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Theorem smofvon2 8362
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 8353 . . . 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 7083 . . . 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 6926 . . . 4 𝐵 ∈ dom 𝐹 → (𝐹𝐵) = ∅)
7 0elon 6418 . . . 4 ∅ ∈ On
86, 7eqeltrdi 2840 . . 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 2105  wral 3060  c0 4322  dom cdm 5676  Ord word 6363  Oncon0 6364  wf 6539  cfv 6543  Smo wsmo 8351
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-tr 5266  df-id 5574  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-ord 6367  df-on 6368  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-smo 8352
This theorem is referenced by:  smo11  8370  smoord  8371  smoword  8372  smogt  8373
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