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Theorem smofvon2 7317
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 7308 . . . 4 (Smo 𝐹 ↔ (𝐹:dom 𝐹⟶On ∧ Ord dom 𝐹 ∧ ∀𝑥 ∈ dom 𝐹𝑦𝑥 (𝐹𝑦) ∈ (𝐹𝑥)))
21simp1bi 1068 . . 3 (Smo 𝐹𝐹:dom 𝐹⟶On)
3 ffvelrn 6250 . . . 4 ((𝐹:dom 𝐹⟶On ∧ 𝐵 ∈ dom 𝐹) → (𝐹𝐵) ∈ On)
43expcom 449 . . 3 (𝐵 ∈ dom 𝐹 → (𝐹:dom 𝐹⟶On → (𝐹𝐵) ∈ On))
52, 4syl5 33 . 2 (𝐵 ∈ dom 𝐹 → (Smo 𝐹 → (𝐹𝐵) ∈ On))
6 ndmfv 6113 . . . 4 𝐵 ∈ dom 𝐹 → (𝐹𝐵) = ∅)
7 0elon 5681 . . . 4 ∅ ∈ On
86, 7syl6eqel 2695 . . 3 𝐵 ∈ dom 𝐹 → (𝐹𝐵) ∈ On)
98a1d 25 . 2 𝐵 ∈ dom 𝐹 → (Smo 𝐹 → (𝐹𝐵) ∈ On))
105, 9pm2.61i 174 1 (Smo 𝐹 → (𝐹𝐵) ∈ On)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wcel 1976  wral 2895  c0 3873  dom cdm 5028  Ord word 5625  Oncon0 5626  wf 5786  cfv 5790  Smo wsmo 7306
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-opab 4638  df-tr 4675  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-ord 5629  df-on 5630  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-fv 5798  df-smo 7307
This theorem is referenced by:  smo11  7325  smoord  7326  smoword  7327  smogt  7328
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