Theorem smodm2 8320

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 8316	. 2 ⊢ (Smo 𝐹 → Ord dom 𝐹)
2		fndm 6619	. . . 4 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
3		ordeq 6348	. . . 4 ⊢ (dom 𝐹 = 𝐴 → (Ord dom 𝐹 ↔ Ord 𝐴))
4	2, 3	syl 17	. . 3 ⊢ (𝐹 Fn 𝐴 → (Ord dom 𝐹 ↔ Ord 𝐴))
5	4	biimpa 480	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ Ord dom 𝐹) → Ord 𝐴)
6	1, 5	sylan2 602	1 ⊢ ((𝐹 Fn 𝐴 ∧ Smo 𝐹) → Ord 𝐴)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1559  dom cdm 5643  Ord word 6340   Fn wfn 6511  Smo wsmo 8310

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-3an 1099  df-tru 1562  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3076  df-v 3455  df-ss 3919  df-uni 4863  df-tr 5205  df-po 5551  df-so 5552  df-fr 5596  df-we 5598  df-ord 6344  df-fn 6519  df-smo 8311

This theorem is referenced by:  smo11  8329  smoord  8330  smoword  8331  smogt  8332  smocdmdom  8333  coftr  10224