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Theorem smofvon2dm 5798
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 5789 . . 3 (Smo 𝐹 ↔ (𝐹:dom 𝐹⟶On Ord dom 𝐹 x dom 𝐹y x (𝐹y) (𝐹x)))
21simp1bi 909 . 2 (Smo 𝐹𝐹:dom 𝐹⟶On)
32ffvelrnda 5194 1 ((Smo 𝐹 B dom 𝐹) → (𝐹B) On)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   wcel 1375  wral 2282  Ord word 4023  Oncon0 4024  dom cdm 4238  wf 4792  cfv 4796  Smo wsmo 5787
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 1315  ax-7 1316  ax-gen 1317  ax-ie1 1362  ax-ie2 1363  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 2004  ax-sep 3827  ax-pow 3879  ax-pr 3896
This theorem depends on definitions:  df-bi 110  df-3an 877  df-tru 1231  df-nf 1329  df-sb 1628  df-eu 1884  df-mo 1885  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2287  df-rex 2288  df-v 2535  df-sbc 2740  df-un 2900  df-in 2902  df-ss 2909  df-pw 3313  df-sn 3333  df-pr 3334  df-op 3336  df-uni 3533  df-br 3717  df-opab 3771  df-tr 3807  df-id 3983  df-iord 4027  df-xp 4244  df-rel 4245  df-cnv 4246  df-co 4247  df-dm 4248  df-rn 4249  df-iota 4761  df-fun 4798  df-fn 4799  df-f 4800  df-fv 4804  df-smo 5788
This theorem is referenced by: (None)
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