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Theorem smofvon2dm 5831
 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 5822 . . 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 1374  ∀wral 2282  Ord word 4046  Oncon0 4047  dom cdm 4270  ⟶wf 4823  ‘cfv 4827  Smo wsmo 5820 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 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3847  ax-pow 3899  ax-pr 3916 This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  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 2897  df-in 2899  df-ss 2906  df-pw 3334  df-sn 3354  df-pr 3355  df-op 3357  df-uni 3553  df-br 3737  df-opab 3791  df-tr 3827  df-id 4002  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 5821 This theorem is referenced by: (None)
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