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Theorem smofvon2dm 5849
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 5840 . . 3 (Smo 𝐹 ↔ (𝐹:dom 𝐹⟶On Ord dom 𝐹 x dom 𝐹y x (𝐹y) (𝐹x)))
21simp1bi 918 . 2 (Smo 𝐹𝐹:dom 𝐹⟶On)
32ffvelrnda 5243 1 ((Smo 𝐹 B dom 𝐹) → (𝐹B) On)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   wcel 1390  wral 2300  Ord word 4064  Oncon0 4065  dom cdm 4287  wf 4840  cfv 4844  Smo wsmo 5838
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 3865  ax-pow 3917  ax-pr 3934
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 3352  df-sn 3372  df-pr 3373  df-op 3375  df-uni 3571  df-br 3755  df-opab 3809  df-tr 3845  df-id 4020  df-iord 4068  df-xp 4293  df-rel 4294  df-cnv 4295  df-co 4296  df-dm 4297  df-rn 4298  df-iota 4809  df-fun 4846  df-fn 4847  df-f 4848  df-fv 4852  df-smo 5839
This theorem is referenced by: (None)
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