Metamath Proof Explorer


Theorem cfle

Description: Cofinality is bounded by its argument. Exercise 1 of TakeutiZaring p. 102. (Contributed by NM, 26-Apr-2004) (Revised by Mario Carneiro, 15-Sep-2013)

Ref Expression
Assertion cfle ( cf ‘ 𝐴 ) ⊆ 𝐴

Proof

Step Hyp Ref Expression
1 cflecard ( cf ‘ 𝐴 ) ⊆ ( card ‘ 𝐴 )
2 cardonle ( 𝐴 ∈ On → ( card ‘ 𝐴 ) ⊆ 𝐴 )
3 1 2 sstrid ( 𝐴 ∈ On → ( cf ‘ 𝐴 ) ⊆ 𝐴 )
4 cff cf : On ⟶ On
5 4 fdmi dom cf = On
6 5 eleq2i ( 𝐴 ∈ dom cf ↔ 𝐴 ∈ On )
7 ndmfv ( ¬ 𝐴 ∈ dom cf → ( cf ‘ 𝐴 ) = ∅ )
8 6 7 sylnbir ( ¬ 𝐴 ∈ On → ( cf ‘ 𝐴 ) = ∅ )
9 0ss ∅ ⊆ 𝐴
10 8 9 eqsstrdi ( ¬ 𝐴 ∈ On → ( cf ‘ 𝐴 ) ⊆ 𝐴 )
11 3 10 pm2.61i ( cf ‘ 𝐴 ) ⊆ 𝐴