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Theorem dfsmo2 7331
 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 7330 . 2 (Smo 𝐹 ↔ (𝐹:dom 𝐹⟶On ∧ Ord dom 𝐹 ∧ ∀𝑦 ∈ dom 𝐹𝑥 ∈ dom 𝐹(𝑦𝑥 → (𝐹𝑦) ∈ (𝐹𝑥))))
2 ralcom 3079 . . . . . 6 (∀𝑦 ∈ dom 𝐹𝑥 ∈ dom 𝐹(𝑦𝑥 → (𝐹𝑦) ∈ (𝐹𝑥)) ↔ ∀𝑥 ∈ dom 𝐹𝑦 ∈ dom 𝐹(𝑦𝑥 → (𝐹𝑦) ∈ (𝐹𝑥)))
3 impexp 461 . . . . . . . . 9 (((𝑦 ∈ dom 𝐹𝑦𝑥) → (𝐹𝑦) ∈ (𝐹𝑥)) ↔ (𝑦 ∈ dom 𝐹 → (𝑦𝑥 → (𝐹𝑦) ∈ (𝐹𝑥))))
4 simpr 476 . . . . . . . . . . 11 ((𝑦 ∈ dom 𝐹𝑦𝑥) → 𝑦𝑥)
5 ordtr1 5684 . . . . . . . . . . . . . . 15 (Ord dom 𝐹 → ((𝑦𝑥𝑥 ∈ dom 𝐹) → 𝑦 ∈ dom 𝐹))
653impib 1254 . . . . . . . . . . . . . 14 ((Ord dom 𝐹𝑦𝑥𝑥 ∈ dom 𝐹) → 𝑦 ∈ dom 𝐹)
763com23 1263 . . . . . . . . . . . . 13 ((Ord dom 𝐹𝑥 ∈ dom 𝐹𝑦𝑥) → 𝑦 ∈ dom 𝐹)
8 simp3 1056 . . . . . . . . . . . . 13 ((Ord dom 𝐹𝑥 ∈ dom 𝐹𝑦𝑥) → 𝑦𝑥)
97, 8jca 553 . . . . . . . . . . . 12 ((Ord dom 𝐹𝑥 ∈ dom 𝐹𝑦𝑥) → (𝑦 ∈ dom 𝐹𝑦𝑥))
1093expia 1259 . . . . . . . . . . 11 ((Ord dom 𝐹𝑥 ∈ dom 𝐹) → (𝑦𝑥 → (𝑦 ∈ dom 𝐹𝑦𝑥)))
114, 10impbid2 215 . . . . . . . . . 10 ((Ord dom 𝐹𝑥 ∈ dom 𝐹) → ((𝑦 ∈ dom 𝐹𝑦𝑥) ↔ 𝑦𝑥))
1211imbi1d 330 . . . . . . . . 9 ((Ord dom 𝐹𝑥 ∈ dom 𝐹) → (((𝑦 ∈ dom 𝐹𝑦𝑥) → (𝐹𝑦) ∈ (𝐹𝑥)) ↔ (𝑦𝑥 → (𝐹𝑦) ∈ (𝐹𝑥))))
133, 12syl5bbr 273 . . . . . . . 8 ((Ord dom 𝐹𝑥 ∈ dom 𝐹) → ((𝑦 ∈ dom 𝐹 → (𝑦𝑥 → (𝐹𝑦) ∈ (𝐹𝑥))) ↔ (𝑦𝑥 → (𝐹𝑦) ∈ (𝐹𝑥))))
1413ralbidv2 2967 . . . . . . 7 ((Ord dom 𝐹𝑥 ∈ dom 𝐹) → (∀𝑦 ∈ dom 𝐹(𝑦𝑥 → (𝐹𝑦) ∈ (𝐹𝑥)) ↔ ∀𝑦𝑥 (𝐹𝑦) ∈ (𝐹𝑥)))
1514ralbidva 2968 . . . . . 6 (Ord dom 𝐹 → (∀𝑥 ∈ dom 𝐹𝑦 ∈ dom 𝐹(𝑦𝑥 → (𝐹𝑦) ∈ (𝐹𝑥)) ↔ ∀𝑥 ∈ dom 𝐹𝑦𝑥 (𝐹𝑦) ∈ (𝐹𝑥)))
162, 15syl5bb 271 . . . . 5 (Ord dom 𝐹 → (∀𝑦 ∈ dom 𝐹𝑥 ∈ dom 𝐹(𝑦𝑥 → (𝐹𝑦) ∈ (𝐹𝑥)) ↔ ∀𝑥 ∈ dom 𝐹𝑦𝑥 (𝐹𝑦) ∈ (𝐹𝑥)))
1716pm5.32i 667 . . . 4 ((Ord dom 𝐹 ∧ ∀𝑦 ∈ dom 𝐹𝑥 ∈ dom 𝐹(𝑦𝑥 → (𝐹𝑦) ∈ (𝐹𝑥))) ↔ (Ord dom 𝐹 ∧ ∀𝑥 ∈ dom 𝐹𝑦𝑥 (𝐹𝑦) ∈ (𝐹𝑥)))
1817anbi2i 726 . . 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 1035 . . 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 1035 . . 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 291 . 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 263 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 195   ∧ wa 383   ∧ w3a 1031   ∈ wcel 1977  ∀wral 2896  dom cdm 5038  Ord word 5639  Oncon0 5640  ⟶wf 5800  ‘cfv 5804  Smo wsmo 7329 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-v 3175  df-in 3547  df-ss 3554  df-uni 4373  df-tr 4681  df-ord 5643  df-smo 7330 This theorem is referenced by:  issmo2  7333  smores2  7338  smofvon2  7340
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