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Theorem smodm2 5851
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 5847 . 2 (Smo 𝐹 → Ord dom 𝐹)
2 fndm 4941 . . . 4 (𝐹 Fn A → dom 𝐹 = A)
3 ordeq 4075 . . . 4 (dom 𝐹 = A → (Ord dom 𝐹 ↔ Ord A))
42, 3syl 14 . . 3 (𝐹 Fn A → (Ord dom 𝐹 ↔ Ord A))
54biimpa 280 . 2 ((𝐹 Fn A Ord dom 𝐹) → Ord A)
61, 5sylan2 270 1 ((𝐹 Fn A Smo 𝐹) → Ord A)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1242  Ord word 4065  dom cdm 4288   Fn wfn 4840  Smo wsmo 5841
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-in 2918  df-ss 2925  df-uni 3572  df-tr 3846  df-iord 4069  df-fn 4848  df-smo 5842
This theorem is referenced by: (None)
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