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Theorem smofvon2 8395
Description: The function values of a strictly monotone ordinal function are ordinals. (Contributed by Mario Carneiro, 12-Mar-2013.)
Assertion
Ref Expression
smofvon2 (Smo 𝐹 → (𝐹𝐵) ∈ On)

Proof of Theorem smofvon2
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dfsmo2 8386 . . . 4 (Smo 𝐹 ↔ (𝐹:dom 𝐹⟶On ∧ Ord dom 𝐹 ∧ ∀𝑥 ∈ dom 𝐹𝑦𝑥 (𝐹𝑦) ∈ (𝐹𝑥)))
21simp1bi 1144 . . 3 (Smo 𝐹𝐹:dom 𝐹⟶On)
3 ffvelcdm 7101 . . . 4 ((𝐹:dom 𝐹⟶On ∧ 𝐵 ∈ dom 𝐹) → (𝐹𝐵) ∈ On)
43expcom 413 . . 3 (𝐵 ∈ dom 𝐹 → (𝐹:dom 𝐹⟶On → (𝐹𝐵) ∈ On))
52, 4syl5 34 . 2 (𝐵 ∈ dom 𝐹 → (Smo 𝐹 → (𝐹𝐵) ∈ On))
6 ndmfv 6942 . . . 4 𝐵 ∈ dom 𝐹 → (𝐹𝐵) = ∅)
7 0elon 6440 . . . 4 ∅ ∈ On
86, 7eqeltrdi 2847 . . 3 𝐵 ∈ dom 𝐹 → (𝐹𝐵) ∈ On)
98a1d 25 . 2 𝐵 ∈ dom 𝐹 → (Smo 𝐹 → (𝐹𝐵) ∈ On))
105, 9pm2.61i 182 1 (Smo 𝐹 → (𝐹𝐵) ∈ On)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wcel 2106  wral 3059  c0 4339  dom cdm 5689  Ord word 6385  Oncon0 6386  wf 6559  cfv 6563  Smo wsmo 8384
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-tr 5266  df-id 5583  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-ord 6389  df-on 6390  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-smo 8385
This theorem is referenced by:  smo11  8403  smoord  8404  smoword  8405  smogt  8406
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