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Theorem omsmo 7779
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 807 . 2 (((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹𝑥) ∈ (𝐹‘suc 𝑥)) → 𝐹:ω⟶𝐴)
2 omsmolem 7778 . . . . . . . . 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 444 . . . . . . 7 (((𝑦 ∈ ω ∧ 𝑧 ∈ ω) ∧ ((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹𝑥) ∈ (𝐹‘suc 𝑥))) → (𝑦𝑧 → (𝐹𝑦) ∈ (𝐹𝑧)))
5 omsmolem 7778 . . . . . . . . 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 444 . . . . . . 7 (((𝑦 ∈ ω ∧ 𝑧 ∈ ω) ∧ ((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹𝑥) ∈ (𝐹‘suc 𝑥))) → (𝑧𝑦 → (𝐹𝑧) ∈ (𝐹𝑦)))
84, 7orim12d 901 . . . . . 6 (((𝑦 ∈ ω ∧ 𝑧 ∈ ω) ∧ ((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹𝑥) ∈ (𝐹‘suc 𝑥))) → ((𝑦𝑧𝑧𝑦) → ((𝐹𝑦) ∈ (𝐹𝑧) ∨ (𝐹𝑧) ∈ (𝐹𝑦))))
98ancoms 468 . . . . 5 ((((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹𝑥) ∈ (𝐹‘suc 𝑥)) ∧ (𝑦 ∈ ω ∧ 𝑧 ∈ ω)) → ((𝑦𝑧𝑧𝑦) → ((𝐹𝑦) ∈ (𝐹𝑧) ∨ (𝐹𝑧) ∈ (𝐹𝑦))))
109con3d 148 . . . 4 ((((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹𝑥) ∈ (𝐹‘suc 𝑥)) ∧ (𝑦 ∈ ω ∧ 𝑧 ∈ ω)) → (¬ ((𝐹𝑦) ∈ (𝐹𝑧) ∨ (𝐹𝑧) ∈ (𝐹𝑦)) → ¬ (𝑦𝑧𝑧𝑦)))
11 ffvelrn 6397 . . . . . . . . . . 11 ((𝐹:ω⟶𝐴𝑦 ∈ ω) → (𝐹𝑦) ∈ 𝐴)
12 ssel 3630 . . . . . . . . . . 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 5771 . . . . . . . . 9 ((𝐹𝑦) ∈ On → Ord (𝐹𝑦))
1614, 15syl6 35 . . . . . . . 8 ((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) → (𝑦 ∈ ω → Ord (𝐹𝑦)))
17 ffvelrn 6397 . . . . . . . . . . 11 ((𝐹:ω⟶𝐴𝑧 ∈ ω) → (𝐹𝑧) ∈ 𝐴)
18 ssel 3630 . . . . . . . . . . 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 5771 . . . . . . . . 9 ((𝐹𝑧) ∈ On → Ord (𝐹𝑧))
2220, 21syl6 35 . . . . . . . 8 ((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) → (𝑧 ∈ ω → Ord (𝐹𝑧)))
2316, 22anim12d 585 . . . . . . 7 ((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) → ((𝑦 ∈ ω ∧ 𝑧 ∈ ω) → (Ord (𝐹𝑦) ∧ Ord (𝐹𝑧))))
2423imp 444 . . . . . 6 (((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ (𝑦 ∈ ω ∧ 𝑧 ∈ ω)) → (Ord (𝐹𝑦) ∧ Ord (𝐹𝑧)))
25 ordtri3 5797 . . . . . 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 751 . . . 4 ((((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹𝑥) ∈ (𝐹‘suc 𝑥)) ∧ (𝑦 ∈ ω ∧ 𝑧 ∈ ω)) → ((𝐹𝑦) = (𝐹𝑧) ↔ ¬ ((𝐹𝑦) ∈ (𝐹𝑧) ∨ (𝐹𝑧) ∈ (𝐹𝑦))))
28 nnord 7115 . . . . . 6 (𝑦 ∈ ω → Ord 𝑦)
29 nnord 7115 . . . . . 6 (𝑧 ∈ ω → Ord 𝑧)
30 ordtri3 5797 . . . . . 6 ((Ord 𝑦 ∧ Ord 𝑧) → (𝑦 = 𝑧 ↔ ¬ (𝑦𝑧𝑧𝑦)))
3128, 29, 30syl2an 493 . . . . 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 283 . . 3 ((((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹𝑥) ∈ (𝐹‘suc 𝑥)) ∧ (𝑦 ∈ ω ∧ 𝑧 ∈ ω)) → ((𝐹𝑦) = (𝐹𝑧) → 𝑦 = 𝑧))
3433ralrimivva 3000 . 2 (((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹𝑥) ∈ (𝐹‘suc 𝑥)) → ∀𝑦 ∈ ω ∀𝑧 ∈ ω ((𝐹𝑦) = (𝐹𝑧) → 𝑦 = 𝑧))
35 dff13 6552 . 2 (𝐹:ω–1-1𝐴 ↔ (𝐹:ω⟶𝐴 ∧ ∀𝑦 ∈ ω ∀𝑧 ∈ ω ((𝐹𝑦) = (𝐹𝑧) → 𝑦 = 𝑧)))
361, 34, 35sylanbrc 699 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 196  wo 382  wa 383   = wceq 1523  wcel 2030  wral 2941  wss 3607  Ord word 5760  Oncon0 5761  suc csuc 5763  wf 5922  1-1wf1 5923  cfv 5926  ωcom 7107
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fv 5934  df-om 7108
This theorem is referenced by:  unblem4  8256
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