Theorem smodm2 6388

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 6384	. 2 ⊢ (Smo 𝐹 → Ord dom 𝐹)
2		fndm 5378	. . . 4 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
3		ordeq 4423	. . . 4 ⊢ (dom 𝐹 = 𝐴 → (Ord dom 𝐹 ↔ Ord 𝐴))
4	2, 3	syl 14	. . 3 ⊢ (𝐹 Fn 𝐴 → (Ord dom 𝐹 ↔ Ord 𝐴))
5	4	biimpa 296	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ Ord dom 𝐹) → Ord 𝐴)
6	1, 5	sylan2 286	1 ⊢ ((𝐹 Fn 𝐴 ∧ Smo 𝐹) → Ord 𝐴)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1373  Ord word 4413  dom cdm 4679   Fn wfn 5271  Smo wsmo 6378

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-in 3173  df-ss 3180  df-uni 3853  df-tr 4147  df-iord 4417  df-fn 5279  df-smo 6379

This theorem is referenced by: (None)