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Theorem issmo2 6092
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 5207 . . . . 5 ((𝐹:𝐴𝐵𝐵 ⊆ On) → 𝐹:𝐴⟶On)
21ex 114 . . . 4 (𝐹:𝐴𝐵 → (𝐵 ⊆ On → 𝐹:𝐴⟶On))
3 fdm 5201 . . . . 5 (𝐹:𝐴𝐵 → dom 𝐹 = 𝐴)
43feq2d 5184 . . . 4 (𝐹:𝐴𝐵 → (𝐹:dom 𝐹⟶On ↔ 𝐹:𝐴⟶On))
52, 4sylibrd 168 . . 3 (𝐹:𝐴𝐵 → (𝐵 ⊆ On → 𝐹:dom 𝐹⟶On))
6 ordeq 4223 . . . . 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 2582 . . . 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 1262 . 2 (𝐹:𝐴𝐵 → ((𝐵 ⊆ On ∧ Ord 𝐴 ∧ ∀𝑥𝐴𝑦𝑥 (𝐹𝑦) ∈ (𝐹𝑥)) → (𝐹:dom 𝐹⟶On ∧ Ord dom 𝐹 ∧ ∀𝑥 ∈ dom 𝐹𝑦𝑥 (𝐹𝑦) ∈ (𝐹𝑥))))
12 dfsmo2 6090 . 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 927   = wceq 1296  wcel 1445  wral 2370  wss 3013  Ord word 4213  Oncon0 4214  dom cdm 4467  wf 5045  cfv 5049  Smo wsmo 6088
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077
This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-v 2635  df-in 3019  df-ss 3026  df-uni 3676  df-tr 3959  df-iord 4217  df-fn 5052  df-f 5053  df-smo 6089
This theorem is referenced by: (None)
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