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Theorem smofvon2dm 5828
Description: The function values of a strictly monotone ordinal function are ordinals. (Contributed by Mario Carneiro, 12-Mar-2013.)
Assertion
Ref Expression
smofvon2dm ((Smo 𝐹 B dom 𝐹) → (𝐹B) On)

Proof of Theorem smofvon2dm
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dfsmo2 5819 . . 3 (Smo 𝐹 ↔ (𝐹:dom 𝐹⟶On Ord dom 𝐹 x dom 𝐹y x (𝐹y) (𝐹x)))
21simp1bi 907 . 2 (Smo 𝐹𝐹:dom 𝐹⟶On)
32ffvelrnda 5225 1 ((Smo 𝐹 B dom 𝐹) → (𝐹B) On)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   wcel 1375  wral 2283  Ord word 4046  Oncon0 4047  dom cdm 4270  wf 4823  cfv 4827  Smo wsmo 5817
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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1364  ax-ie2 1365  ax-8 1377  ax-10 1378  ax-11 1379  ax-i12 1380  ax-bnd 1381  ax-4 1382  ax-14 1387  ax-17 1401  ax-i9 1405  ax-ial 1410  ax-i5r 1411  ax-ext 2005  ax-sep 3848  ax-pow 3900  ax-pr 3917
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1629  df-eu 1886  df-mo 1887  df-clab 2010  df-cleq 2016  df-clel 2019  df-nfc 2150  df-ral 2288  df-rex 2289  df-v 2536  df-sbc 2741  df-un 2898  df-in 2900  df-ss 2907  df-pw 3335  df-sn 3355  df-pr 3356  df-op 3358  df-uni 3554  df-br 3738  df-opab 3792  df-tr 3828  df-id 4003  df-iord 4050  df-xp 4276  df-rel 4277  df-cnv 4278  df-co 4279  df-dm 4280  df-rn 4281  df-iota 4792  df-fun 4829  df-fn 4830  df-f 4831  df-fv 4835  df-smo 5818
This theorem is referenced by: (None)
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