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Theorem smofvon2dm 6256
Description: The function values of a strictly monotone ordinal function are ordinals. (Contributed by Mario Carneiro, 12-Mar-2013.)
Assertion
Ref Expression
smofvon2dm ((Smo 𝐹𝐵 ∈ 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 6247 . . 3 (Smo 𝐹 ↔ (𝐹:dom 𝐹⟶On ∧ Ord dom 𝐹 ∧ ∀𝑥 ∈ dom 𝐹𝑦𝑥 (𝐹𝑦) ∈ (𝐹𝑥)))
21simp1bi 1001 . 2 (Smo 𝐹𝐹:dom 𝐹⟶On)
32ffvelrnda 5615 1 ((Smo 𝐹𝐵 ∈ dom 𝐹) → (𝐹𝐵) ∈ On)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wcel 2135  wral 2442  Ord word 4335  Oncon0 4336  dom cdm 4599  wf 5179  cfv 5183  Smo wsmo 6245
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-14 2138  ax-ext 2146  ax-sep 4095  ax-pow 4148  ax-pr 4182
This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ral 2447  df-rex 2448  df-v 2724  df-sbc 2948  df-un 3116  df-in 3118  df-ss 3125  df-pw 3556  df-sn 3577  df-pr 3578  df-op 3580  df-uni 3785  df-br 3978  df-opab 4039  df-tr 4076  df-id 4266  df-iord 4339  df-xp 4605  df-rel 4606  df-cnv 4607  df-co 4608  df-dm 4609  df-rn 4610  df-iota 5148  df-fun 5185  df-fn 5186  df-f 5187  df-fv 5191  df-smo 6246
This theorem is referenced by: (None)
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