Metamath Proof Explorer


Theorem isotone2

Description: Two different ways to say subset relation persists across applications of a function. (Contributed by RP, 31-May-2021)

Ref Expression
Assertion isotone2 ( ∀ 𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴 ( 𝑎𝑏 → ( 𝐹𝑎 ) ⊆ ( 𝐹𝑏 ) ) ↔ ∀ 𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴 ( 𝐹 ‘ ( 𝑎𝑏 ) ) ⊆ ( ( 𝐹𝑎 ) ∩ ( 𝐹𝑏 ) ) )

Proof

Step Hyp Ref Expression
1 sseq1 ( 𝑎 = 𝑐 → ( 𝑎𝑏𝑐𝑏 ) )
2 fveq2 ( 𝑎 = 𝑐 → ( 𝐹𝑎 ) = ( 𝐹𝑐 ) )
3 2 sseq1d ( 𝑎 = 𝑐 → ( ( 𝐹𝑎 ) ⊆ ( 𝐹𝑏 ) ↔ ( 𝐹𝑐 ) ⊆ ( 𝐹𝑏 ) ) )
4 1 3 imbi12d ( 𝑎 = 𝑐 → ( ( 𝑎𝑏 → ( 𝐹𝑎 ) ⊆ ( 𝐹𝑏 ) ) ↔ ( 𝑐𝑏 → ( 𝐹𝑐 ) ⊆ ( 𝐹𝑏 ) ) ) )
5 sseq2 ( 𝑏 = 𝑑 → ( 𝑐𝑏𝑐𝑑 ) )
6 fveq2 ( 𝑏 = 𝑑 → ( 𝐹𝑏 ) = ( 𝐹𝑑 ) )
7 6 sseq2d ( 𝑏 = 𝑑 → ( ( 𝐹𝑐 ) ⊆ ( 𝐹𝑏 ) ↔ ( 𝐹𝑐 ) ⊆ ( 𝐹𝑑 ) ) )
8 5 7 imbi12d ( 𝑏 = 𝑑 → ( ( 𝑐𝑏 → ( 𝐹𝑐 ) ⊆ ( 𝐹𝑏 ) ) ↔ ( 𝑐𝑑 → ( 𝐹𝑐 ) ⊆ ( 𝐹𝑑 ) ) ) )
9 4 8 cbvral2vw ( ∀ 𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴 ( 𝑎𝑏 → ( 𝐹𝑎 ) ⊆ ( 𝐹𝑏 ) ) ↔ ∀ 𝑐 ∈ 𝒫 𝐴𝑑 ∈ 𝒫 𝐴 ( 𝑐𝑑 → ( 𝐹𝑐 ) ⊆ ( 𝐹𝑑 ) ) )
10 inss1 ( 𝑎𝑏 ) ⊆ 𝑎
11 inss2 ( 𝑎𝑏 ) ⊆ 𝑏
12 elpwi ( 𝑏 ∈ 𝒫 𝐴𝑏𝐴 )
13 11 12 sstrid ( 𝑏 ∈ 𝒫 𝐴 → ( 𝑎𝑏 ) ⊆ 𝐴 )
14 vex 𝑏 ∈ V
15 14 inex2 ( 𝑎𝑏 ) ∈ V
16 15 elpw ( ( 𝑎𝑏 ) ∈ 𝒫 𝐴 ↔ ( 𝑎𝑏 ) ⊆ 𝐴 )
17 13 16 sylibr ( 𝑏 ∈ 𝒫 𝐴 → ( 𝑎𝑏 ) ∈ 𝒫 𝐴 )
18 17 ad2antll ( ( ∀ 𝑐 ∈ 𝒫 𝐴𝑑 ∈ 𝒫 𝐴 ( 𝑐𝑑 → ( 𝐹𝑐 ) ⊆ ( 𝐹𝑑 ) ) ∧ ( 𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴 ) ) → ( 𝑎𝑏 ) ∈ 𝒫 𝐴 )
19 simprl ( ( ∀ 𝑐 ∈ 𝒫 𝐴𝑑 ∈ 𝒫 𝐴 ( 𝑐𝑑 → ( 𝐹𝑐 ) ⊆ ( 𝐹𝑑 ) ) ∧ ( 𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴 ) ) → 𝑎 ∈ 𝒫 𝐴 )
20 simpl ( ( ∀ 𝑐 ∈ 𝒫 𝐴𝑑 ∈ 𝒫 𝐴 ( 𝑐𝑑 → ( 𝐹𝑐 ) ⊆ ( 𝐹𝑑 ) ) ∧ ( 𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴 ) ) → ∀ 𝑐 ∈ 𝒫 𝐴𝑑 ∈ 𝒫 𝐴 ( 𝑐𝑑 → ( 𝐹𝑐 ) ⊆ ( 𝐹𝑑 ) ) )
21 sseq1 ( 𝑐 = ( 𝑎𝑏 ) → ( 𝑐𝑑 ↔ ( 𝑎𝑏 ) ⊆ 𝑑 ) )
22 fveq2 ( 𝑐 = ( 𝑎𝑏 ) → ( 𝐹𝑐 ) = ( 𝐹 ‘ ( 𝑎𝑏 ) ) )
23 22 sseq1d ( 𝑐 = ( 𝑎𝑏 ) → ( ( 𝐹𝑐 ) ⊆ ( 𝐹𝑑 ) ↔ ( 𝐹 ‘ ( 𝑎𝑏 ) ) ⊆ ( 𝐹𝑑 ) ) )
24 21 23 imbi12d ( 𝑐 = ( 𝑎𝑏 ) → ( ( 𝑐𝑑 → ( 𝐹𝑐 ) ⊆ ( 𝐹𝑑 ) ) ↔ ( ( 𝑎𝑏 ) ⊆ 𝑑 → ( 𝐹 ‘ ( 𝑎𝑏 ) ) ⊆ ( 𝐹𝑑 ) ) ) )
25 sseq2 ( 𝑑 = 𝑎 → ( ( 𝑎𝑏 ) ⊆ 𝑑 ↔ ( 𝑎𝑏 ) ⊆ 𝑎 ) )
26 fveq2 ( 𝑑 = 𝑎 → ( 𝐹𝑑 ) = ( 𝐹𝑎 ) )
27 26 sseq2d ( 𝑑 = 𝑎 → ( ( 𝐹 ‘ ( 𝑎𝑏 ) ) ⊆ ( 𝐹𝑑 ) ↔ ( 𝐹 ‘ ( 𝑎𝑏 ) ) ⊆ ( 𝐹𝑎 ) ) )
28 25 27 imbi12d ( 𝑑 = 𝑎 → ( ( ( 𝑎𝑏 ) ⊆ 𝑑 → ( 𝐹 ‘ ( 𝑎𝑏 ) ) ⊆ ( 𝐹𝑑 ) ) ↔ ( ( 𝑎𝑏 ) ⊆ 𝑎 → ( 𝐹 ‘ ( 𝑎𝑏 ) ) ⊆ ( 𝐹𝑎 ) ) ) )
29 24 28 rspc2va ( ( ( ( 𝑎𝑏 ) ∈ 𝒫 𝐴𝑎 ∈ 𝒫 𝐴 ) ∧ ∀ 𝑐 ∈ 𝒫 𝐴𝑑 ∈ 𝒫 𝐴 ( 𝑐𝑑 → ( 𝐹𝑐 ) ⊆ ( 𝐹𝑑 ) ) ) → ( ( 𝑎𝑏 ) ⊆ 𝑎 → ( 𝐹 ‘ ( 𝑎𝑏 ) ) ⊆ ( 𝐹𝑎 ) ) )
30 18 19 20 29 syl21anc ( ( ∀ 𝑐 ∈ 𝒫 𝐴𝑑 ∈ 𝒫 𝐴 ( 𝑐𝑑 → ( 𝐹𝑐 ) ⊆ ( 𝐹𝑑 ) ) ∧ ( 𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴 ) ) → ( ( 𝑎𝑏 ) ⊆ 𝑎 → ( 𝐹 ‘ ( 𝑎𝑏 ) ) ⊆ ( 𝐹𝑎 ) ) )
31 10 30 mpi ( ( ∀ 𝑐 ∈ 𝒫 𝐴𝑑 ∈ 𝒫 𝐴 ( 𝑐𝑑 → ( 𝐹𝑐 ) ⊆ ( 𝐹𝑑 ) ) ∧ ( 𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴 ) ) → ( 𝐹 ‘ ( 𝑎𝑏 ) ) ⊆ ( 𝐹𝑎 ) )
32 simprr ( ( ∀ 𝑐 ∈ 𝒫 𝐴𝑑 ∈ 𝒫 𝐴 ( 𝑐𝑑 → ( 𝐹𝑐 ) ⊆ ( 𝐹𝑑 ) ) ∧ ( 𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴 ) ) → 𝑏 ∈ 𝒫 𝐴 )
33 sseq2 ( 𝑑 = 𝑏 → ( ( 𝑎𝑏 ) ⊆ 𝑑 ↔ ( 𝑎𝑏 ) ⊆ 𝑏 ) )
34 fveq2 ( 𝑑 = 𝑏 → ( 𝐹𝑑 ) = ( 𝐹𝑏 ) )
35 34 sseq2d ( 𝑑 = 𝑏 → ( ( 𝐹 ‘ ( 𝑎𝑏 ) ) ⊆ ( 𝐹𝑑 ) ↔ ( 𝐹 ‘ ( 𝑎𝑏 ) ) ⊆ ( 𝐹𝑏 ) ) )
36 33 35 imbi12d ( 𝑑 = 𝑏 → ( ( ( 𝑎𝑏 ) ⊆ 𝑑 → ( 𝐹 ‘ ( 𝑎𝑏 ) ) ⊆ ( 𝐹𝑑 ) ) ↔ ( ( 𝑎𝑏 ) ⊆ 𝑏 → ( 𝐹 ‘ ( 𝑎𝑏 ) ) ⊆ ( 𝐹𝑏 ) ) ) )
37 24 36 rspc2va ( ( ( ( 𝑎𝑏 ) ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴 ) ∧ ∀ 𝑐 ∈ 𝒫 𝐴𝑑 ∈ 𝒫 𝐴 ( 𝑐𝑑 → ( 𝐹𝑐 ) ⊆ ( 𝐹𝑑 ) ) ) → ( ( 𝑎𝑏 ) ⊆ 𝑏 → ( 𝐹 ‘ ( 𝑎𝑏 ) ) ⊆ ( 𝐹𝑏 ) ) )
38 18 32 20 37 syl21anc ( ( ∀ 𝑐 ∈ 𝒫 𝐴𝑑 ∈ 𝒫 𝐴 ( 𝑐𝑑 → ( 𝐹𝑐 ) ⊆ ( 𝐹𝑑 ) ) ∧ ( 𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴 ) ) → ( ( 𝑎𝑏 ) ⊆ 𝑏 → ( 𝐹 ‘ ( 𝑎𝑏 ) ) ⊆ ( 𝐹𝑏 ) ) )
39 11 38 mpi ( ( ∀ 𝑐 ∈ 𝒫 𝐴𝑑 ∈ 𝒫 𝐴 ( 𝑐𝑑 → ( 𝐹𝑐 ) ⊆ ( 𝐹𝑑 ) ) ∧ ( 𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴 ) ) → ( 𝐹 ‘ ( 𝑎𝑏 ) ) ⊆ ( 𝐹𝑏 ) )
40 31 39 ssind ( ( ∀ 𝑐 ∈ 𝒫 𝐴𝑑 ∈ 𝒫 𝐴 ( 𝑐𝑑 → ( 𝐹𝑐 ) ⊆ ( 𝐹𝑑 ) ) ∧ ( 𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴 ) ) → ( 𝐹 ‘ ( 𝑎𝑏 ) ) ⊆ ( ( 𝐹𝑎 ) ∩ ( 𝐹𝑏 ) ) )
41 40 ralrimivva ( ∀ 𝑐 ∈ 𝒫 𝐴𝑑 ∈ 𝒫 𝐴 ( 𝑐𝑑 → ( 𝐹𝑐 ) ⊆ ( 𝐹𝑑 ) ) → ∀ 𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴 ( 𝐹 ‘ ( 𝑎𝑏 ) ) ⊆ ( ( 𝐹𝑎 ) ∩ ( 𝐹𝑏 ) ) )
42 dfss ( 𝑐𝑑𝑐 = ( 𝑐𝑑 ) )
43 fveq2 ( 𝑐 = ( 𝑐𝑑 ) → ( 𝐹𝑐 ) = ( 𝐹 ‘ ( 𝑐𝑑 ) ) )
44 43 adantl ( ( ( ∀ 𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴 ( 𝐹 ‘ ( 𝑎𝑏 ) ) ⊆ ( ( 𝐹𝑎 ) ∩ ( 𝐹𝑏 ) ) ∧ ( 𝑐 ∈ 𝒫 𝐴𝑑 ∈ 𝒫 𝐴 ) ) ∧ 𝑐 = ( 𝑐𝑑 ) ) → ( 𝐹𝑐 ) = ( 𝐹 ‘ ( 𝑐𝑑 ) ) )
45 ineq1 ( 𝑎 = 𝑐 → ( 𝑎𝑏 ) = ( 𝑐𝑏 ) )
46 45 fveq2d ( 𝑎 = 𝑐 → ( 𝐹 ‘ ( 𝑎𝑏 ) ) = ( 𝐹 ‘ ( 𝑐𝑏 ) ) )
47 2 ineq1d ( 𝑎 = 𝑐 → ( ( 𝐹𝑎 ) ∩ ( 𝐹𝑏 ) ) = ( ( 𝐹𝑐 ) ∩ ( 𝐹𝑏 ) ) )
48 46 47 sseq12d ( 𝑎 = 𝑐 → ( ( 𝐹 ‘ ( 𝑎𝑏 ) ) ⊆ ( ( 𝐹𝑎 ) ∩ ( 𝐹𝑏 ) ) ↔ ( 𝐹 ‘ ( 𝑐𝑏 ) ) ⊆ ( ( 𝐹𝑐 ) ∩ ( 𝐹𝑏 ) ) ) )
49 ineq2 ( 𝑏 = 𝑑 → ( 𝑐𝑏 ) = ( 𝑐𝑑 ) )
50 49 fveq2d ( 𝑏 = 𝑑 → ( 𝐹 ‘ ( 𝑐𝑏 ) ) = ( 𝐹 ‘ ( 𝑐𝑑 ) ) )
51 6 ineq2d ( 𝑏 = 𝑑 → ( ( 𝐹𝑐 ) ∩ ( 𝐹𝑏 ) ) = ( ( 𝐹𝑐 ) ∩ ( 𝐹𝑑 ) ) )
52 50 51 sseq12d ( 𝑏 = 𝑑 → ( ( 𝐹 ‘ ( 𝑐𝑏 ) ) ⊆ ( ( 𝐹𝑐 ) ∩ ( 𝐹𝑏 ) ) ↔ ( 𝐹 ‘ ( 𝑐𝑑 ) ) ⊆ ( ( 𝐹𝑐 ) ∩ ( 𝐹𝑑 ) ) ) )
53 48 52 rspc2va ( ( ( 𝑐 ∈ 𝒫 𝐴𝑑 ∈ 𝒫 𝐴 ) ∧ ∀ 𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴 ( 𝐹 ‘ ( 𝑎𝑏 ) ) ⊆ ( ( 𝐹𝑎 ) ∩ ( 𝐹𝑏 ) ) ) → ( 𝐹 ‘ ( 𝑐𝑑 ) ) ⊆ ( ( 𝐹𝑐 ) ∩ ( 𝐹𝑑 ) ) )
54 53 ancoms ( ( ∀ 𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴 ( 𝐹 ‘ ( 𝑎𝑏 ) ) ⊆ ( ( 𝐹𝑎 ) ∩ ( 𝐹𝑏 ) ) ∧ ( 𝑐 ∈ 𝒫 𝐴𝑑 ∈ 𝒫 𝐴 ) ) → ( 𝐹 ‘ ( 𝑐𝑑 ) ) ⊆ ( ( 𝐹𝑐 ) ∩ ( 𝐹𝑑 ) ) )
55 inss2 ( ( 𝐹𝑐 ) ∩ ( 𝐹𝑑 ) ) ⊆ ( 𝐹𝑑 )
56 54 55 sstrdi ( ( ∀ 𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴 ( 𝐹 ‘ ( 𝑎𝑏 ) ) ⊆ ( ( 𝐹𝑎 ) ∩ ( 𝐹𝑏 ) ) ∧ ( 𝑐 ∈ 𝒫 𝐴𝑑 ∈ 𝒫 𝐴 ) ) → ( 𝐹 ‘ ( 𝑐𝑑 ) ) ⊆ ( 𝐹𝑑 ) )
57 56 adantr ( ( ( ∀ 𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴 ( 𝐹 ‘ ( 𝑎𝑏 ) ) ⊆ ( ( 𝐹𝑎 ) ∩ ( 𝐹𝑏 ) ) ∧ ( 𝑐 ∈ 𝒫 𝐴𝑑 ∈ 𝒫 𝐴 ) ) ∧ 𝑐 = ( 𝑐𝑑 ) ) → ( 𝐹 ‘ ( 𝑐𝑑 ) ) ⊆ ( 𝐹𝑑 ) )
58 44 57 eqsstrd ( ( ( ∀ 𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴 ( 𝐹 ‘ ( 𝑎𝑏 ) ) ⊆ ( ( 𝐹𝑎 ) ∩ ( 𝐹𝑏 ) ) ∧ ( 𝑐 ∈ 𝒫 𝐴𝑑 ∈ 𝒫 𝐴 ) ) ∧ 𝑐 = ( 𝑐𝑑 ) ) → ( 𝐹𝑐 ) ⊆ ( 𝐹𝑑 ) )
59 58 ex ( ( ∀ 𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴 ( 𝐹 ‘ ( 𝑎𝑏 ) ) ⊆ ( ( 𝐹𝑎 ) ∩ ( 𝐹𝑏 ) ) ∧ ( 𝑐 ∈ 𝒫 𝐴𝑑 ∈ 𝒫 𝐴 ) ) → ( 𝑐 = ( 𝑐𝑑 ) → ( 𝐹𝑐 ) ⊆ ( 𝐹𝑑 ) ) )
60 42 59 syl5bi ( ( ∀ 𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴 ( 𝐹 ‘ ( 𝑎𝑏 ) ) ⊆ ( ( 𝐹𝑎 ) ∩ ( 𝐹𝑏 ) ) ∧ ( 𝑐 ∈ 𝒫 𝐴𝑑 ∈ 𝒫 𝐴 ) ) → ( 𝑐𝑑 → ( 𝐹𝑐 ) ⊆ ( 𝐹𝑑 ) ) )
61 60 ralrimivva ( ∀ 𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴 ( 𝐹 ‘ ( 𝑎𝑏 ) ) ⊆ ( ( 𝐹𝑎 ) ∩ ( 𝐹𝑏 ) ) → ∀ 𝑐 ∈ 𝒫 𝐴𝑑 ∈ 𝒫 𝐴 ( 𝑐𝑑 → ( 𝐹𝑐 ) ⊆ ( 𝐹𝑑 ) ) )
62 41 61 impbii ( ∀ 𝑐 ∈ 𝒫 𝐴𝑑 ∈ 𝒫 𝐴 ( 𝑐𝑑 → ( 𝐹𝑐 ) ⊆ ( 𝐹𝑑 ) ) ↔ ∀ 𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴 ( 𝐹 ‘ ( 𝑎𝑏 ) ) ⊆ ( ( 𝐹𝑎 ) ∩ ( 𝐹𝑏 ) ) )
63 9 62 bitri ( ∀ 𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴 ( 𝑎𝑏 → ( 𝐹𝑎 ) ⊆ ( 𝐹𝑏 ) ) ↔ ∀ 𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴 ( 𝐹 ‘ ( 𝑎𝑏 ) ) ⊆ ( ( 𝐹𝑎 ) ∩ ( 𝐹𝑏 ) ) )