Theorem issmo2 8332

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 6722	. . . . 5 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐵 ⊆ On) → 𝐹:𝐴⟶On)
2	1	ex 417	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → (𝐵 ⊆ On → 𝐹:𝐴⟶On))
3		fdm 6715	. . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 = 𝐴)
4	3	feq2d 6689	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹:dom 𝐹⟶On ↔ 𝐹:𝐴⟶On))
5	2, 4	sylibrd 262	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → (𝐵 ⊆ On → 𝐹:dom 𝐹⟶On))
6		ordeq 6367	. . . . 5 ⊢ (dom 𝐹 = 𝐴 → (Ord dom 𝐹 ↔ Ord 𝐴))
7	3, 6	syl 18	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → (Ord dom 𝐹 ↔ Ord 𝐴))
8	7	biimprd 251	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → (Ord 𝐴 → Ord dom 𝐹))
9	3	raleqdv 3323	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → (∀𝑥 ∈ dom 𝐹∀𝑦 ∈ 𝑥 (𝐹‘𝑦) ∈ (𝐹‘𝑥) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 (𝐹‘𝑦) ∈ (𝐹‘𝑥)))
10	9	biimprd 251	. . 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 1471	. 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 8330	. 2 ⊢ (Smo 𝐹 ↔ (𝐹:dom 𝐹⟶On ∧ Ord dom 𝐹 ∧ ∀𝑥 ∈ dom 𝐹∀𝑦 ∈ 𝑥 (𝐹‘𝑦) ∈ (𝐹‘𝑥)))
13	11, 12	imbitrrdi 255	1 ⊢ (𝐹:𝐴⟶𝐵 → ((𝐵 ⊆ On ∧ Ord 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 (𝐹‘𝑦) ∈ (𝐹‘𝑥)) → Smo 𝐹))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143  ∀wral 3079   ⊆ wss 3905  dom cdm 5661  Ord word 6359  Oncon0 6360  ⟶wf 6532  ‘cfv 6536  Smo wsmo 8328

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-11 2192  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 401  df-3an 1105  df-tru 1573  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rex 3090  df-v 3457  df-ss 3922  df-uni 4873  df-tr 5219  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-ord 6363  df-fn 6539  df-f 6540  df-smo 8329

This theorem is used by:  alephsmo  10091  cofsmo  10257  cfsmolem  10258