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Theorem issmo2 7310
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 5954 . . . . 5 ((𝐹:𝐴𝐵𝐵 ⊆ On) → 𝐹:𝐴⟶On)
21ex 448 . . . 4 (𝐹:𝐴𝐵 → (𝐵 ⊆ On → 𝐹:𝐴⟶On))
3 fdm 5949 . . . . 5 (𝐹:𝐴𝐵 → dom 𝐹 = 𝐴)
43feq2d 5929 . . . 4 (𝐹:𝐴𝐵 → (𝐹:dom 𝐹⟶On ↔ 𝐹:𝐴⟶On))
52, 4sylibrd 247 . . 3 (𝐹:𝐴𝐵 → (𝐵 ⊆ On → 𝐹:dom 𝐹⟶On))
6 ordeq 5632 . . . . 5 (dom 𝐹 = 𝐴 → (Ord dom 𝐹 ↔ Ord 𝐴))
73, 6syl 17 . . . 4 (𝐹:𝐴𝐵 → (Ord dom 𝐹 ↔ Ord 𝐴))
87biimprd 236 . . 3 (𝐹:𝐴𝐵 → (Ord 𝐴 → Ord dom 𝐹))
93raleqdv 3120 . . . 4 (𝐹:𝐴𝐵 → (∀𝑥 ∈ dom 𝐹𝑦𝑥 (𝐹𝑦) ∈ (𝐹𝑥) ↔ ∀𝑥𝐴𝑦𝑥 (𝐹𝑦) ∈ (𝐹𝑥)))
109biimprd 236 . . 3 (𝐹:𝐴𝐵 → (∀𝑥𝐴𝑦𝑥 (𝐹𝑦) ∈ (𝐹𝑥) → ∀𝑥 ∈ dom 𝐹𝑦𝑥 (𝐹𝑦) ∈ (𝐹𝑥)))
115, 8, 103anim123d 1397 . 2 (𝐹:𝐴𝐵 → ((𝐵 ⊆ On ∧ Ord 𝐴 ∧ ∀𝑥𝐴𝑦𝑥 (𝐹𝑦) ∈ (𝐹𝑥)) → (𝐹:dom 𝐹⟶On ∧ Ord dom 𝐹 ∧ ∀𝑥 ∈ dom 𝐹𝑦𝑥 (𝐹𝑦) ∈ (𝐹𝑥))))
12 dfsmo2 7308 . 2 (Smo 𝐹 ↔ (𝐹:dom 𝐹⟶On ∧ Ord dom 𝐹 ∧ ∀𝑥 ∈ dom 𝐹𝑦𝑥 (𝐹𝑦) ∈ (𝐹𝑥)))
1311, 12syl6ibr 240 1 (𝐹:𝐴𝐵 → ((𝐵 ⊆ On ∧ Ord 𝐴 ∧ ∀𝑥𝐴𝑦𝑥 (𝐹𝑦) ∈ (𝐹𝑥)) → Smo 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  w3a 1030   = wceq 1474  wcel 1976  wral 2895  wss 3539  dom cdm 5027  Ord word 5624  Oncon0 5625  wf 5785  cfv 5789  Smo wsmo 7306
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-v 3174  df-in 3546  df-ss 3553  df-uni 4367  df-tr 4675  df-po 4948  df-so 4949  df-fr 4986  df-we 4988  df-ord 5628  df-fn 5792  df-f 5793  df-smo 7307
This theorem is referenced by:  alephsmo  8785  cofsmo  8951  cfsmolem  8952
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