Metamath Proof Explorer

Theorem iunincfi

Description: Given a sequence of increasing sets, the union of a finite subsequence, is its last element. (Contributed by Glauco Siliprandi, 8-Apr-2021)

Ref Expression
Hypotheses iunincfi.1 ⊢ ( 𝜑 → 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) )
iunincfi.2 ⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( 𝑀 ..^ 𝑁 ) ) → ( 𝐹 ‘ 𝑛 ) ⊆ ( 𝐹 ‘ ( 𝑛 + 1 ) ) )
Assertion iunincfi ( 𝜑 → ∪ 𝑛 ∈ ( 𝑀 ... 𝑁 ) ( 𝐹 ‘ 𝑛 ) = ( 𝐹 ‘ 𝑁 ) )

Proof

Step Hyp Ref Expression
1 iunincfi.1 ⊢ ( 𝜑 → 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) )
2 iunincfi.2 ⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( 𝑀 ..^ 𝑁 ) ) → ( 𝐹 ‘ 𝑛 ) ⊆ ( 𝐹 ‘ ( 𝑛 + 1 ) ) )
3 eliun ⊢ ( 𝑥 ∈ ∪ 𝑛 ∈ ( 𝑀 ... 𝑁 ) ( 𝐹 ‘ 𝑛 ) ↔ ∃ 𝑛 ∈ ( 𝑀 ... 𝑁 ) 𝑥 ∈ ( 𝐹 ‘ 𝑛 ) )
4 3 bilani ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ∪ 𝑛 ∈ ( 𝑀 ... 𝑁 ) ( 𝐹 ‘ 𝑛 ) ) → ∃ 𝑛 ∈ ( 𝑀 ... 𝑁 ) 𝑥 ∈ ( 𝐹 ‘ 𝑛 ) )
5 elfzuz3 ⊢ ( 𝑛 ∈ ( 𝑀 ... 𝑁 ) → 𝑁 ∈ ( ℤ≥ ‘ 𝑛 ) )
6 5 adantl ⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( 𝑀 ... 𝑁 ) ) → 𝑁 ∈ ( ℤ≥ ‘ 𝑛 ) )
7 simpll ⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ ( 𝑀 ... 𝑁 ) ) ∧ 𝑚 ∈ ( 𝑛 ..^ 𝑁 ) ) → 𝜑 )
8 elfzuz ⊢ ( 𝑛 ∈ ( 𝑀 ... 𝑁 ) → 𝑛 ∈ ( ℤ≥ ‘ 𝑀 ) )
9 fzoss1 ⊢ ( 𝑛 ∈ ( ℤ≥ ‘ 𝑀 ) → ( 𝑛 ..^ 𝑁 ) ⊆ ( 𝑀 ..^ 𝑁 ) )
10 8 9 syl ⊢ ( 𝑛 ∈ ( 𝑀 ... 𝑁 ) → ( 𝑛 ..^ 𝑁 ) ⊆ ( 𝑀 ..^ 𝑁 ) )
11 10 adantr ⊢ ( ( 𝑛 ∈ ( 𝑀 ... 𝑁 ) ∧ 𝑚 ∈ ( 𝑛 ..^ 𝑁 ) ) → ( 𝑛 ..^ 𝑁 ) ⊆ ( 𝑀 ..^ 𝑁 ) )
12 simpr ⊢ ( ( 𝑛 ∈ ( 𝑀 ... 𝑁 ) ∧ 𝑚 ∈ ( 𝑛 ..^ 𝑁 ) ) → 𝑚 ∈ ( 𝑛 ..^ 𝑁 ) )
13 11 12 sseldd ⊢ ( ( 𝑛 ∈ ( 𝑀 ... 𝑁 ) ∧ 𝑚 ∈ ( 𝑛 ..^ 𝑁 ) ) → 𝑚 ∈ ( 𝑀 ..^ 𝑁 ) )
14 13 adantll ⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ ( 𝑀 ... 𝑁 ) ) ∧ 𝑚 ∈ ( 𝑛 ..^ 𝑁 ) ) → 𝑚 ∈ ( 𝑀 ..^ 𝑁 ) )
15 eleq1w ⊢ ( 𝑛 = 𝑚 → ( 𝑛 ∈ ( 𝑀 ..^ 𝑁 ) ↔ 𝑚 ∈ ( 𝑀 ..^ 𝑁 ) ) )
16 15 anbi2d ⊢ ( 𝑛 = 𝑚 → ( ( 𝜑 ∧ 𝑛 ∈ ( 𝑀 ..^ 𝑁 ) ) ↔ ( 𝜑 ∧ 𝑚 ∈ ( 𝑀 ..^ 𝑁 ) ) ) )
17 fveq2 ⊢ ( 𝑛 = 𝑚 → ( 𝐹 ‘ 𝑛 ) = ( 𝐹 ‘ 𝑚 ) )
18 fvoveq1 ⊢ ( 𝑛 = 𝑚 → ( 𝐹 ‘ ( 𝑛 + 1 ) ) = ( 𝐹 ‘ ( 𝑚 + 1 ) ) )
19 17 18 sseq12d ⊢ ( 𝑛 = 𝑚 → ( ( 𝐹 ‘ 𝑛 ) ⊆ ( 𝐹 ‘ ( 𝑛 + 1 ) ) ↔ ( 𝐹 ‘ 𝑚 ) ⊆ ( 𝐹 ‘ ( 𝑚 + 1 ) ) ) )
20 16 19 imbi12d ⊢ ( 𝑛 = 𝑚 → ( ( ( 𝜑 ∧ 𝑛 ∈ ( 𝑀 ..^ 𝑁 ) ) → ( 𝐹 ‘ 𝑛 ) ⊆ ( 𝐹 ‘ ( 𝑛 + 1 ) ) ) ↔ ( ( 𝜑 ∧ 𝑚 ∈ ( 𝑀 ..^ 𝑁 ) ) → ( 𝐹 ‘ 𝑚 ) ⊆ ( 𝐹 ‘ ( 𝑚 + 1 ) ) ) ) )
21 20 2 chvarvv ⊢ ( ( 𝜑 ∧ 𝑚 ∈ ( 𝑀 ..^ 𝑁 ) ) → ( 𝐹 ‘ 𝑚 ) ⊆ ( 𝐹 ‘ ( 𝑚 + 1 ) ) )
22 7 14 21 syl2anc ⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ ( 𝑀 ... 𝑁 ) ) ∧ 𝑚 ∈ ( 𝑛 ..^ 𝑁 ) ) → ( 𝐹 ‘ 𝑚 ) ⊆ ( 𝐹 ‘ ( 𝑚 + 1 ) ) )
23 6 22 ssinc ⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( 𝑀 ... 𝑁 ) ) → ( 𝐹 ‘ 𝑛 ) ⊆ ( 𝐹 ‘ 𝑁 ) )
24 23 3adant3 ⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( 𝑀 ... 𝑁 ) ∧ 𝑥 ∈ ( 𝐹 ‘ 𝑛 ) ) → ( 𝐹 ‘ 𝑛 ) ⊆ ( 𝐹 ‘ 𝑁 ) )
25 simp3 ⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( 𝑀 ... 𝑁 ) ∧ 𝑥 ∈ ( 𝐹 ‘ 𝑛 ) ) → 𝑥 ∈ ( 𝐹 ‘ 𝑛 ) )
26 24 25 sseldd ⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( 𝑀 ... 𝑁 ) ∧ 𝑥 ∈ ( 𝐹 ‘ 𝑛 ) ) → 𝑥 ∈ ( 𝐹 ‘ 𝑁 ) )
27 26 3exp ⊢ ( 𝜑 → ( 𝑛 ∈ ( 𝑀 ... 𝑁 ) → ( 𝑥 ∈ ( 𝐹 ‘ 𝑛 ) → 𝑥 ∈ ( 𝐹 ‘ 𝑁 ) ) ) )
28 27 rexlimdv ⊢ ( 𝜑 → ( ∃ 𝑛 ∈ ( 𝑀 ... 𝑁 ) 𝑥 ∈ ( 𝐹 ‘ 𝑛 ) → 𝑥 ∈ ( 𝐹 ‘ 𝑁 ) ) )
29 28 imp ⊢ ( ( 𝜑 ∧ ∃ 𝑛 ∈ ( 𝑀 ... 𝑁 ) 𝑥 ∈ ( 𝐹 ‘ 𝑛 ) ) → 𝑥 ∈ ( 𝐹 ‘ 𝑁 ) )
30 4 29 syldan ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ∪ 𝑛 ∈ ( 𝑀 ... 𝑁 ) ( 𝐹 ‘ 𝑛 ) ) → 𝑥 ∈ ( 𝐹 ‘ 𝑁 ) )
31 30 ralrimiva ⊢ ( 𝜑 → ∀ 𝑥 ∈ ∪ 𝑛 ∈ ( 𝑀 ... 𝑁 ) ( 𝐹 ‘ 𝑛 ) 𝑥 ∈ ( 𝐹 ‘ 𝑁 ) )
32 dfss3 ⊢ ( ∪ 𝑛 ∈ ( 𝑀 ... 𝑁 ) ( 𝐹 ‘ 𝑛 ) ⊆ ( 𝐹 ‘ 𝑁 ) ↔ ∀ 𝑥 ∈ ∪ 𝑛 ∈ ( 𝑀 ... 𝑁 ) ( 𝐹 ‘ 𝑛 ) 𝑥 ∈ ( 𝐹 ‘ 𝑁 ) )
33 31 32 sylibr ⊢ ( 𝜑 → ∪ 𝑛 ∈ ( 𝑀 ... 𝑁 ) ( 𝐹 ‘ 𝑛 ) ⊆ ( 𝐹 ‘ 𝑁 ) )
34 eluzfz2 ⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) → 𝑁 ∈ ( 𝑀 ... 𝑁 ) )
35 1 34 syl ⊢ ( 𝜑 → 𝑁 ∈ ( 𝑀 ... 𝑁 ) )
36 fveq2 ⊢ ( 𝑛 = 𝑁 → ( 𝐹 ‘ 𝑛 ) = ( 𝐹 ‘ 𝑁 ) )
37 36 ssiun2s ⊢ ( 𝑁 ∈ ( 𝑀 ... 𝑁 ) → ( 𝐹 ‘ 𝑁 ) ⊆ ∪ 𝑛 ∈ ( 𝑀 ... 𝑁 ) ( 𝐹 ‘ 𝑛 ) )
38 35 37 syl ⊢ ( 𝜑 → ( 𝐹 ‘ 𝑁 ) ⊆ ∪ 𝑛 ∈ ( 𝑀 ... 𝑁 ) ( 𝐹 ‘ 𝑛 ) )
39 33 38 eqssd ⊢ ( 𝜑 → ∪ 𝑛 ∈ ( 𝑀 ... 𝑁 ) ( 𝐹 ‘ 𝑛 ) = ( 𝐹 ‘ 𝑁 ) )