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Theorem smodm2 7609
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
StepHypRef Expression
1 smodm 7605 . 2 (Smo 𝐹 → Ord dom 𝐹)
2 fndm 6139 . . . 4 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
3 ordeq 5879 . . . 4 (dom 𝐹 = 𝐴 → (Ord dom 𝐹 ↔ Ord 𝐴))
42, 3syl 17 . . 3 (𝐹 Fn 𝐴 → (Ord dom 𝐹 ↔ Ord 𝐴))
54biimpa 502 . 2 ((𝐹 Fn 𝐴 ∧ Ord dom 𝐹) → Ord 𝐴)
61, 5sylan2 492 1 ((𝐹 Fn 𝐴 ∧ Smo 𝐹) → Ord 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1620  dom cdm 5254  Ord word 5871   Fn wfn 6032  Smo wsmo 7599
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ral 3043  df-rex 3044  df-in 3710  df-ss 3717  df-uni 4577  df-tr 4893  df-po 5175  df-so 5176  df-fr 5213  df-we 5215  df-ord 5875  df-fn 6040  df-smo 7600
This theorem is referenced by:  smo11  7618  smoord  7619  smoword  7620  smogt  7621  smorndom  7622  coftr  9258
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