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Theorem issmo2 5935
Description: Alternative 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 5082 . . . . 5 ((𝐹:𝐴𝐵𝐵 ⊆ On) → 𝐹:𝐴⟶On)
21ex 112 . . . 4 (𝐹:𝐴𝐵 → (𝐵 ⊆ On → 𝐹:𝐴⟶On))
3 fdm 5078 . . . . 5 (𝐹:𝐴𝐵 → dom 𝐹 = 𝐴)
43feq2d 5063 . . . 4 (𝐹:𝐴𝐵 → (𝐹:dom 𝐹⟶On ↔ 𝐹:𝐴⟶On))
52, 4sylibrd 162 . . 3 (𝐹:𝐴𝐵 → (𝐵 ⊆ On → 𝐹:dom 𝐹⟶On))
6 ordeq 4137 . . . . 5 (dom 𝐹 = 𝐴 → (Ord dom 𝐹 ↔ Ord 𝐴))
73, 6syl 14 . . . 4 (𝐹:𝐴𝐵 → (Ord dom 𝐹 ↔ Ord 𝐴))
87biimprd 151 . . 3 (𝐹:𝐴𝐵 → (Ord 𝐴 → Ord dom 𝐹))
93raleqdv 2528 . . . 4 (𝐹:𝐴𝐵 → (∀𝑥 ∈ dom 𝐹𝑦𝑥 (𝐹𝑦) ∈ (𝐹𝑥) ↔ ∀𝑥𝐴𝑦𝑥 (𝐹𝑦) ∈ (𝐹𝑥)))
109biimprd 151 . . 3 (𝐹:𝐴𝐵 → (∀𝑥𝐴𝑦𝑥 (𝐹𝑦) ∈ (𝐹𝑥) → ∀𝑥 ∈ dom 𝐹𝑦𝑥 (𝐹𝑦) ∈ (𝐹𝑥)))
115, 8, 103anim123d 1225 . 2 (𝐹:𝐴𝐵 → ((𝐵 ⊆ On ∧ Ord 𝐴 ∧ ∀𝑥𝐴𝑦𝑥 (𝐹𝑦) ∈ (𝐹𝑥)) → (𝐹:dom 𝐹⟶On ∧ Ord dom 𝐹 ∧ ∀𝑥 ∈ dom 𝐹𝑦𝑥 (𝐹𝑦) ∈ (𝐹𝑥))))
12 dfsmo2 5933 . 2 (Smo 𝐹 ↔ (𝐹:dom 𝐹⟶On ∧ Ord dom 𝐹 ∧ ∀𝑥 ∈ dom 𝐹𝑦𝑥 (𝐹𝑦) ∈ (𝐹𝑥)))
1311, 12syl6ibr 155 1 (𝐹:𝐴𝐵 → ((𝐵 ⊆ On ∧ Ord 𝐴 ∧ ∀𝑥𝐴𝑦𝑥 (𝐹𝑦) ∈ (𝐹𝑥)) → Smo 𝐹))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 102  w3a 896   = wceq 1259  wcel 1409  wral 2323  wss 2945  Ord word 4127  Oncon0 4128  dom cdm 4373  wf 4926  cfv 4930  Smo wsmo 5931
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-in 2952  df-ss 2959  df-uni 3609  df-tr 3883  df-iord 4131  df-fn 4933  df-f 4934  df-smo 5932
This theorem is referenced by: (None)
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