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Theorem smofvon2 8004
 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 7995 . . . 4 (Smo 𝐹 ↔ (𝐹:dom 𝐹⟶On ∧ Ord dom 𝐹 ∧ ∀𝑥 ∈ dom 𝐹𝑦𝑥 (𝐹𝑦) ∈ (𝐹𝑥)))
21simp1bi 1143 . . 3 (Smo 𝐹𝐹:dom 𝐹⟶On)
3 ffvelrn 6841 . . . 4 ((𝐹:dom 𝐹⟶On ∧ 𝐵 ∈ dom 𝐹) → (𝐹𝐵) ∈ On)
43expcom 418 . . 3 (𝐵 ∈ dom 𝐹 → (𝐹:dom 𝐹⟶On → (𝐹𝐵) ∈ On))
52, 4syl5 34 . 2 (𝐵 ∈ dom 𝐹 → (Smo 𝐹 → (𝐹𝐵) ∈ On))
6 ndmfv 6689 . . . 4 𝐵 ∈ dom 𝐹 → (𝐹𝐵) = ∅)
7 0elon 6223 . . . 4 ∅ ∈ On
86, 7eqeltrdi 2861 . . 3 𝐵 ∈ dom 𝐹 → (𝐹𝐵) ∈ On)
98a1d 25 . 2 𝐵 ∈ dom 𝐹 → (Smo 𝐹 → (𝐹𝐵) ∈ On))
105, 9pm2.61i 185 1 (Smo 𝐹 → (𝐹𝐵) ∈ On)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2112  ∀wral 3071  ∅c0 4226  dom cdm 5525  Ord word 6169  Oncon0 6170  ⟶wf 6332  ‘cfv 6336  Smo wsmo 7993 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5170  ax-nul 5177  ax-pr 5299 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-ral 3076  df-rex 3077  df-rab 3080  df-v 3412  df-sbc 3698  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-nul 4227  df-if 4422  df-pw 4497  df-sn 4524  df-pr 4526  df-op 4530  df-uni 4800  df-br 5034  df-opab 5096  df-tr 5140  df-id 5431  df-po 5444  df-so 5445  df-fr 5484  df-we 5486  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-ord 6173  df-on 6174  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-fv 6344  df-smo 7994 This theorem is referenced by:  smo11  8012  smoord  8013  smoword  8014  smogt  8015
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