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Theorem smodm2 7622
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 7618 . 2 (Smo 𝐹 → Ord dom 𝐹)
2 fndm 6151 . . . 4 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
3 ordeq 5891 . . . 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 1632  dom cdm 5266  Ord word 5883   Fn wfn 6044  Smo wsmo 7612
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-in 3722  df-ss 3729  df-uni 4589  df-tr 4905  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-ord 5887  df-fn 6052  df-smo 7613
This theorem is referenced by:  smo11  7631  smoord  7632  smoword  7633  smogt  7634  smorndom  7635  coftr  9307
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