simpl ⊢ ( ( 𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ) → 𝜑 )
16 2 3 ntrclsbex ⊢ ( 𝜑 → 𝐵 ∈ V )
17 15 16 syl ⊢ ( ( 𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ) → 𝐵 ∈ V )
18 difssd ⊢ ( ( 𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ) → ( 𝐵 ∖ 𝑠 ) ⊆ 𝐵 )
19 17 18 sselpwd ⊢ ( ( 𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ) → ( 𝐵 ∖ 𝑠 ) ∈ 𝒫 𝐵 )
20 elpwi ⊢ ( 𝑎 ∈ 𝒫 𝐵 → 𝑎 ⊆ 𝐵 )
21 simpl ⊢ ( ( 𝐵 ∈ V ∧ 𝑎 ⊆ 𝐵 ) → 𝐵 ∈ V )
22 difssd ⊢ ( ( 𝐵 ∈ V ∧ 𝑎 ⊆ 𝐵 ) → ( 𝐵 ∖ 𝑎 ) ⊆ 𝐵 )
23 21 22 sselpwd ⊢ ( ( 𝐵 ∈ V ∧ 𝑎 ⊆ 𝐵 ) → ( 𝐵 ∖ 𝑎 ) ∈ 𝒫 𝐵 )
24 simpr ⊢ ( ( ( 𝐵 ∈ V ∧ 𝑎 ⊆ 𝐵 ) ∧ 𝑠 = ( 𝐵 ∖ 𝑎 ) ) → 𝑠 = ( 𝐵 ∖ 𝑎 ) )
25 24 difeq2d ⊢ ( ( ( 𝐵 ∈ V ∧ 𝑎 ⊆ 𝐵 ) ∧ 𝑠 = ( 𝐵 ∖ 𝑎 ) ) → ( 𝐵 ∖ 𝑠 ) = ( 𝐵 ∖ ( 𝐵 ∖ 𝑎 ) ) )
26 25 eqeq2d ⊢ ( ( ( 𝐵 ∈ V ∧ 𝑎 ⊆ 𝐵 ) ∧ 𝑠 = ( 𝐵 ∖ 𝑎 ) ) → ( 𝑎 = ( 𝐵 ∖ 𝑠 ) ↔ 𝑎 = ( 𝐵 ∖ ( 𝐵 ∖ 𝑎 ) ) ) )
27 eqcom ⊢ ( 𝑎 = ( 𝐵 ∖ ( 𝐵 ∖ 𝑎 ) ) ↔ ( 𝐵 ∖ ( 𝐵 ∖ 𝑎 ) ) = 𝑎 )
28 26 27 bitrdi ⊢ ( ( ( 𝐵 ∈ V ∧ 𝑎 ⊆ 𝐵 ) ∧ 𝑠 = ( 𝐵 ∖ 𝑎 ) ) → ( 𝑎 = ( 𝐵 ∖ 𝑠 ) ↔ ( 𝐵 ∖ ( 𝐵 ∖ 𝑎 ) ) = 𝑎 ) )
29 dfss4 ⊢ ( 𝑎 ⊆ 𝐵 ↔ ( 𝐵 ∖ ( 𝐵 ∖ 𝑎 ) ) = 𝑎 )
30 29 bilani ⊢ ( ( 𝐵 ∈ V ∧ 𝑎 ⊆ 𝐵 ) → ( 𝐵 ∖ ( 𝐵 ∖ 𝑎 ) ) = 𝑎 )
31 23 28 30 rspcedvd ⊢ ( ( 𝐵 ∈ V ∧ 𝑎 ⊆ 𝐵 ) → ∃ 𝑠 ∈ 𝒫 𝐵 𝑎 = ( 𝐵 ∖ 𝑠 ) )
32 16 20 31 syl2an ⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝒫 𝐵 ) → ∃ 𝑠 ∈ 𝒫 𝐵 𝑎 = ( 𝐵 ∖ 𝑠 ) )
33 simpl1 ⊢ ( ( ( 𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = ( 𝐵 ∖ 𝑠 ) ) ∧ 𝑡 ∈ 𝒫 𝐵 ) → 𝜑 )
34 33 16 syl ⊢ ( ( ( 𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = ( 𝐵 ∖ 𝑠 ) ) ∧ 𝑡 ∈ 𝒫 𝐵 ) → 𝐵 ∈ V )
35 difssd ⊢ ( ( ( 𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = ( 𝐵 ∖ 𝑠 ) ) ∧ 𝑡 ∈ 𝒫 𝐵 ) → ( 𝐵 ∖ 𝑡 ) ⊆ 𝐵 )
36 34 35 sselpwd ⊢ ( ( ( 𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = ( 𝐵 ∖ 𝑠 ) ) ∧ 𝑡 ∈ 𝒫 𝐵 ) → ( 𝐵 ∖ 𝑡 ) ∈ 𝒫 𝐵 )
37 elpwi ⊢ ( 𝑏 ∈ 𝒫 𝐵 → 𝑏 ⊆ 𝐵 )
38 simpl ⊢ ( ( 𝐵 ∈ V ∧ 𝑏 ⊆ 𝐵 ) → 𝐵 ∈ V )
39 difssd ⊢ ( ( 𝐵 ∈ V ∧ 𝑏 ⊆ 𝐵 ) → ( 𝐵 ∖ 𝑏 ) ⊆ 𝐵 )
40 38 39 sselpwd ⊢ ( ( 𝐵 ∈ V ∧ 𝑏 ⊆ 𝐵 ) → ( 𝐵 ∖ 𝑏 ) ∈ 𝒫 𝐵 )
41 simpr ⊢ ( ( ( 𝐵 ∈ V ∧ 𝑏 ⊆ 𝐵 ) ∧ 𝑡 = ( 𝐵 ∖ 𝑏 ) ) → 𝑡 = ( 𝐵 ∖ 𝑏 ) )
42 41 difeq2d ⊢ ( ( ( 𝐵 ∈ V ∧ 𝑏 ⊆ 𝐵 ) ∧ 𝑡 = ( 𝐵 ∖ 𝑏 ) ) → ( 𝐵 ∖ 𝑡 ) = ( 𝐵 ∖ ( 𝐵 ∖ 𝑏 ) ) )
43 42 eqeq2d ⊢ ( ( ( 𝐵 ∈ V ∧ 𝑏 ⊆ 𝐵 ) ∧ 𝑡 = ( 𝐵 ∖ 𝑏 ) ) → ( 𝑏 = ( 𝐵 ∖ 𝑡 ) ↔ 𝑏 = ( 𝐵 ∖ ( 𝐵 ∖ 𝑏 ) ) ) )
44 eqcom ⊢ ( 𝑏 = ( 𝐵 ∖ ( 𝐵 ∖ 𝑏 ) ) ↔ ( 𝐵 ∖ ( 𝐵 ∖ 𝑏 ) ) = 𝑏 )
45 43 44 bitrdi ⊢ ( ( ( 𝐵 ∈ V ∧ 𝑏 ⊆ 𝐵 ) ∧ 𝑡 = ( 𝐵 ∖ 𝑏 ) ) → ( 𝑏 = ( 𝐵 ∖ 𝑡 ) ↔ ( 𝐵 ∖ ( 𝐵 ∖ 𝑏 ) ) = 𝑏 ) )
46 dfss4 ⊢ ( 𝑏 ⊆ 𝐵 ↔ ( 𝐵 ∖ ( 𝐵 ∖ 𝑏 ) ) = 𝑏 )
47 46 bilani ⊢ ( ( 𝐵 ∈ V ∧ 𝑏 ⊆ 𝐵 ) → ( 𝐵 ∖ ( 𝐵 ∖ 𝑏 ) ) = 𝑏 )
48 40 45 47 rspcedvd ⊢ ( ( 𝐵 ∈ V ∧ 𝑏 ⊆ 𝐵 ) → ∃ 𝑡 ∈ 𝒫 𝐵 𝑏 = ( 𝐵 ∖ 𝑡 ) )
49 16 37 48 syl2an ⊢ ( ( 𝜑 ∧ 𝑏 ∈ 𝒫 𝐵 ) → ∃ 𝑡 ∈ 𝒫 𝐵 𝑏 = ( 𝐵 ∖ 𝑡 ) )
50 49 3ad2antl1 ⊢ ( ( ( 𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = ( 𝐵 ∖ 𝑠 ) ) ∧ 𝑏 ∈ 𝒫 𝐵 ) → ∃ 𝑡 ∈ 𝒫 𝐵 𝑏 = ( 𝐵 ∖ 𝑡 ) )
51 simp12 ⊢ ( ( ( 𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = ( 𝐵 ∖ 𝑠 ) ) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = ( 𝐵 ∖ 𝑡 ) ) → 𝑠 ∈ 𝒫 𝐵 )
52 51 elpwid ⊢ ( ( ( 𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = ( 𝐵 ∖ 𝑠 ) ) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = ( 𝐵 ∖ 𝑡 ) ) → 𝑠 ⊆ 𝐵 )
53 simp2 ⊢ ( ( ( 𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = ( 𝐵 ∖ 𝑠 ) ) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = ( 𝐵 ∖ 𝑡 ) ) → 𝑡 ∈ 𝒫 𝐵 )
54 53 elpwid ⊢ ( ( ( 𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = ( 𝐵 ∖ 𝑠 ) ) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = ( 𝐵 ∖ 𝑡 ) ) → 𝑡 ⊆ 𝐵 )
55 sscon34b ⊢ ( ( 𝑠 ⊆ 𝐵 ∧ 𝑡 ⊆ 𝐵 ) → ( 𝑠 ⊆ 𝑡 ↔ ( 𝐵 ∖ 𝑡 ) ⊆ ( 𝐵 ∖ 𝑠 ) ) )
56 52 54 55 syl2anc ⊢ ( ( ( 𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = ( 𝐵 ∖ 𝑠 ) ) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = ( 𝐵 ∖ 𝑡 ) ) → ( 𝑠 ⊆ 𝑡 ↔ ( 𝐵 ∖ 𝑡 ) ⊆ ( 𝐵 ∖ 𝑠 ) ) )
57 56 bicomd ⊢ ( ( ( 𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = ( 𝐵 ∖ 𝑠 ) ) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = ( 𝐵 ∖ 𝑡 ) ) → ( ( 𝐵 ∖ 𝑡 ) ⊆ ( 𝐵 ∖ 𝑠 ) ↔ 𝑠 ⊆ 𝑡 ) )
58 simp11 ⊢ ( ( ( 𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = ( 𝐵 ∖ 𝑠 ) ) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = ( 𝐵 ∖ 𝑡 ) ) → 𝜑 )
59 1 2 3 ntrclsiex ⊢ ( 𝜑 → 𝐼 ∈ ( 𝒫 𝐵 ↑m 𝒫 𝐵 ) )
60 58 59 syl ⊢ ( ( ( 𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = ( 𝐵 ∖ 𝑠 ) ) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = ( 𝐵 ∖ 𝑡 ) ) → 𝐼 ∈ ( 𝒫 𝐵 ↑m 𝒫 𝐵 ) )
61 elmapi ⊢ ( 𝐼 ∈ ( 𝒫 𝐵 ↑m 𝒫 𝐵 ) → 𝐼 : 𝒫 𝐵 ⟶ 𝒫 𝐵 )
62 60 61 syl ⊢ ( ( ( 𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = ( 𝐵 ∖ 𝑠 ) ) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = ( 𝐵 ∖ 𝑡 ) ) → 𝐼 : 𝒫 𝐵 ⟶ 𝒫 𝐵 )
63 58 16 syl ⊢ ( ( ( 𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = ( 𝐵 ∖ 𝑠 ) ) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = ( 𝐵 ∖ 𝑡 ) ) → 𝐵 ∈ V )
64 difssd ⊢ ( ( ( 𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = ( 𝐵 ∖ 𝑠 ) ) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = ( 𝐵 ∖ 𝑡 ) ) → ( 𝐵 ∖ 𝑡 ) ⊆ 𝐵 )
65 63 64 sselpwd ⊢ ( ( ( 𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = ( 𝐵 ∖ 𝑠 ) ) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = ( 𝐵 ∖ 𝑡 ) ) → ( 𝐵 ∖ 𝑡 ) ∈ 𝒫 𝐵 )
66 62 65 ffvelcdmd ⊢ ( ( ( 𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = ( 𝐵 ∖ 𝑠 ) ) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = ( 𝐵 ∖ 𝑡 ) ) → ( 𝐼 ‘ ( 𝐵 ∖ 𝑡 ) ) ∈ 𝒫 𝐵 )
67 66 elpwid ⊢ ( ( ( 𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = ( 𝐵 ∖ 𝑠 ) ) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = ( 𝐵 ∖ 𝑡 ) ) → ( 𝐼 ‘ ( 𝐵 ∖ 𝑡 ) ) ⊆ 𝐵 )
68 difssd ⊢ ( ( ( 𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = ( 𝐵 ∖ 𝑠 ) ) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = ( 𝐵 ∖ 𝑡 ) ) → ( 𝐵 ∖ 𝑠 ) ⊆ 𝐵 )
69 63 68 sselpwd ⊢ ( ( ( 𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = ( 𝐵 ∖ 𝑠 ) ) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = ( 𝐵 ∖ 𝑡 ) ) → ( 𝐵 ∖ 𝑠 ) ∈ 𝒫 𝐵 )
70 62 69 ffvelcdmd ⊢ ( ( ( 𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = ( 𝐵 ∖ 𝑠 ) ) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = ( 𝐵 ∖ 𝑡 ) ) → ( 𝐼 ‘ ( 𝐵 ∖ 𝑠 ) ) ∈ 𝒫 𝐵 )
71 70 elpwid ⊢ ( ( ( 𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = ( 𝐵 ∖ 𝑠 ) ) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = ( 𝐵 ∖ 𝑡 ) ) → ( 𝐼 ‘ ( 𝐵 ∖ 𝑠 ) ) ⊆ 𝐵 )
72 sscon34b ⊢ ( ( ( 𝐼 ‘ ( 𝐵 ∖ 𝑡 ) ) ⊆ 𝐵 ∧ ( 𝐼 ‘ ( 𝐵 ∖ 𝑠 ) ) ⊆ 𝐵 ) → ( ( 𝐼 ‘ ( 𝐵 ∖ 𝑡 ) ) ⊆ ( 𝐼 ‘ ( 𝐵 ∖ 𝑠 ) ) ↔ ( 𝐵 ∖ ( 𝐼 ‘ ( 𝐵 ∖ 𝑠 ) ) ) ⊆ ( 𝐵 ∖ ( 𝐼 ‘ ( 𝐵 ∖ 𝑡 ) ) ) ) )
73 67 71 72 syl2anc ⊢ ( ( ( 𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = ( 𝐵 ∖ 𝑠 ) ) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = ( 𝐵 ∖ 𝑡 ) ) → ( ( 𝐼 ‘ ( 𝐵 ∖ 𝑡 ) ) ⊆ ( 𝐼 ‘ ( 𝐵 ∖ 𝑠 ) ) ↔ ( 𝐵 ∖ ( 𝐼 ‘ ( 𝐵 ∖ 𝑠 ) ) ) ⊆ ( 𝐵 ∖ ( 𝐼 ‘ ( 𝐵 ∖ 𝑡 ) ) ) ) )
74 57 73 imbi12d ⊢ ( ( ( 𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = ( 𝐵 ∖ 𝑠 ) ) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = ( 𝐵 ∖ 𝑡 ) ) → ( ( ( 𝐵 ∖ 𝑡 ) ⊆ ( 𝐵 ∖ 𝑠 ) → ( 𝐼 ‘ ( 𝐵 ∖ 𝑡 ) ) ⊆ ( 𝐼 ‘ ( 𝐵 ∖ 𝑠 ) ) ) ↔ ( 𝑠 ⊆ 𝑡 → ( 𝐵 ∖ ( 𝐼 ‘ ( 𝐵 ∖ 𝑠 ) ) ) ⊆ ( 𝐵 ∖ ( 𝐼 ‘ ( 𝐵 ∖ 𝑡 ) ) ) ) ) )
75 simp3 ⊢ ( ( ( 𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = ( 𝐵 ∖ 𝑠 ) ) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = ( 𝐵 ∖ 𝑡 ) ) → 𝑏 = ( 𝐵 ∖ 𝑡 ) )
76 simp13 ⊢ ( ( ( 𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = ( 𝐵 ∖ 𝑠 ) ) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = ( 𝐵 ∖ 𝑡 ) ) → 𝑎 = ( 𝐵 ∖ 𝑠 ) )
77 75 76 sseq12d ⊢ ( ( ( 𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = ( 𝐵 ∖ 𝑠 ) ) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = ( 𝐵 ∖ 𝑡 ) ) → ( 𝑏 ⊆ 𝑎 ↔ ( 𝐵 ∖ 𝑡 ) ⊆ ( 𝐵 ∖ 𝑠 ) ) )
78 75 fveq2d ⊢ ( ( ( 𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = ( 𝐵 ∖ 𝑠 ) ) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = ( 𝐵 ∖ 𝑡 ) ) → ( 𝐼 ‘ 𝑏 ) = ( 𝐼 ‘ ( 𝐵 ∖ 𝑡 ) ) )
79 76 fveq2d ⊢ ( ( ( 𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = ( 𝐵 ∖ 𝑠 ) ) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = ( 𝐵 ∖ 𝑡 ) ) → ( 𝐼 ‘ 𝑎 ) = ( 𝐼 ‘ ( 𝐵 ∖ 𝑠 ) ) )
80 78 79 sseq12d ⊢ ( ( ( 𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = ( 𝐵 ∖ 𝑠 ) ) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = ( 𝐵 ∖ 𝑡 ) ) → ( ( 𝐼 ‘ 𝑏 ) ⊆ ( 𝐼 ‘ 𝑎 ) ↔ ( 𝐼 ‘ ( 𝐵 ∖ 𝑡 ) ) ⊆ ( 𝐼 ‘ ( 𝐵 ∖ 𝑠 ) ) ) )
81 77 80 imbi12d ⊢ ( ( ( 𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = ( 𝐵 ∖ 𝑠 ) ) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = ( 𝐵 ∖ 𝑡 ) ) → ( ( 𝑏 ⊆ 𝑎 → ( 𝐼 ‘ 𝑏 ) ⊆ ( 𝐼 ‘ 𝑎 ) ) ↔ ( ( 𝐵 ∖ 𝑡 ) ⊆ ( 𝐵 ∖ 𝑠 ) → ( 𝐼 ‘ ( 𝐵 ∖ 𝑡 ) ) ⊆ ( 𝐼 ‘ ( 𝐵 ∖ 𝑠 ) ) ) ) )
82 1 2 3 ntrclsfv1 ⊢ ( 𝜑 → ( 𝐷 ‘ 𝐼 ) = 𝐾 )
83 58 82 syl ⊢ ( ( ( 𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = ( 𝐵 ∖ 𝑠 ) ) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = ( 𝐵 ∖ 𝑡 ) ) → ( 𝐷 ‘ 𝐼 ) = 𝐾 )
84 83 fveq1d ⊢ ( ( ( 𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = ( 𝐵 ∖ 𝑠 ) ) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = ( 𝐵 ∖ 𝑡 ) ) → ( ( 𝐷 ‘ 𝐼 ) ‘ 𝑠 ) = ( 𝐾 ‘ 𝑠 ) )
85 eqid ⊢ ( 𝐷 ‘ 𝐼 ) = ( 𝐷 ‘ 𝐼 )
86 eqid ⊢ ( ( 𝐷 ‘ 𝐼 ) ‘ 𝑠 ) = ( ( 𝐷 ‘ 𝐼 ) ‘ 𝑠 )
87 1 2 63 60 85 51 86 dssmapfv3d ⊢ ( ( ( 𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = ( 𝐵 ∖ 𝑠 ) ) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = ( 𝐵 ∖ 𝑡 ) ) → ( ( 𝐷 ‘ 𝐼 ) ‘ 𝑠 ) = ( 𝐵 ∖ ( 𝐼 ‘ ( 𝐵 ∖ 𝑠 ) ) ) )
88 84 87 eqtr3d ⊢ ( ( ( 𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = ( 𝐵 ∖ 𝑠 ) ) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = ( 𝐵 ∖ 𝑡 ) ) → ( 𝐾 ‘ 𝑠 ) = ( 𝐵 ∖ ( 𝐼 ‘ ( 𝐵 ∖ 𝑠 ) ) ) )
89 58 3 syl ⊢ ( ( ( 𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = ( 𝐵 ∖ 𝑠 ) ) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = ( 𝐵 ∖ 𝑡 ) ) → 𝐼 𝐷 𝐾 )
90 1 2 89 ntrclsfv1 ⊢ ( ( ( 𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = ( 𝐵 ∖ 𝑠 ) ) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = ( 𝐵 ∖ 𝑡 ) ) → ( 𝐷 ‘ 𝐼 ) = 𝐾 )
91 90 fveq1d ⊢ ( ( ( 𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = ( 𝐵 ∖ 𝑠 ) ) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = ( 𝐵 ∖ 𝑡 ) ) → ( ( 𝐷 ‘ 𝐼 ) ‘ 𝑡 ) = ( 𝐾 ‘ 𝑡 ) )
92 eqid ⊢ ( ( 𝐷 ‘ 𝐼 ) ‘ 𝑡 ) = ( ( 𝐷 ‘ 𝐼 ) ‘ 𝑡 )
93 1 2 63 60 85 53 92 dssmapfv3d ⊢ ( ( ( 𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = ( 𝐵 ∖ 𝑠 ) ) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = ( 𝐵 ∖ 𝑡 ) ) → ( ( 𝐷 ‘ 𝐼 ) ‘ 𝑡 ) = ( 𝐵 ∖ ( 𝐼 ‘ ( 𝐵 ∖ 𝑡 ) ) ) )
94 91 93 eqtr3d ⊢ ( ( ( 𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = ( 𝐵 ∖ 𝑠 ) ) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = ( 𝐵 ∖ 𝑡 ) ) → ( 𝐾 ‘ 𝑡 ) = ( 𝐵 ∖ ( 𝐼 ‘ ( 𝐵 ∖ 𝑡 ) ) ) )
95 88 94 sseq12d ⊢ ( ( ( 𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = ( 𝐵 ∖ 𝑠 ) ) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = ( 𝐵 ∖ 𝑡 ) ) → ( ( 𝐾 ‘ 𝑠 ) ⊆ ( 𝐾 ‘ 𝑡 ) ↔ ( 𝐵 ∖ ( 𝐼 ‘ ( 𝐵 ∖ 𝑠 ) ) ) ⊆ ( 𝐵 ∖ ( 𝐼 ‘ ( 𝐵 ∖ 𝑡 ) ) ) ) )
96 95 imbi2d ⊢ ( ( ( 𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = ( 𝐵 ∖ 𝑠 ) ) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = ( 𝐵 ∖ 𝑡 ) ) → ( ( 𝑠 ⊆ 𝑡 → ( 𝐾 ‘ 𝑠 ) ⊆ ( 𝐾 ‘ 𝑡 ) ) ↔ ( 𝑠 ⊆ 𝑡 → ( 𝐵 ∖ ( 𝐼 ‘ ( 𝐵 ∖ 𝑠 ) ) ) ⊆ ( 𝐵 ∖ ( 𝐼 ‘ ( 𝐵 ∖ 𝑡 ) ) ) ) ) )
97 74 81 96 3bitr4d ⊢ ( ( ( 𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = ( 𝐵 ∖ 𝑠 ) ) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = ( 𝐵 ∖ 𝑡 ) ) → ( ( 𝑏 ⊆ 𝑎 → ( 𝐼 ‘ 𝑏 ) ⊆ ( 𝐼 ‘ 𝑎 ) ) ↔ ( 𝑠 ⊆ 𝑡 → ( 𝐾 ‘ 𝑠 ) ⊆ ( 𝐾 ‘ 𝑡 ) ) ) )
98 36 50 97 ralxfrd2 ⊢ ( ( 𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = ( 𝐵 ∖ 𝑠 ) ) → ( ∀ 𝑏 ∈ 𝒫 𝐵 ( 𝑏 ⊆ 𝑎 → ( 𝐼 ‘ 𝑏 ) ⊆ ( 𝐼 ‘ 𝑎 ) ) ↔ ∀ 𝑡 ∈ 𝒫 𝐵 ( 𝑠 ⊆ 𝑡 → ( 𝐾 ‘ 𝑠 ) ⊆ ( 𝐾 ‘ 𝑡 ) ) ) )
99 19 32 98 ralxfrd2 ⊢ ( 𝜑 → ( ∀ 𝑎 ∈ 𝒫 𝐵 ∀ 𝑏 ∈ 𝒫 𝐵 ( 𝑏 ⊆ 𝑎 → ( 𝐼 ‘ 𝑏 ) ⊆ ( 𝐼 ‘ 𝑎 ) ) ↔ ∀ 𝑠 ∈ 𝒫 𝐵 ∀ 𝑡 ∈ 𝒫 𝐵 ( 𝑠 ⊆ 𝑡 → ( 𝐾 ‘ 𝑠 ) ⊆ ( 𝐾 ‘ 𝑡 ) ) ) )
100 14 99 bitrid ⊢ ( 𝜑 → ( ∀ 𝑠 ∈ 𝒫 𝐵 ∀ 𝑡 ∈ 𝒫 𝐵 ( 𝑠 ⊆ 𝑡 → ( 𝐼 ‘ 𝑠 ) ⊆ ( 𝐼 ‘ 𝑡 ) ) ↔ ∀ 𝑠 ∈ 𝒫 𝐵 ∀ 𝑡 ∈ 𝒫 𝐵 ( 𝑠 ⊆ 𝑡 → ( 𝐾 ‘ 𝑠 ) ⊆ ( 𝐾 ‘ 𝑡 ) ) ) )