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Theorem dfsmo2 6313
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 6312 . 2 (Smo 𝐹 ↔ (𝐹:dom 𝐹⟶On ∧ Ord dom 𝐹 ∧ ∀𝑦 ∈ dom 𝐹𝑥 ∈ dom 𝐹(𝑦𝑥 → (𝐹𝑦) ∈ (𝐹𝑥))))
2 ralcom 2653 . . . . . 6 (∀𝑦 ∈ dom 𝐹𝑥 ∈ dom 𝐹(𝑦𝑥 → (𝐹𝑦) ∈ (𝐹𝑥)) ↔ ∀𝑥 ∈ dom 𝐹𝑦 ∈ dom 𝐹(𝑦𝑥 → (𝐹𝑦) ∈ (𝐹𝑥)))
3 impexp 263 . . . . . . . . 9 (((𝑦 ∈ dom 𝐹𝑦𝑥) → (𝐹𝑦) ∈ (𝐹𝑥)) ↔ (𝑦 ∈ dom 𝐹 → (𝑦𝑥 → (𝐹𝑦) ∈ (𝐹𝑥))))
4 simpr 110 . . . . . . . . . . 11 ((𝑦 ∈ dom 𝐹𝑦𝑥) → 𝑦𝑥)
5 ordtr1 4406 . . . . . . . . . . . . . . 15 (Ord dom 𝐹 → ((𝑦𝑥𝑥 ∈ dom 𝐹) → 𝑦 ∈ dom 𝐹))
653impib 1203 . . . . . . . . . . . . . 14 ((Ord dom 𝐹𝑦𝑥𝑥 ∈ dom 𝐹) → 𝑦 ∈ dom 𝐹)
763com23 1211 . . . . . . . . . . . . 13 ((Ord dom 𝐹𝑥 ∈ dom 𝐹𝑦𝑥) → 𝑦 ∈ dom 𝐹)
8 simp3 1001 . . . . . . . . . . . . 13 ((Ord dom 𝐹𝑥 ∈ dom 𝐹𝑦𝑥) → 𝑦𝑥)
97, 8jca 306 . . . . . . . . . . . 12 ((Ord dom 𝐹𝑥 ∈ dom 𝐹𝑦𝑥) → (𝑦 ∈ dom 𝐹𝑦𝑥))
1093expia 1207 . . . . . . . . . . 11 ((Ord dom 𝐹𝑥 ∈ dom 𝐹) → (𝑦𝑥 → (𝑦 ∈ dom 𝐹𝑦𝑥)))
114, 10impbid2 143 . . . . . . . . . 10 ((Ord dom 𝐹𝑥 ∈ dom 𝐹) → ((𝑦 ∈ dom 𝐹𝑦𝑥) ↔ 𝑦𝑥))
1211imbi1d 231 . . . . . . . . 9 ((Ord dom 𝐹𝑥 ∈ dom 𝐹) → (((𝑦 ∈ dom 𝐹𝑦𝑥) → (𝐹𝑦) ∈ (𝐹𝑥)) ↔ (𝑦𝑥 → (𝐹𝑦) ∈ (𝐹𝑥))))
133, 12bitr3id 194 . . . . . . . 8 ((Ord dom 𝐹𝑥 ∈ dom 𝐹) → ((𝑦 ∈ dom 𝐹 → (𝑦𝑥 → (𝐹𝑦) ∈ (𝐹𝑥))) ↔ (𝑦𝑥 → (𝐹𝑦) ∈ (𝐹𝑥))))
1413ralbidv2 2492 . . . . . . 7 ((Ord dom 𝐹𝑥 ∈ dom 𝐹) → (∀𝑦 ∈ dom 𝐹(𝑦𝑥 → (𝐹𝑦) ∈ (𝐹𝑥)) ↔ ∀𝑦𝑥 (𝐹𝑦) ∈ (𝐹𝑥)))
1514ralbidva 2486 . . . . . 6 (Ord dom 𝐹 → (∀𝑥 ∈ dom 𝐹𝑦 ∈ dom 𝐹(𝑦𝑥 → (𝐹𝑦) ∈ (𝐹𝑥)) ↔ ∀𝑥 ∈ dom 𝐹𝑦𝑥 (𝐹𝑦) ∈ (𝐹𝑥)))
162, 15bitrid 192 . . . . 5 (Ord dom 𝐹 → (∀𝑦 ∈ dom 𝐹𝑥 ∈ dom 𝐹(𝑦𝑥 → (𝐹𝑦) ∈ (𝐹𝑥)) ↔ ∀𝑥 ∈ dom 𝐹𝑦𝑥 (𝐹𝑦) ∈ (𝐹𝑥)))
1716pm5.32i 454 . . . 4 ((Ord dom 𝐹 ∧ ∀𝑦 ∈ dom 𝐹𝑥 ∈ dom 𝐹(𝑦𝑥 → (𝐹𝑦) ∈ (𝐹𝑥))) ↔ (Ord dom 𝐹 ∧ ∀𝑥 ∈ dom 𝐹𝑦𝑥 (𝐹𝑦) ∈ (𝐹𝑥)))
1817anbi2i 457 . . 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 984 . . 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 984 . . 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 212 . 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 184 1 (Smo 𝐹 ↔ (𝐹:dom 𝐹⟶On ∧ Ord dom 𝐹 ∧ ∀𝑥 ∈ dom 𝐹𝑦𝑥 (𝐹𝑦) ∈ (𝐹𝑥)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  w3a 980  wcel 2160  wral 2468  Ord word 4380  Oncon0 4381  dom cdm 4644  wf 5231  cfv 5235  Smo wsmo 6311
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-v 2754  df-in 3150  df-ss 3157  df-uni 3825  df-tr 4117  df-iord 4384  df-smo 6312
This theorem is referenced by:  issmo2  6315  smores2  6320  smofvon2dm  6322
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