Theorem smodm2 8285

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 8281	. 2 ⊢ (Smo 𝐹 → Ord dom 𝐹)
2		fndm 6589	. . . 4 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
3		ordeq 6318	. . . 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 476	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ Ord dom 𝐹) → Ord 𝐴)
6	1, 5	sylan2 593	1 ⊢ ((𝐹 Fn 𝐴 ∧ Smo 𝐹) → Ord 𝐴)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540  dom cdm 5623  Ord word 6310   Fn wfn 6481  Smo wsmo 8275

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-v 3440  df-ss 3922  df-uni 4862  df-tr 5203  df-po 5531  df-so 5532  df-fr 5576  df-we 5578  df-ord 6314  df-fn 6489  df-smo 8276

This theorem is referenced by:  smo11  8294  smoord  8295  smoword  8296  smogt  8297  smocdmdom  8298  coftr  10186