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Metamath Proof Explorer

Definition df-smo

Description: Definition of a strictly monotone ordinal function. Definition 7.46 in TakeutiZaring p. 50. (Contributed by Andrew Salmon, 15-Nov-2011)

Ref Expression
Assertion df-smo ( Smo 𝐴 ↔ ( 𝐴 : dom 𝐴 ⟶ On ∧ Ord dom 𝐴 ∧ ∀ 𝑥 ∈ dom 𝐴 ∀ 𝑦 ∈ dom 𝐴 ( 𝑥 ∈ 𝑦 → ( 𝐴 ‘ 𝑥 ) ∈ ( 𝐴 ‘ 𝑦 ) ) ) )

Detailed syntax breakdown

Step Hyp Ref Expression
0 cA ⊢ 𝐴
1 0 wsmo ⊢ Smo 𝐴
2 0 cdm ⊢ dom 𝐴
3 con0 ⊢ On
4 2 3 0 wf ⊢ 𝐴 : dom 𝐴 ⟶ On
5 2 word ⊢ Ord dom 𝐴
6 vx ⊢ 𝑥
7 vy ⊢ 𝑦
8 6 cv ⊢ 𝑥
9 7 cv ⊢ 𝑦
10 8 9 wcel ⊢ 𝑥 ∈ 𝑦
11 8 0 cfv ⊢ ( 𝐴 ‘ 𝑥 )
12 9 0 cfv ⊢ ( 𝐴 ‘ 𝑦 )
13 11 12 wcel ⊢ ( 𝐴 ‘ 𝑥 ) ∈ ( 𝐴 ‘ 𝑦 )
14 10 13 wi ⊢ ( 𝑥 ∈ 𝑦 → ( 𝐴 ‘ 𝑥 ) ∈ ( 𝐴 ‘ 𝑦 ) )
15 14 7 2 wral ⊢ ∀ 𝑦 ∈ dom 𝐴 ( 𝑥 ∈ 𝑦 → ( 𝐴 ‘ 𝑥 ) ∈ ( 𝐴 ‘ 𝑦 ) )
16 15 6 2 wral ⊢ ∀ 𝑥 ∈ dom 𝐴 ∀ 𝑦 ∈ dom 𝐴 ( 𝑥 ∈ 𝑦 → ( 𝐴 ‘ 𝑥 ) ∈ ( 𝐴 ‘ 𝑦 ) )
17 4 5 16 w3a ⊢ ( 𝐴 : dom 𝐴 ⟶ On ∧ Ord dom 𝐴 ∧ ∀ 𝑥 ∈ dom 𝐴 ∀ 𝑦 ∈ dom 𝐴 ( 𝑥 ∈ 𝑦 → ( 𝐴 ‘ 𝑥 ) ∈ ( 𝐴 ‘ 𝑦 ) ) )
18 1 17 wb ⊢ ( Smo 𝐴 ↔ ( 𝐴 : dom 𝐴 ⟶ On ∧ Ord dom 𝐴 ∧ ∀ 𝑥 ∈ dom 𝐴 ∀ 𝑦 ∈ dom 𝐴 ( 𝑥 ∈ 𝑦 → ( 𝐴 ‘ 𝑥 ) ∈ ( 𝐴 ‘ 𝑦 ) ) ) )