Theorem issmo2 8371

Description: Alternate definition of a strictly monotone ordinal function. (Contributed by Mario Carneiro, 12-Mar-2013.)

Assertion

Ref	Expression
issmo2	⊢ (𝐹:𝐴⟶𝐵 → ((𝐵 ⊆ On ∧ Ord 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 (𝐹‘𝑦) ∈ (𝐹‘𝑥)) → Smo 𝐹))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐹,𝑦

Allowed substitution hints:   𝐴(𝑦)   𝐵(𝑥,𝑦)

Proof of Theorem issmo2

Step	Hyp	Ref	Expression
1		fss 6732	. . . . 5 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐵 ⊆ On) → 𝐹:𝐴⟶On)
2	1	ex 412	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → (𝐵 ⊆ On → 𝐹:𝐴⟶On))
3		fdm 6725	. . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 = 𝐴)
4	3	feq2d 6702	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹:dom 𝐹⟶On ↔ 𝐹:𝐴⟶On))
5	2, 4	sylibrd 259	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → (𝐵 ⊆ On → 𝐹:dom 𝐹⟶On))
6		ordeq 6370	. . . . 5 ⊢ (dom 𝐹 = 𝐴 → (Ord dom 𝐹 ↔ Ord 𝐴))
7	3, 6	syl 17	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → (Ord dom 𝐹 ↔ Ord 𝐴))
8	7	biimprd 248	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → (Ord 𝐴 → Ord dom 𝐹))
9	3	raleqdv 3309	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → (∀𝑥 ∈ dom 𝐹∀𝑦 ∈ 𝑥 (𝐹‘𝑦) ∈ (𝐹‘𝑥) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 (𝐹‘𝑦) ∈ (𝐹‘𝑥)))
10	9	biimprd 248	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 (𝐹‘𝑦) ∈ (𝐹‘𝑥) → ∀𝑥 ∈ dom 𝐹∀𝑦 ∈ 𝑥 (𝐹‘𝑦) ∈ (𝐹‘𝑥)))
11	5, 8, 10	3anim123d 1444	. 2 ⊢ (𝐹:𝐴⟶𝐵 → ((𝐵 ⊆ On ∧ Ord 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 (𝐹‘𝑦) ∈ (𝐹‘𝑥)) → (𝐹:dom 𝐹⟶On ∧ Ord dom 𝐹 ∧ ∀𝑥 ∈ dom 𝐹∀𝑦 ∈ 𝑥 (𝐹‘𝑦) ∈ (𝐹‘𝑥))))
12		dfsmo2 8369	. 2 ⊢ (Smo 𝐹 ↔ (𝐹:dom 𝐹⟶On ∧ Ord dom 𝐹 ∧ ∀𝑥 ∈ dom 𝐹∀𝑦 ∈ 𝑥 (𝐹‘𝑦) ∈ (𝐹‘𝑥)))
13	11, 12	imbitrrdi 252	1 ⊢ (𝐹:𝐴⟶𝐵 → ((𝐵 ⊆ On ∧ Ord 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 (𝐹‘𝑦) ∈ (𝐹‘𝑥)) → Smo 𝐹))

Syntax hints:   → wi 4   ↔ wb 206   ∧ w3a 1086   = wceq 1539   ∈ wcel 2107  ∀wral 3050   ⊆ wss 3931  dom cdm 5665  Ord word 6362  Oncon0 6363  ⟶wf 6537  ‘cfv 6541  Smo wsmo 8367

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-11 2156  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1542  df-ex 1779  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-ral 3051  df-rex 3060  df-v 3465  df-ss 3948  df-uni 4888  df-tr 5240  df-po 5572  df-so 5573  df-fr 5617  df-we 5619  df-ord 6366  df-fn 6544  df-f 6545  df-smo 8368

This theorem is referenced by:  alephsmo  10124  cofsmo  10291  cfsmolem  10292