MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  omsmo Structured version   Visualization version   GIF version

Theorem omsmo 7974
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 786 . 2 (((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹𝑥) ∈ (𝐹‘suc 𝑥)) → 𝐹:ω⟶𝐴)
2 omsmolem 7973 . . . . . . . . 9 (𝑧 ∈ ω → (((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹𝑥) ∈ (𝐹‘suc 𝑥)) → (𝑦𝑧 → (𝐹𝑦) ∈ (𝐹𝑧))))
32adantl 474 . . . . . . . 8 ((𝑦 ∈ ω ∧ 𝑧 ∈ ω) → (((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹𝑥) ∈ (𝐹‘suc 𝑥)) → (𝑦𝑧 → (𝐹𝑦) ∈ (𝐹𝑧))))
43imp 396 . . . . . . 7 (((𝑦 ∈ ω ∧ 𝑧 ∈ ω) ∧ ((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹𝑥) ∈ (𝐹‘suc 𝑥))) → (𝑦𝑧 → (𝐹𝑦) ∈ (𝐹𝑧)))
5 omsmolem 7973 . . . . . . . . 9 (𝑦 ∈ ω → (((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹𝑥) ∈ (𝐹‘suc 𝑥)) → (𝑧𝑦 → (𝐹𝑧) ∈ (𝐹𝑦))))
65adantr 473 . . . . . . . 8 ((𝑦 ∈ ω ∧ 𝑧 ∈ ω) → (((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹𝑥) ∈ (𝐹‘suc 𝑥)) → (𝑧𝑦 → (𝐹𝑧) ∈ (𝐹𝑦))))
76imp 396 . . . . . . 7 (((𝑦 ∈ ω ∧ 𝑧 ∈ ω) ∧ ((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹𝑥) ∈ (𝐹‘suc 𝑥))) → (𝑧𝑦 → (𝐹𝑧) ∈ (𝐹𝑦)))
84, 7orim12d 988 . . . . . 6 (((𝑦 ∈ ω ∧ 𝑧 ∈ ω) ∧ ((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹𝑥) ∈ (𝐹‘suc 𝑥))) → ((𝑦𝑧𝑧𝑦) → ((𝐹𝑦) ∈ (𝐹𝑧) ∨ (𝐹𝑧) ∈ (𝐹𝑦))))
98ancoms 451 . . . . 5 ((((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹𝑥) ∈ (𝐹‘suc 𝑥)) ∧ (𝑦 ∈ ω ∧ 𝑧 ∈ ω)) → ((𝑦𝑧𝑧𝑦) → ((𝐹𝑦) ∈ (𝐹𝑧) ∨ (𝐹𝑧) ∈ (𝐹𝑦))))
109con3d 150 . . . 4 ((((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹𝑥) ∈ (𝐹‘suc 𝑥)) ∧ (𝑦 ∈ ω ∧ 𝑧 ∈ ω)) → (¬ ((𝐹𝑦) ∈ (𝐹𝑧) ∨ (𝐹𝑧) ∈ (𝐹𝑦)) → ¬ (𝑦𝑧𝑧𝑦)))
11 ffvelrn 6583 . . . . . . . . . . 11 ((𝐹:ω⟶𝐴𝑦 ∈ ω) → (𝐹𝑦) ∈ 𝐴)
12 ssel 3792 . . . . . . . . . . 11 (𝐴 ⊆ On → ((𝐹𝑦) ∈ 𝐴 → (𝐹𝑦) ∈ On))
1311, 12syl5 34 . . . . . . . . . 10 (𝐴 ⊆ On → ((𝐹:ω⟶𝐴𝑦 ∈ ω) → (𝐹𝑦) ∈ On))
1413expdimp 445 . . . . . . . . 9 ((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) → (𝑦 ∈ ω → (𝐹𝑦) ∈ On))
15 eloni 5951 . . . . . . . . 9 ((𝐹𝑦) ∈ On → Ord (𝐹𝑦))
1614, 15syl6 35 . . . . . . . 8 ((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) → (𝑦 ∈ ω → Ord (𝐹𝑦)))
17 ffvelrn 6583 . . . . . . . . . . 11 ((𝐹:ω⟶𝐴𝑧 ∈ ω) → (𝐹𝑧) ∈ 𝐴)
18 ssel 3792 . . . . . . . . . . 11 (𝐴 ⊆ On → ((𝐹𝑧) ∈ 𝐴 → (𝐹𝑧) ∈ On))
1917, 18syl5 34 . . . . . . . . . 10 (𝐴 ⊆ On → ((𝐹:ω⟶𝐴𝑧 ∈ ω) → (𝐹𝑧) ∈ On))
2019expdimp 445 . . . . . . . . 9 ((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) → (𝑧 ∈ ω → (𝐹𝑧) ∈ On))
21 eloni 5951 . . . . . . . . 9 ((𝐹𝑧) ∈ On → Ord (𝐹𝑧))
2220, 21syl6 35 . . . . . . . 8 ((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) → (𝑧 ∈ ω → Ord (𝐹𝑧)))
2316, 22anim12d 603 . . . . . . 7 ((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) → ((𝑦 ∈ ω ∧ 𝑧 ∈ ω) → (Ord (𝐹𝑦) ∧ Ord (𝐹𝑧))))
2423imp 396 . . . . . 6 (((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ (𝑦 ∈ ω ∧ 𝑧 ∈ ω)) → (Ord (𝐹𝑦) ∧ Ord (𝐹𝑧)))
25 ordtri3 5977 . . . . . 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 707 . . . 4 ((((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹𝑥) ∈ (𝐹‘suc 𝑥)) ∧ (𝑦 ∈ ω ∧ 𝑧 ∈ ω)) → ((𝐹𝑦) = (𝐹𝑧) ↔ ¬ ((𝐹𝑦) ∈ (𝐹𝑧) ∨ (𝐹𝑧) ∈ (𝐹𝑦))))
28 nnord 7307 . . . . . 6 (𝑦 ∈ ω → Ord 𝑦)
29 nnord 7307 . . . . . 6 (𝑧 ∈ ω → Ord 𝑧)
30 ordtri3 5977 . . . . . 6 ((Ord 𝑦 ∧ Ord 𝑧) → (𝑦 = 𝑧 ↔ ¬ (𝑦𝑧𝑧𝑦)))
3128, 29, 30syl2an 590 . . . . 5 ((𝑦 ∈ ω ∧ 𝑧 ∈ ω) → (𝑦 = 𝑧 ↔ ¬ (𝑦𝑧𝑧𝑦)))
3231adantl 474 . . . 4 ((((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹𝑥) ∈ (𝐹‘suc 𝑥)) ∧ (𝑦 ∈ ω ∧ 𝑧 ∈ ω)) → (𝑦 = 𝑧 ↔ ¬ (𝑦𝑧𝑧𝑦)))
3310, 27, 323imtr4d 286 . . 3 ((((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹𝑥) ∈ (𝐹‘suc 𝑥)) ∧ (𝑦 ∈ ω ∧ 𝑧 ∈ ω)) → ((𝐹𝑦) = (𝐹𝑧) → 𝑦 = 𝑧))
3433ralrimivva 3152 . 2 (((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹𝑥) ∈ (𝐹‘suc 𝑥)) → ∀𝑦 ∈ ω ∀𝑧 ∈ ω ((𝐹𝑦) = (𝐹𝑧) → 𝑦 = 𝑧))
35 dff13 6740 . 2 (𝐹:ω–1-1𝐴 ↔ (𝐹:ω⟶𝐴 ∧ ∀𝑦 ∈ ω ∀𝑧 ∈ ω ((𝐹𝑦) = (𝐹𝑧) → 𝑦 = 𝑧)))
361, 34, 35sylanbrc 579 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 198  wa 385  wo 874   = wceq 1653  wcel 2157  wral 3089  wss 3769  Ord word 5940  Oncon0 5941  suc csuc 5943  wf 6097  1-1wf1 6098  cfv 6101  ωcom 7299
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097  ax-un 7183
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-tp 4373  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-tr 4946  df-id 5220  df-eprel 5225  df-po 5233  df-so 5234  df-fr 5271  df-we 5273  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-ord 5944  df-on 5945  df-lim 5946  df-suc 5947  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fv 6109  df-om 7300
This theorem is referenced by:  unblem4  8457
  Copyright terms: Public domain W3C validator