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Metamath Proof Explorer

Theorem iccen

Description: Any nontrivial closed interval is equinumerous to the unit interval. (Contributed by Mario Carneiro, 26-Jul-2014) (Revised by Mario Carneiro, 8-Sep-2015)

Ref Expression
Assertion iccen ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵 ) → ( 0 [,] 1 ) ≈ ( 𝐴 [,] 𝐵 ) )

Proof

Step Hyp Ref Expression
1 ovex ⊢ ( 0 [,] 1 ) ∈ V
2 ovex ⊢ ( 𝐴 [,] 𝐵 ) ∈ V
3 eqid ⊢ ( 𝑥 ∈ ( 0 [,] 1 ) ↦ ( ( 𝑥 · 𝐵 ) + ( ( 1 − 𝑥 ) · 𝐴 ) ) ) = ( 𝑥 ∈ ( 0 [,] 1 ) ↦ ( ( 𝑥 · 𝐵 ) + ( ( 1 − 𝑥 ) · 𝐴 ) ) )
4 3 iccf1o ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵 ) → ( ( 𝑥 ∈ ( 0 [,] 1 ) ↦ ( ( 𝑥 · 𝐵 ) + ( ( 1 − 𝑥 ) · 𝐴 ) ) ) : ( 0 [,] 1 ) –1-1-onto→ ( 𝐴 [,] 𝐵 ) ∧ ◡ ( 𝑥 ∈ ( 0 [,] 1 ) ↦ ( ( 𝑥 · 𝐵 ) + ( ( 1 − 𝑥 ) · 𝐴 ) ) ) = ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ↦ ( ( 𝑦 − 𝐴 ) / ( 𝐵 − 𝐴 ) ) ) ) )
5 4 simpld ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵 ) → ( 𝑥 ∈ ( 0 [,] 1 ) ↦ ( ( 𝑥 · 𝐵 ) + ( ( 1 − 𝑥 ) · 𝐴 ) ) ) : ( 0 [,] 1 ) –1-1-onto→ ( 𝐴 [,] 𝐵 ) )
6 f1oen2g ⊢ ( ( ( 0 [,] 1 ) ∈ V ∧ ( 𝐴 [,] 𝐵 ) ∈ V ∧ ( 𝑥 ∈ ( 0 [,] 1 ) ↦ ( ( 𝑥 · 𝐵 ) + ( ( 1 − 𝑥 ) · 𝐴 ) ) ) : ( 0 [,] 1 ) –1-1-onto→ ( 𝐴 [,] 𝐵 ) ) → ( 0 [,] 1 ) ≈ ( 𝐴 [,] 𝐵 ) )
7 1 2 5 6 mp3an12i ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵 ) → ( 0 [,] 1 ) ≈ ( 𝐴 [,] 𝐵 ) )