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Theorem smodm2 5910
 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 5906 . 2 (Smo 𝐹 → Ord dom 𝐹)
2 fndm 4998 . . . 4 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
3 ordeq 4109 . . . 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 1243  Ord word 4099  dom cdm 4345   Fn wfn 4897  Smo wsmo 5900 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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-in 2924  df-ss 2931  df-uni 3581  df-tr 3855  df-iord 4103  df-fn 4905  df-smo 5901 This theorem is referenced by: (None)
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