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 𝐴𝑦 ∈ dom 𝐴 ( 𝑥𝑦 → ( 𝐴𝑥 ) ∈ ( 𝐴𝑦 ) ) ) )

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 𝑥𝑦
11 8 0 cfv ( 𝐴𝑥 )
12 9 0 cfv ( 𝐴𝑦 )
13 11 12 wcel ( 𝐴𝑥 ) ∈ ( 𝐴𝑦 )
14 10 13 wi ( 𝑥𝑦 → ( 𝐴𝑥 ) ∈ ( 𝐴𝑦 ) )
15 14 7 2 wral 𝑦 ∈ dom 𝐴 ( 𝑥𝑦 → ( 𝐴𝑥 ) ∈ ( 𝐴𝑦 ) )
16 15 6 2 wral 𝑥 ∈ dom 𝐴𝑦 ∈ dom 𝐴 ( 𝑥𝑦 → ( 𝐴𝑥 ) ∈ ( 𝐴𝑦 ) )
17 4 5 16 w3a ( 𝐴 : dom 𝐴 ⟶ On ∧ Ord dom 𝐴 ∧ ∀ 𝑥 ∈ dom 𝐴𝑦 ∈ dom 𝐴 ( 𝑥𝑦 → ( 𝐴𝑥 ) ∈ ( 𝐴𝑦 ) ) )
18 1 17 wb ( Smo 𝐴 ↔ ( 𝐴 : dom 𝐴 ⟶ On ∧ Ord dom 𝐴 ∧ ∀ 𝑥 ∈ dom 𝐴𝑦 ∈ dom 𝐴 ( 𝑥𝑦 → ( 𝐴𝑥 ) ∈ ( 𝐴𝑦 ) ) ) )