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Metamath Proof Explorer

Theorem cardfz

Description: The cardinality of a finite set of sequential integers. (See om2uz0i for a description of the hypothesis.) (Contributed by NM, 7-Nov-2008) (Revised by Mario Carneiro, 15-Sep-2013)

Ref Expression
Hypothesis fzennn.1 ⊢ 𝐺 = ( rec ( ( 𝑥 ∈ V ↦ ( 𝑥 + 1 ) ) , 0 ) ↾ ω )
Assertion cardfz ( 𝑁 ∈ ℕ0 → ( card ‘ ( 1 ... 𝑁 ) ) = ( ◡ 𝐺 ‘ 𝑁 ) )

Proof

Step Hyp Ref Expression
1 fzennn.1 ⊢ 𝐺 = ( rec ( ( 𝑥 ∈ V ↦ ( 𝑥 + 1 ) ) , 0 ) ↾ ω )
2 1 fzennn ⊢ ( 𝑁 ∈ ℕ0 → ( 1 ... 𝑁 ) ≈ ( ◡ 𝐺 ‘ 𝑁 ) )
3 carden2b ⊢ ( ( 1 ... 𝑁 ) ≈ ( ◡ 𝐺 ‘ 𝑁 ) → ( card ‘ ( 1 ... 𝑁 ) ) = ( card ‘ ( ◡ 𝐺 ‘ 𝑁 ) ) )
4 2 3 syl ⊢ ( 𝑁 ∈ ℕ0 → ( card ‘ ( 1 ... 𝑁 ) ) = ( card ‘ ( ◡ 𝐺 ‘ 𝑁 ) ) )
5 0z ⊢ 0 ∈ ℤ
6 5 1 om2uzf1oi ⊢ 𝐺 : ω –1-1-onto→ ( ℤ≥ ‘ 0 )
7 elnn0uz ⊢ ( 𝑁 ∈ ℕ0 ↔ 𝑁 ∈ ( ℤ≥ ‘ 0 ) )
8 7 biimpi ⊢ ( 𝑁 ∈ ℕ0 → 𝑁 ∈ ( ℤ≥ ‘ 0 ) )
9 f1ocnvdm ⊢ ( ( 𝐺 : ω –1-1-onto→ ( ℤ≥ ‘ 0 ) ∧ 𝑁 ∈ ( ℤ≥ ‘ 0 ) ) → ( ◡ 𝐺 ‘ 𝑁 ) ∈ ω )
10 6 8 9 sylancr ⊢ ( 𝑁 ∈ ℕ0 → ( ◡ 𝐺 ‘ 𝑁 ) ∈ ω )
11 cardnn ⊢ ( ( ◡ 𝐺 ‘ 𝑁 ) ∈ ω → ( card ‘ ( ◡ 𝐺 ‘ 𝑁 ) ) = ( ◡ 𝐺 ‘ 𝑁 ) )
12 10 11 syl ⊢ ( 𝑁 ∈ ℕ0 → ( card ‘ ( ◡ 𝐺 ‘ 𝑁 ) ) = ( ◡ 𝐺 ‘ 𝑁 ) )
13 4 12 eqtrd ⊢ ( 𝑁 ∈ ℕ0 → ( card ‘ ( 1 ... 𝑁 ) ) = ( ◡ 𝐺 ‘ 𝑁 ) )