Theorem smodm2 7991

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 7987	. 2 ⊢ (Smo 𝐹 → Ord dom 𝐹)
2		fndm 6454	. . . 4 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
3		ordeq 6197	. . . 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 479	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ Ord dom 𝐹) → Ord 𝐴)
6	1, 5	sylan2 594	1 ⊢ ((𝐹 Fn 𝐴 ∧ Smo 𝐹) → Ord 𝐴)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1533  dom cdm 5554  Ord word 6189   Fn wfn 6349  Smo wsmo 7981

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-v 3496  df-in 3942  df-ss 3951  df-uni 4838  df-tr 5172  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-ord 6193  df-fn 6357  df-smo 7982

This theorem is referenced by:  smo11  8000  smoord  8001  smoword  8002  smogt  8003  smorndom  8004  coftr  9694