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Definition df-smo 7428
Description: Definition of a strictly monotone ordinal function. Definition 7.46 in [TakeutiZaring] p. 50. (Contributed by Andrew Salmon, 15-Nov-2011.)
Assertion
Ref Expression
df-smo (Smo 𝐴 ↔ (𝐴:dom 𝐴⟶On ∧ Ord dom 𝐴 ∧ ∀𝑥 ∈ dom 𝐴𝑦 ∈ dom 𝐴(𝑥𝑦 → (𝐴𝑥) ∈ (𝐴𝑦))))
Distinct variable group:   𝑥,𝑦,𝐴

Detailed syntax breakdown of Definition df-smo
StepHypRef Expression
1 cA . . 3 class 𝐴
21wsmo 7427 . 2 wff Smo 𝐴
31cdm 5104 . . . 4 class dom 𝐴
4 con0 5711 . . . 4 class On
53, 4, 1wf 5872 . . 3 wff 𝐴:dom 𝐴⟶On
63word 5710 . . 3 wff Ord dom 𝐴
7 vx . . . . . . 7 setvar 𝑥
8 vy . . . . . . 7 setvar 𝑦
97, 8wel 1989 . . . . . 6 wff 𝑥𝑦
107cv 1480 . . . . . . . 8 class 𝑥
1110, 1cfv 5876 . . . . . . 7 class (𝐴𝑥)
128cv 1480 . . . . . . . 8 class 𝑦
1312, 1cfv 5876 . . . . . . 7 class (𝐴𝑦)
1411, 13wcel 1988 . . . . . 6 wff (𝐴𝑥) ∈ (𝐴𝑦)
159, 14wi 4 . . . . 5 wff (𝑥𝑦 → (𝐴𝑥) ∈ (𝐴𝑦))
1615, 8, 3wral 2909 . . . 4 wff 𝑦 ∈ dom 𝐴(𝑥𝑦 → (𝐴𝑥) ∈ (𝐴𝑦))
1716, 7, 3wral 2909 . . 3 wff 𝑥 ∈ dom 𝐴𝑦 ∈ dom 𝐴(𝑥𝑦 → (𝐴𝑥) ∈ (𝐴𝑦))
185, 6, 17w3a 1036 . 2 wff (𝐴:dom 𝐴⟶On ∧ Ord dom 𝐴 ∧ ∀𝑥 ∈ dom 𝐴𝑦 ∈ dom 𝐴(𝑥𝑦 → (𝐴𝑥) ∈ (𝐴𝑦)))
192, 18wb 196 1 wff (Smo 𝐴 ↔ (𝐴:dom 𝐴⟶On ∧ Ord dom 𝐴 ∧ ∀𝑥 ∈ dom 𝐴𝑦 ∈ dom 𝐴(𝑥𝑦 → (𝐴𝑥) ∈ (𝐴𝑦))))
Colors of variables: wff setvar class
This definition is referenced by:  dfsmo2  7429  issmo  7430  smoeq  7432  smodm  7433  smores  7434  smofvon  7441  smoel  7442  smoiso  7444
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