Theorem issmo2 8320

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 6708	. . . . 5 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐵 ⊆ On) → 𝐹:𝐴⟶On)
2	1	ex 416	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → (𝐵 ⊆ On → 𝐹:𝐴⟶On))
3		fdm 6701	. . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 = 𝐴)
4	3	feq2d 6675	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹:dom 𝐹⟶On ↔ 𝐹:𝐴⟶On))
5	2, 4	sylibrd 261	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → (𝐵 ⊆ On → 𝐹:dom 𝐹⟶On))
6		ordeq 6353	. . . . 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 250	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → (Ord 𝐴 → Ord dom 𝐹))
9	3	raleqdv 3320	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → (∀𝑥 ∈ dom 𝐹∀𝑦 ∈ 𝑥 (𝐹‘𝑦) ∈ (𝐹‘𝑥) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 (𝐹‘𝑦) ∈ (𝐹‘𝑥)))
10	9	biimprd 250	. . 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 1464	. 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 8318	. 2 ⊢ (Smo 𝐹 ↔ (𝐹:dom 𝐹⟶On ∧ Ord dom 𝐹 ∧ ∀𝑥 ∈ dom 𝐹∀𝑦 ∈ 𝑥 (𝐹‘𝑦) ∈ (𝐹‘𝑥)))
13	11, 12	imbitrrdi 254	1 ⊢ (𝐹:𝐴⟶𝐵 → ((𝐵 ⊆ On ∧ Ord 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 (𝐹‘𝑦) ∈ (𝐹‘𝑥)) → Smo 𝐹))

Syntax hints:   → wi 4   ↔ wb 208   ∧ w3a 1098   = wceq 1560   ∈ wcel 2142  ∀wral 3076   ⊆ wss 3904  dom cdm 5647  Ord word 6345  Oncon0 6346  ⟶wf 6517  ‘cfv 6521  Smo wsmo 8316

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-11 2191  ax-ext 2734

This theorem depends on definitions:  df-bi 209  df-an 400  df-3an 1100  df-tru 1563  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3077  df-rex 3087  df-v 3456  df-ss 3921  df-uni 4866  df-tr 5208  df-po 5555  df-so 5556  df-fr 5600  df-we 5602  df-ord 6349  df-fn 6524  df-f 6525  df-smo 8317

This theorem is referenced by:  alephsmo  10058  cofsmo  10226  cfsmolem  10227