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

Theorem smofvon2dm 5852
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 5843 . . 3  Smo 
F  F : dom  F --> On  Ord  dom  F  dom  F  F `  F `
21simp1bi 918 . 2  Smo 
F  F : dom  F --> On
32ffvelrnda 5245 1  Smo  F  dom  F  F `  On
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wcel 1390  wral 2300   Ord word 4065   Oncon0 4066   dom cdm 4288   -->wf 4841   ` cfv 4845   Smo wsmo 5841
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-tr 3846  df-id 4021  df-iord 4069  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-fv 4853  df-smo 5842
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator