ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  smofvon2dm Structured version   Unicode version

Theorem smofvon2dm 5833
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  dom  F  F `  On

Proof of Theorem smofvon2dm
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dfsmo2 5824 . . 3  Smo 
F  F : dom  F --> On  Ord  dom  F  dom  F  F `  F `
21simp1bi 907 . 2  Smo 
F  F : dom  F --> On
32ffvelrnda 5227 1  Smo  F  dom  F  F `  On
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wcel 1374  wral 2284   Ord word 4048   Oncon0 4049   dom cdm 4272   -->wf 4825   ` cfv 4829   Smo wsmo 5822
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 3849  ax-pow 3901  ax-pr 3918
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 2289  df-rex 2290  df-v 2537  df-sbc 2742  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-uni 3555  df-br 3739  df-opab 3793  df-tr 3829  df-id 4004  df-iord 4052  df-xp 4278  df-rel 4279  df-cnv 4280  df-co 4281  df-dm 4282  df-rn 4283  df-iota 4794  df-fun 4831  df-fn 4832  df-f 4833  df-fv 4837  df-smo 5823
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator