Theorem issmo2 8264

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 6662	. . . . 5 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐵 ⊆ On) → 𝐹:𝐴⟶On)
2	1	ex 412	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → (𝐵 ⊆ On → 𝐹:𝐴⟶On))
3		fdm 6655	. . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 = 𝐴)
4	3	feq2d 6630	. . . 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 6308	. . . . 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 3292	. . . 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 1445	. 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 8262	. 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 1541   ∈ wcel 2111  ∀wral 3047   ⊆ wss 3897  dom cdm 5611  Ord word 6300  Oncon0 6301  ⟶wf 6472  ‘cfv 6476  Smo wsmo 8260

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-11 2160  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-v 3438  df-ss 3914  df-uni 4855  df-tr 5194  df-po 5519  df-so 5520  df-fr 5564  df-we 5566  df-ord 6304  df-fn 6479  df-f 6480  df-smo 8261

This theorem is referenced by:  alephsmo  9988  cofsmo  10155  cfsmolem  10156