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Theorem issmo2 6193
Description: Alternate definition of a strictly monotone ordinal function. (Contributed by Mario Carneiro, 12-Mar-2013.)
Assertion
Ref Expression
issmo2 (𝐹:𝐴𝐵 → ((𝐵 ⊆ On ∧ Ord 𝐴 ∧ ∀𝑥𝐴𝑦𝑥 (𝐹𝑦) ∈ (𝐹𝑥)) → Smo 𝐹))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐹,𝑦
Allowed substitution hints:   𝐴(𝑦)   𝐵(𝑥,𝑦)

Proof of Theorem issmo2
StepHypRef Expression
1 fss 5291 . . . . 5 ((𝐹:𝐴𝐵𝐵 ⊆ On) → 𝐹:𝐴⟶On)
21ex 114 . . . 4 (𝐹:𝐴𝐵 → (𝐵 ⊆ On → 𝐹:𝐴⟶On))
3 fdm 5285 . . . . 5 (𝐹:𝐴𝐵 → dom 𝐹 = 𝐴)
43feq2d 5267 . . . 4 (𝐹:𝐴𝐵 → (𝐹:dom 𝐹⟶On ↔ 𝐹:𝐴⟶On))
52, 4sylibrd 168 . . 3 (𝐹:𝐴𝐵 → (𝐵 ⊆ On → 𝐹:dom 𝐹⟶On))
6 ordeq 4301 . . . . 5 (dom 𝐹 = 𝐴 → (Ord dom 𝐹 ↔ Ord 𝐴))
73, 6syl 14 . . . 4 (𝐹:𝐴𝐵 → (Ord dom 𝐹 ↔ Ord 𝐴))
87biimprd 157 . . 3 (𝐹:𝐴𝐵 → (Ord 𝐴 → Ord dom 𝐹))
93raleqdv 2635 . . . 4 (𝐹:𝐴𝐵 → (∀𝑥 ∈ dom 𝐹𝑦𝑥 (𝐹𝑦) ∈ (𝐹𝑥) ↔ ∀𝑥𝐴𝑦𝑥 (𝐹𝑦) ∈ (𝐹𝑥)))
109biimprd 157 . . 3 (𝐹:𝐴𝐵 → (∀𝑥𝐴𝑦𝑥 (𝐹𝑦) ∈ (𝐹𝑥) → ∀𝑥 ∈ dom 𝐹𝑦𝑥 (𝐹𝑦) ∈ (𝐹𝑥)))
115, 8, 103anim123d 1298 . 2 (𝐹:𝐴𝐵 → ((𝐵 ⊆ On ∧ Ord 𝐴 ∧ ∀𝑥𝐴𝑦𝑥 (𝐹𝑦) ∈ (𝐹𝑥)) → (𝐹:dom 𝐹⟶On ∧ Ord dom 𝐹 ∧ ∀𝑥 ∈ dom 𝐹𝑦𝑥 (𝐹𝑦) ∈ (𝐹𝑥))))
12 dfsmo2 6191 . 2 (Smo 𝐹 ↔ (𝐹:dom 𝐹⟶On ∧ Ord dom 𝐹 ∧ ∀𝑥 ∈ dom 𝐹𝑦𝑥 (𝐹𝑦) ∈ (𝐹𝑥)))
1311, 12syl6ibr 161 1 (𝐹:𝐴𝐵 → ((𝐵 ⊆ On ∧ Ord 𝐴 ∧ ∀𝑥𝐴𝑦𝑥 (𝐹𝑦) ∈ (𝐹𝑥)) → Smo 𝐹))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  w3a 963   = wceq 1332  wcel 1481  wral 2417  wss 3075  Ord word 4291  Oncon0 4292  dom cdm 4546  wf 5126  cfv 5130  Smo wsmo 6189
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-in 3081  df-ss 3088  df-uni 3744  df-tr 4034  df-iord 4295  df-fn 5133  df-f 5134  df-smo 6190
This theorem is referenced by: (None)
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