Metamath Proof Explorer


Theorem rnmptlb

Description: Boundness below of the range of a function in maps-to notation. (Contributed by Glauco Siliprandi, 23-Oct-2021)

Ref Expression
Hypothesis rnmptlb.1 ( 𝜑 → ∃ 𝑦 ∈ ℝ ∀ 𝑥𝐴 𝑦𝐵 )
Assertion rnmptlb ( 𝜑 → ∃ 𝑦 ∈ ℝ ∀ 𝑧 ∈ ran ( 𝑥𝐴𝐵 ) 𝑦𝑧 )

Proof

Step Hyp Ref Expression
1 rnmptlb.1 ( 𝜑 → ∃ 𝑦 ∈ ℝ ∀ 𝑥𝐴 𝑦𝐵 )
2 eqid ( 𝑥𝐴𝐵 ) = ( 𝑥𝐴𝐵 )
3 2 elrnmpt ( 𝑧 ∈ V → ( 𝑧 ∈ ran ( 𝑥𝐴𝐵 ) ↔ ∃ 𝑥𝐴 𝑧 = 𝐵 ) )
4 3 elv ( 𝑧 ∈ ran ( 𝑥𝐴𝐵 ) ↔ ∃ 𝑥𝐴 𝑧 = 𝐵 )
5 nfra1 𝑥𝑥𝐴 𝑤𝐵
6 nfv 𝑥 𝑤𝑧
7 rspa ( ( ∀ 𝑥𝐴 𝑤𝐵𝑥𝐴 ) → 𝑤𝐵 )
8 7 3adant3 ( ( ∀ 𝑥𝐴 𝑤𝐵𝑥𝐴𝑧 = 𝐵 ) → 𝑤𝐵 )
9 simp3 ( ( ∀ 𝑥𝐴 𝑤𝐵𝑥𝐴𝑧 = 𝐵 ) → 𝑧 = 𝐵 )
10 8 9 breqtrrd ( ( ∀ 𝑥𝐴 𝑤𝐵𝑥𝐴𝑧 = 𝐵 ) → 𝑤𝑧 )
11 10 3exp ( ∀ 𝑥𝐴 𝑤𝐵 → ( 𝑥𝐴 → ( 𝑧 = 𝐵𝑤𝑧 ) ) )
12 5 6 11 rexlimd ( ∀ 𝑥𝐴 𝑤𝐵 → ( ∃ 𝑥𝐴 𝑧 = 𝐵𝑤𝑧 ) )
13 12 imp ( ( ∀ 𝑥𝐴 𝑤𝐵 ∧ ∃ 𝑥𝐴 𝑧 = 𝐵 ) → 𝑤𝑧 )
14 13 adantll ( ( ( ( 𝜑𝑤 ∈ ℝ ) ∧ ∀ 𝑥𝐴 𝑤𝐵 ) ∧ ∃ 𝑥𝐴 𝑧 = 𝐵 ) → 𝑤𝑧 )
15 4 14 sylan2b ( ( ( ( 𝜑𝑤 ∈ ℝ ) ∧ ∀ 𝑥𝐴 𝑤𝐵 ) ∧ 𝑧 ∈ ran ( 𝑥𝐴𝐵 ) ) → 𝑤𝑧 )
16 15 ralrimiva ( ( ( 𝜑𝑤 ∈ ℝ ) ∧ ∀ 𝑥𝐴 𝑤𝐵 ) → ∀ 𝑧 ∈ ran ( 𝑥𝐴𝐵 ) 𝑤𝑧 )
17 breq1 ( 𝑦 = 𝑤 → ( 𝑦𝐵𝑤𝐵 ) )
18 17 ralbidv ( 𝑦 = 𝑤 → ( ∀ 𝑥𝐴 𝑦𝐵 ↔ ∀ 𝑥𝐴 𝑤𝐵 ) )
19 18 cbvrexvw ( ∃ 𝑦 ∈ ℝ ∀ 𝑥𝐴 𝑦𝐵 ↔ ∃ 𝑤 ∈ ℝ ∀ 𝑥𝐴 𝑤𝐵 )
20 1 19 sylib ( 𝜑 → ∃ 𝑤 ∈ ℝ ∀ 𝑥𝐴 𝑤𝐵 )
21 16 20 reximddv3 ( 𝜑 → ∃ 𝑤 ∈ ℝ ∀ 𝑧 ∈ ran ( 𝑥𝐴𝐵 ) 𝑤𝑧 )
22 breq1 ( 𝑤 = 𝑦 → ( 𝑤𝑧𝑦𝑧 ) )
23 22 ralbidv ( 𝑤 = 𝑦 → ( ∀ 𝑧 ∈ ran ( 𝑥𝐴𝐵 ) 𝑤𝑧 ↔ ∀ 𝑧 ∈ ran ( 𝑥𝐴𝐵 ) 𝑦𝑧 ) )
24 23 cbvrexvw ( ∃ 𝑤 ∈ ℝ ∀ 𝑧 ∈ ran ( 𝑥𝐴𝐵 ) 𝑤𝑧 ↔ ∃ 𝑦 ∈ ℝ ∀ 𝑧 ∈ ran ( 𝑥𝐴𝐵 ) 𝑦𝑧 )
25 21 24 sylib ( 𝜑 → ∃ 𝑦 ∈ ℝ ∀ 𝑧 ∈ ran ( 𝑥𝐴𝐵 ) 𝑦𝑧 )