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Theorem smodm2 6292
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 6288 . 2 (Smo 𝐹 → Ord dom 𝐹)
2 fndm 5313 . . . 4 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
3 ordeq 4371 . . . 4 (dom 𝐹 = 𝐴 → (Ord dom 𝐹 ↔ Ord 𝐴))
42, 3syl 14 . . 3 (𝐹 Fn 𝐴 → (Ord dom 𝐹 ↔ Ord 𝐴))
54biimpa 296 . 2 ((𝐹 Fn 𝐴 ∧ Ord dom 𝐹) → Ord 𝐴)
61, 5sylan2 286 1 ((𝐹 Fn 𝐴 ∧ Smo 𝐹) → Ord 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1353  Ord word 4361  dom cdm 4625   Fn wfn 5209  Smo wsmo 6282
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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-in 3135  df-ss 3142  df-uni 3810  df-tr 4101  df-iord 4365  df-fn 5217  df-smo 6283
This theorem is referenced by: (None)
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