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Theorem omsmo 8696
Description: A strictly monotonic ordinal function on the set of natural numbers is one-to-one. (Contributed by NM, 30-Nov-2003.) (Revised by David Abernethy, 1-Jan-2014.)
Assertion
Ref Expression
omsmo (((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹𝑥) ∈ (𝐹‘suc 𝑥)) → 𝐹:ω–1-1𝐴)
Distinct variable group:   𝑥,𝐹
Allowed substitution hint:   𝐴(𝑥)

Proof of Theorem omsmo
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simplr 769 . 2 (((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹𝑥) ∈ (𝐹‘suc 𝑥)) → 𝐹:ω⟶𝐴)
2 omsmolem 8695 . . . . . . . . 9 (𝑧 ∈ ω → (((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹𝑥) ∈ (𝐹‘suc 𝑥)) → (𝑦𝑧 → (𝐹𝑦) ∈ (𝐹𝑧))))
32adantl 481 . . . . . . . 8 ((𝑦 ∈ ω ∧ 𝑧 ∈ ω) → (((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹𝑥) ∈ (𝐹‘suc 𝑥)) → (𝑦𝑧 → (𝐹𝑦) ∈ (𝐹𝑧))))
43imp 406 . . . . . . 7 (((𝑦 ∈ ω ∧ 𝑧 ∈ ω) ∧ ((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹𝑥) ∈ (𝐹‘suc 𝑥))) → (𝑦𝑧 → (𝐹𝑦) ∈ (𝐹𝑧)))
5 omsmolem 8695 . . . . . . . . 9 (𝑦 ∈ ω → (((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹𝑥) ∈ (𝐹‘suc 𝑥)) → (𝑧𝑦 → (𝐹𝑧) ∈ (𝐹𝑦))))
65adantr 480 . . . . . . . 8 ((𝑦 ∈ ω ∧ 𝑧 ∈ ω) → (((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹𝑥) ∈ (𝐹‘suc 𝑥)) → (𝑧𝑦 → (𝐹𝑧) ∈ (𝐹𝑦))))
76imp 406 . . . . . . 7 (((𝑦 ∈ ω ∧ 𝑧 ∈ ω) ∧ ((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹𝑥) ∈ (𝐹‘suc 𝑥))) → (𝑧𝑦 → (𝐹𝑧) ∈ (𝐹𝑦)))
84, 7orim12d 967 . . . . . 6 (((𝑦 ∈ ω ∧ 𝑧 ∈ ω) ∧ ((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹𝑥) ∈ (𝐹‘suc 𝑥))) → ((𝑦𝑧𝑧𝑦) → ((𝐹𝑦) ∈ (𝐹𝑧) ∨ (𝐹𝑧) ∈ (𝐹𝑦))))
98ancoms 458 . . . . 5 ((((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹𝑥) ∈ (𝐹‘suc 𝑥)) ∧ (𝑦 ∈ ω ∧ 𝑧 ∈ ω)) → ((𝑦𝑧𝑧𝑦) → ((𝐹𝑦) ∈ (𝐹𝑧) ∨ (𝐹𝑧) ∈ (𝐹𝑦))))
109con3d 152 . . . 4 ((((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹𝑥) ∈ (𝐹‘suc 𝑥)) ∧ (𝑦 ∈ ω ∧ 𝑧 ∈ ω)) → (¬ ((𝐹𝑦) ∈ (𝐹𝑧) ∨ (𝐹𝑧) ∈ (𝐹𝑦)) → ¬ (𝑦𝑧𝑧𝑦)))
11 ffvelcdm 7101 . . . . . . . . . . 11 ((𝐹:ω⟶𝐴𝑦 ∈ ω) → (𝐹𝑦) ∈ 𝐴)
12 ssel 3977 . . . . . . . . . . 11 (𝐴 ⊆ On → ((𝐹𝑦) ∈ 𝐴 → (𝐹𝑦) ∈ On))
1311, 12syl5 34 . . . . . . . . . 10 (𝐴 ⊆ On → ((𝐹:ω⟶𝐴𝑦 ∈ ω) → (𝐹𝑦) ∈ On))
1413expdimp 452 . . . . . . . . 9 ((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) → (𝑦 ∈ ω → (𝐹𝑦) ∈ On))
15 eloni 6394 . . . . . . . . 9 ((𝐹𝑦) ∈ On → Ord (𝐹𝑦))
1614, 15syl6 35 . . . . . . . 8 ((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) → (𝑦 ∈ ω → Ord (𝐹𝑦)))
17 ffvelcdm 7101 . . . . . . . . . . 11 ((𝐹:ω⟶𝐴𝑧 ∈ ω) → (𝐹𝑧) ∈ 𝐴)
18 ssel 3977 . . . . . . . . . . 11 (𝐴 ⊆ On → ((𝐹𝑧) ∈ 𝐴 → (𝐹𝑧) ∈ On))
1917, 18syl5 34 . . . . . . . . . 10 (𝐴 ⊆ On → ((𝐹:ω⟶𝐴𝑧 ∈ ω) → (𝐹𝑧) ∈ On))
2019expdimp 452 . . . . . . . . 9 ((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) → (𝑧 ∈ ω → (𝐹𝑧) ∈ On))
21 eloni 6394 . . . . . . . . 9 ((𝐹𝑧) ∈ On → Ord (𝐹𝑧))
2220, 21syl6 35 . . . . . . . 8 ((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) → (𝑧 ∈ ω → Ord (𝐹𝑧)))
2316, 22anim12d 609 . . . . . . 7 ((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) → ((𝑦 ∈ ω ∧ 𝑧 ∈ ω) → (Ord (𝐹𝑦) ∧ Ord (𝐹𝑧))))
2423imp 406 . . . . . 6 (((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ (𝑦 ∈ ω ∧ 𝑧 ∈ ω)) → (Ord (𝐹𝑦) ∧ Ord (𝐹𝑧)))
25 ordtri3 6420 . . . . . 6 ((Ord (𝐹𝑦) ∧ Ord (𝐹𝑧)) → ((𝐹𝑦) = (𝐹𝑧) ↔ ¬ ((𝐹𝑦) ∈ (𝐹𝑧) ∨ (𝐹𝑧) ∈ (𝐹𝑦))))
2624, 25syl 17 . . . . 5 (((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ (𝑦 ∈ ω ∧ 𝑧 ∈ ω)) → ((𝐹𝑦) = (𝐹𝑧) ↔ ¬ ((𝐹𝑦) ∈ (𝐹𝑧) ∨ (𝐹𝑧) ∈ (𝐹𝑦))))
2726adantlr 715 . . . 4 ((((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹𝑥) ∈ (𝐹‘suc 𝑥)) ∧ (𝑦 ∈ ω ∧ 𝑧 ∈ ω)) → ((𝐹𝑦) = (𝐹𝑧) ↔ ¬ ((𝐹𝑦) ∈ (𝐹𝑧) ∨ (𝐹𝑧) ∈ (𝐹𝑦))))
28 nnord 7895 . . . . . 6 (𝑦 ∈ ω → Ord 𝑦)
29 nnord 7895 . . . . . 6 (𝑧 ∈ ω → Ord 𝑧)
30 ordtri3 6420 . . . . . 6 ((Ord 𝑦 ∧ Ord 𝑧) → (𝑦 = 𝑧 ↔ ¬ (𝑦𝑧𝑧𝑦)))
3128, 29, 30syl2an 596 . . . . 5 ((𝑦 ∈ ω ∧ 𝑧 ∈ ω) → (𝑦 = 𝑧 ↔ ¬ (𝑦𝑧𝑧𝑦)))
3231adantl 481 . . . 4 ((((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹𝑥) ∈ (𝐹‘suc 𝑥)) ∧ (𝑦 ∈ ω ∧ 𝑧 ∈ ω)) → (𝑦 = 𝑧 ↔ ¬ (𝑦𝑧𝑧𝑦)))
3310, 27, 323imtr4d 294 . . 3 ((((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹𝑥) ∈ (𝐹‘suc 𝑥)) ∧ (𝑦 ∈ ω ∧ 𝑧 ∈ ω)) → ((𝐹𝑦) = (𝐹𝑧) → 𝑦 = 𝑧))
3433ralrimivva 3202 . 2 (((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹𝑥) ∈ (𝐹‘suc 𝑥)) → ∀𝑦 ∈ ω ∀𝑧 ∈ ω ((𝐹𝑦) = (𝐹𝑧) → 𝑦 = 𝑧))
35 dff13 7275 . 2 (𝐹:ω–1-1𝐴 ↔ (𝐹:ω⟶𝐴 ∧ ∀𝑦 ∈ ω ∀𝑧 ∈ ω ((𝐹𝑦) = (𝐹𝑧) → 𝑦 = 𝑧)))
361, 34, 35sylanbrc 583 1 (((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹𝑥) ∈ (𝐹‘suc 𝑥)) → 𝐹:ω–1-1𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wo 848   = wceq 1540  wcel 2108  wral 3061  wss 3951  Ord word 6383  Oncon0 6384  suc csuc 6386  wf 6557  1-1wf1 6558  cfv 6561  ωcom 7887
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fv 6569  df-om 7888
This theorem is referenced by:  unblem4  9331
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