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Theorem issmo2 7616
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 6217 . . . . 5 ((𝐹:𝐴𝐵𝐵 ⊆ On) → 𝐹:𝐴⟶On)
21ex 449 . . . 4 (𝐹:𝐴𝐵 → (𝐵 ⊆ On → 𝐹:𝐴⟶On))
3 fdm 6212 . . . . 5 (𝐹:𝐴𝐵 → dom 𝐹 = 𝐴)
43feq2d 6192 . . . 4 (𝐹:𝐴𝐵 → (𝐹:dom 𝐹⟶On ↔ 𝐹:𝐴⟶On))
52, 4sylibrd 249 . . 3 (𝐹:𝐴𝐵 → (𝐵 ⊆ On → 𝐹:dom 𝐹⟶On))
6 ordeq 5891 . . . . 5 (dom 𝐹 = 𝐴 → (Ord dom 𝐹 ↔ Ord 𝐴))
73, 6syl 17 . . . 4 (𝐹:𝐴𝐵 → (Ord dom 𝐹 ↔ Ord 𝐴))
87biimprd 238 . . 3 (𝐹:𝐴𝐵 → (Ord 𝐴 → Ord dom 𝐹))
93raleqdv 3283 . . . 4 (𝐹:𝐴𝐵 → (∀𝑥 ∈ dom 𝐹𝑦𝑥 (𝐹𝑦) ∈ (𝐹𝑥) ↔ ∀𝑥𝐴𝑦𝑥 (𝐹𝑦) ∈ (𝐹𝑥)))
109biimprd 238 . . 3 (𝐹:𝐴𝐵 → (∀𝑥𝐴𝑦𝑥 (𝐹𝑦) ∈ (𝐹𝑥) → ∀𝑥 ∈ dom 𝐹𝑦𝑥 (𝐹𝑦) ∈ (𝐹𝑥)))
115, 8, 103anim123d 1555 . 2 (𝐹:𝐴𝐵 → ((𝐵 ⊆ On ∧ Ord 𝐴 ∧ ∀𝑥𝐴𝑦𝑥 (𝐹𝑦) ∈ (𝐹𝑥)) → (𝐹:dom 𝐹⟶On ∧ Ord dom 𝐹 ∧ ∀𝑥 ∈ dom 𝐹𝑦𝑥 (𝐹𝑦) ∈ (𝐹𝑥))))
12 dfsmo2 7614 . 2 (Smo 𝐹 ↔ (𝐹:dom 𝐹⟶On ∧ Ord dom 𝐹 ∧ ∀𝑥 ∈ dom 𝐹𝑦𝑥 (𝐹𝑦) ∈ (𝐹𝑥)))
1311, 12syl6ibr 242 1 (𝐹:𝐴𝐵 → ((𝐵 ⊆ On ∧ Ord 𝐴 ∧ ∀𝑥𝐴𝑦𝑥 (𝐹𝑦) ∈ (𝐹𝑥)) → Smo 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  w3a 1072   = wceq 1632  wcel 2139  wral 3050  wss 3715  dom cdm 5266  Ord word 5883  Oncon0 5884  wf 6045  cfv 6049  Smo wsmo 7612
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-v 3342  df-in 3722  df-ss 3729  df-uni 4589  df-tr 4905  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-ord 5887  df-fn 6052  df-f 6053  df-smo 7613
This theorem is referenced by:  alephsmo  9135  cofsmo  9303  cfsmolem  9304
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