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Theorem smofvon2 7438
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 7429 . . . 4 (Smo 𝐹 ↔ (𝐹:dom 𝐹⟶On ∧ Ord dom 𝐹 ∧ ∀𝑥 ∈ dom 𝐹𝑦𝑥 (𝐹𝑦) ∈ (𝐹𝑥)))
21simp1bi 1074 . . 3 (Smo 𝐹𝐹:dom 𝐹⟶On)
3 ffvelrn 6343 . . . 4 ((𝐹:dom 𝐹⟶On ∧ 𝐵 ∈ dom 𝐹) → (𝐹𝐵) ∈ On)
43expcom 451 . . 3 (𝐵 ∈ dom 𝐹 → (𝐹:dom 𝐹⟶On → (𝐹𝐵) ∈ On))
52, 4syl5 34 . 2 (𝐵 ∈ dom 𝐹 → (Smo 𝐹 → (𝐹𝐵) ∈ On))
6 ndmfv 6205 . . . 4 𝐵 ∈ dom 𝐹 → (𝐹𝐵) = ∅)
7 0elon 5766 . . . 4 ∅ ∈ On
86, 7syl6eqel 2707 . . 3 𝐵 ∈ dom 𝐹 → (𝐹𝐵) ∈ On)
98a1d 25 . 2 𝐵 ∈ dom 𝐹 → (Smo 𝐹 → (𝐹𝐵) ∈ On))
105, 9pm2.61i 176 1 (Smo 𝐹 → (𝐹𝐵) ∈ On)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wcel 1988  wral 2909  c0 3907  dom cdm 5104  Ord word 5710  Oncon0 5711  wf 5872  cfv 5876  Smo wsmo 7427
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-tr 4744  df-id 5014  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-ord 5714  df-on 5715  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-fv 5884  df-smo 7428
This theorem is referenced by:  smo11  7446  smoord  7447  smoword  7448  smogt  7449
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