Theorem smodm2 8344

Description: The domain of a strictly monotone ordinal function is an ordinal. (Contributed by Mario Carneiro, 12-Mar-2013.)

Assertion

Ref	Expression
smodm2	⊢ ((𝐹 Fn 𝐴 ∧ Smo 𝐹) → Ord 𝐴)

Proof of Theorem smodm2

Step	Hyp	Ref	Expression
1		smodm 8340	. 2 ⊢ (Smo 𝐹 → Ord dom 𝐹)
2		fndm 6642	. . . 4 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
3		ordeq 6371	. . . 4 ⊢ (dom 𝐹 = 𝐴 → (Ord dom 𝐹 ↔ Ord 𝐴))
4	2, 3	syl 18	. . 3 ⊢ (𝐹 Fn 𝐴 → (Ord dom 𝐹 ↔ Ord 𝐴))
5	4	biimpa 482	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ Ord dom 𝐹) → Ord 𝐴)
6	1, 5	sylan2 605	1 ⊢ ((𝐹 Fn 𝐴 ∧ Smo 𝐹) → Ord 𝐴)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  dom cdm 5663  Ord word 6363   Fn wfn 6535  Smo wsmo 8334

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3082  df-v 3459  df-ss 3923  df-uni 4875  df-tr 5221  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-ord 6367  df-fn 6543  df-smo 8335

This theorem is used by:  smo11  8353  smoord  8354  smoword  8355  smogt  8356  smocdmdom  8357  coftr  10268