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Theorem dfsmo2 7597
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 7596 . 2 (Smo 𝐹 ↔ (𝐹:dom 𝐹⟶On ∧ Ord dom 𝐹 ∧ ∀𝑦 ∈ dom 𝐹𝑥 ∈ dom 𝐹(𝑦𝑥 → (𝐹𝑦) ∈ (𝐹𝑥))))
2 ralcom 3246 . . . . . 6 (∀𝑦 ∈ dom 𝐹𝑥 ∈ dom 𝐹(𝑦𝑥 → (𝐹𝑦) ∈ (𝐹𝑥)) ↔ ∀𝑥 ∈ dom 𝐹𝑦 ∈ dom 𝐹(𝑦𝑥 → (𝐹𝑦) ∈ (𝐹𝑥)))
3 impexp 437 . . . . . . . . 9 (((𝑦 ∈ dom 𝐹𝑦𝑥) → (𝐹𝑦) ∈ (𝐹𝑥)) ↔ (𝑦 ∈ dom 𝐹 → (𝑦𝑥 → (𝐹𝑦) ∈ (𝐹𝑥))))
4 simpr 471 . . . . . . . . . . 11 ((𝑦 ∈ dom 𝐹𝑦𝑥) → 𝑦𝑥)
5 ordtr1 5910 . . . . . . . . . . . . . . 15 (Ord dom 𝐹 → ((𝑦𝑥𝑥 ∈ dom 𝐹) → 𝑦 ∈ dom 𝐹))
653impib 1108 . . . . . . . . . . . . . 14 ((Ord dom 𝐹𝑦𝑥𝑥 ∈ dom 𝐹) → 𝑦 ∈ dom 𝐹)
763com23 1120 . . . . . . . . . . . . 13 ((Ord dom 𝐹𝑥 ∈ dom 𝐹𝑦𝑥) → 𝑦 ∈ dom 𝐹)
8 simp3 1132 . . . . . . . . . . . . 13 ((Ord dom 𝐹𝑥 ∈ dom 𝐹𝑦𝑥) → 𝑦𝑥)
97, 8jca 501 . . . . . . . . . . . 12 ((Ord dom 𝐹𝑥 ∈ dom 𝐹𝑦𝑥) → (𝑦 ∈ dom 𝐹𝑦𝑥))
1093expia 1114 . . . . . . . . . . 11 ((Ord dom 𝐹𝑥 ∈ dom 𝐹) → (𝑦𝑥 → (𝑦 ∈ dom 𝐹𝑦𝑥)))
114, 10impbid2 216 . . . . . . . . . 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 274 . . . . . . . 8 ((Ord dom 𝐹𝑥 ∈ dom 𝐹) → ((𝑦 ∈ dom 𝐹 → (𝑦𝑥 → (𝐹𝑦) ∈ (𝐹𝑥))) ↔ (𝑦𝑥 → (𝐹𝑦) ∈ (𝐹𝑥))))
1413ralbidv2 3133 . . . . . . 7 ((Ord dom 𝐹𝑥 ∈ dom 𝐹) → (∀𝑦 ∈ dom 𝐹(𝑦𝑥 → (𝐹𝑦) ∈ (𝐹𝑥)) ↔ ∀𝑦𝑥 (𝐹𝑦) ∈ (𝐹𝑥)))
1514ralbidva 3134 . . . . . 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 564 . . . 4 ((Ord dom 𝐹 ∧ ∀𝑦 ∈ dom 𝐹𝑥 ∈ dom 𝐹(𝑦𝑥 → (𝐹𝑦) ∈ (𝐹𝑥))) ↔ (Ord dom 𝐹 ∧ ∀𝑥 ∈ dom 𝐹𝑦𝑥 (𝐹𝑦) ∈ (𝐹𝑥)))
1817anbi2i 609 . . 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 1080 . . 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 1080 . . 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 382  w3a 1071  wcel 2145  wral 3061  dom cdm 5249  Ord word 5865  Oncon0 5866  wf 6027  cfv 6031  Smo wsmo 7595
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-v 3353  df-in 3730  df-ss 3737  df-uni 4575  df-tr 4887  df-ord 5869  df-smo 7596
This theorem is referenced by:  issmo2  7599  smores2  7604  smofvon2  7606
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