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Theorem dfsmo2 7441
Description: Alternate definition of a strictly monotone ordinal function. (Contributed by Mario Carneiro, 4-Mar-2013.)
Assertion
Ref Expression
dfsmo2 (Smo 𝐹 ↔ (𝐹:dom 𝐹⟶On ∧ Ord dom 𝐹 ∧ ∀𝑥 ∈ dom 𝐹𝑦𝑥 (𝐹𝑦) ∈ (𝐹𝑥)))
Distinct variable group:   𝑥,𝐹,𝑦

Proof of Theorem dfsmo2
StepHypRef Expression
1 df-smo 7440 . 2 (Smo 𝐹 ↔ (𝐹:dom 𝐹⟶On ∧ Ord dom 𝐹 ∧ ∀𝑦 ∈ dom 𝐹𝑥 ∈ dom 𝐹(𝑦𝑥 → (𝐹𝑦) ∈ (𝐹𝑥))))
2 ralcom 3096 . . . . . 6 (∀𝑦 ∈ dom 𝐹𝑥 ∈ dom 𝐹(𝑦𝑥 → (𝐹𝑦) ∈ (𝐹𝑥)) ↔ ∀𝑥 ∈ dom 𝐹𝑦 ∈ dom 𝐹(𝑦𝑥 → (𝐹𝑦) ∈ (𝐹𝑥)))
3 impexp 462 . . . . . . . . 9 (((𝑦 ∈ dom 𝐹𝑦𝑥) → (𝐹𝑦) ∈ (𝐹𝑥)) ↔ (𝑦 ∈ dom 𝐹 → (𝑦𝑥 → (𝐹𝑦) ∈ (𝐹𝑥))))
4 simpr 477 . . . . . . . . . . 11 ((𝑦 ∈ dom 𝐹𝑦𝑥) → 𝑦𝑥)
5 ordtr1 5765 . . . . . . . . . . . . . . 15 (Ord dom 𝐹 → ((𝑦𝑥𝑥 ∈ dom 𝐹) → 𝑦 ∈ dom 𝐹))
653impib 1261 . . . . . . . . . . . . . 14 ((Ord dom 𝐹𝑦𝑥𝑥 ∈ dom 𝐹) → 𝑦 ∈ dom 𝐹)
763com23 1270 . . . . . . . . . . . . 13 ((Ord dom 𝐹𝑥 ∈ dom 𝐹𝑦𝑥) → 𝑦 ∈ dom 𝐹)
8 simp3 1062 . . . . . . . . . . . . 13 ((Ord dom 𝐹𝑥 ∈ dom 𝐹𝑦𝑥) → 𝑦𝑥)
97, 8jca 554 . . . . . . . . . . . 12 ((Ord dom 𝐹𝑥 ∈ dom 𝐹𝑦𝑥) → (𝑦 ∈ dom 𝐹𝑦𝑥))
1093expia 1266 . . . . . . . . . . 11 ((Ord dom 𝐹𝑥 ∈ dom 𝐹) → (𝑦𝑥 → (𝑦 ∈ dom 𝐹𝑦𝑥)))
114, 10impbid2 216 . . . . . . . . . 10 ((Ord dom 𝐹𝑥 ∈ dom 𝐹) → ((𝑦 ∈ dom 𝐹𝑦𝑥) ↔ 𝑦𝑥))
1211imbi1d 331 . . . . . . . . 9 ((Ord dom 𝐹𝑥 ∈ dom 𝐹) → (((𝑦 ∈ dom 𝐹𝑦𝑥) → (𝐹𝑦) ∈ (𝐹𝑥)) ↔ (𝑦𝑥 → (𝐹𝑦) ∈ (𝐹𝑥))))
133, 12syl5bbr 274 . . . . . . . 8 ((Ord dom 𝐹𝑥 ∈ dom 𝐹) → ((𝑦 ∈ dom 𝐹 → (𝑦𝑥 → (𝐹𝑦) ∈ (𝐹𝑥))) ↔ (𝑦𝑥 → (𝐹𝑦) ∈ (𝐹𝑥))))
1413ralbidv2 2983 . . . . . . 7 ((Ord dom 𝐹𝑥 ∈ dom 𝐹) → (∀𝑦 ∈ dom 𝐹(𝑦𝑥 → (𝐹𝑦) ∈ (𝐹𝑥)) ↔ ∀𝑦𝑥 (𝐹𝑦) ∈ (𝐹𝑥)))
1514ralbidva 2984 . . . . . 6 (Ord dom 𝐹 → (∀𝑥 ∈ dom 𝐹𝑦 ∈ dom 𝐹(𝑦𝑥 → (𝐹𝑦) ∈ (𝐹𝑥)) ↔ ∀𝑥 ∈ dom 𝐹𝑦𝑥 (𝐹𝑦) ∈ (𝐹𝑥)))
162, 15syl5bb 272 . . . . 5 (Ord dom 𝐹 → (∀𝑦 ∈ dom 𝐹𝑥 ∈ dom 𝐹(𝑦𝑥 → (𝐹𝑦) ∈ (𝐹𝑥)) ↔ ∀𝑥 ∈ dom 𝐹𝑦𝑥 (𝐹𝑦) ∈ (𝐹𝑥)))
1716pm5.32i 669 . . . 4 ((Ord dom 𝐹 ∧ ∀𝑦 ∈ dom 𝐹𝑥 ∈ dom 𝐹(𝑦𝑥 → (𝐹𝑦) ∈ (𝐹𝑥))) ↔ (Ord dom 𝐹 ∧ ∀𝑥 ∈ dom 𝐹𝑦𝑥 (𝐹𝑦) ∈ (𝐹𝑥)))
1817anbi2i 730 . . 3 ((𝐹:dom 𝐹⟶On ∧ (Ord dom 𝐹 ∧ ∀𝑦 ∈ dom 𝐹𝑥 ∈ dom 𝐹(𝑦𝑥 → (𝐹𝑦) ∈ (𝐹𝑥)))) ↔ (𝐹:dom 𝐹⟶On ∧ (Ord dom 𝐹 ∧ ∀𝑥 ∈ dom 𝐹𝑦𝑥 (𝐹𝑦) ∈ (𝐹𝑥))))
19 3anass 1041 . . 3 ((𝐹:dom 𝐹⟶On ∧ Ord dom 𝐹 ∧ ∀𝑦 ∈ dom 𝐹𝑥 ∈ dom 𝐹(𝑦𝑥 → (𝐹𝑦) ∈ (𝐹𝑥))) ↔ (𝐹:dom 𝐹⟶On ∧ (Ord dom 𝐹 ∧ ∀𝑦 ∈ dom 𝐹𝑥 ∈ dom 𝐹(𝑦𝑥 → (𝐹𝑦) ∈ (𝐹𝑥)))))
20 3anass 1041 . . 3 ((𝐹:dom 𝐹⟶On ∧ Ord dom 𝐹 ∧ ∀𝑥 ∈ dom 𝐹𝑦𝑥 (𝐹𝑦) ∈ (𝐹𝑥)) ↔ (𝐹:dom 𝐹⟶On ∧ (Ord dom 𝐹 ∧ ∀𝑥 ∈ dom 𝐹𝑦𝑥 (𝐹𝑦) ∈ (𝐹𝑥))))
2118, 19, 203bitr4i 292 . 2 ((𝐹:dom 𝐹⟶On ∧ Ord dom 𝐹 ∧ ∀𝑦 ∈ dom 𝐹𝑥 ∈ dom 𝐹(𝑦𝑥 → (𝐹𝑦) ∈ (𝐹𝑥))) ↔ (𝐹:dom 𝐹⟶On ∧ Ord dom 𝐹 ∧ ∀𝑥 ∈ dom 𝐹𝑦𝑥 (𝐹𝑦) ∈ (𝐹𝑥)))
221, 21bitri 264 1 (Smo 𝐹 ↔ (𝐹:dom 𝐹⟶On ∧ Ord dom 𝐹 ∧ ∀𝑥 ∈ dom 𝐹𝑦𝑥 (𝐹𝑦) ∈ (𝐹𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1037  wcel 1989  wral 2911  dom cdm 5112  Ord word 5720  Oncon0 5721  wf 5882  cfv 5886  Smo wsmo 7439
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ral 2916  df-v 3200  df-in 3579  df-ss 3586  df-uni 4435  df-tr 4751  df-ord 5724  df-smo 7440
This theorem is referenced by:  issmo2  7443  smores2  7448  smofvon2  7450
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