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Theorem smodm2 5941
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 5937 . 2 (Smo 𝐹 → Ord dom 𝐹)
2 fndm 5026 . . . 4 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
3 ordeq 4137 . . . 4 (dom 𝐹 = 𝐴 → (Ord dom 𝐹 ↔ Ord 𝐴))
42, 3syl 14 . . 3 (𝐹 Fn 𝐴 → (Ord dom 𝐹 ↔ Ord 𝐴))
54biimpa 284 . 2 ((𝐹 Fn 𝐴 ∧ Ord dom 𝐹) → Ord 𝐴)
61, 5sylan2 274 1 ((𝐹 Fn 𝐴 ∧ Smo 𝐹) → Ord 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102   = wceq 1259  Ord word 4127  dom cdm 4373   Fn wfn 4925  Smo wsmo 5931
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-in 2952  df-ss 2959  df-uni 3609  df-tr 3883  df-iord 4131  df-fn 4933  df-smo 5932
This theorem is referenced by: (None)
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