Theorem issmo2 6388

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 5447	. . . . 5 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐵 ⊆ On) → 𝐹:𝐴⟶On)
2	1	ex 115	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → (𝐵 ⊆ On → 𝐹:𝐴⟶On))
3		fdm 5441	. . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 = 𝐴)
4	3	feq2d 5423	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹:dom 𝐹⟶On ↔ 𝐹:𝐴⟶On))
5	2, 4	sylibrd 169	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → (𝐵 ⊆ On → 𝐹:dom 𝐹⟶On))
6		ordeq 4427	. . . . 5 ⊢ (dom 𝐹 = 𝐴 → (Ord dom 𝐹 ↔ Ord 𝐴))
7	3, 6	syl 14	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → (Ord dom 𝐹 ↔ Ord 𝐴))
8	7	biimprd 158	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → (Ord 𝐴 → Ord dom 𝐹))
9	3	raleqdv 2709	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → (∀𝑥 ∈ dom 𝐹∀𝑦 ∈ 𝑥 (𝐹‘𝑦) ∈ (𝐹‘𝑥) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 (𝐹‘𝑦) ∈ (𝐹‘𝑥)))
10	9	biimprd 158	. . 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 1332	. 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 6386	. 2 ⊢ (Smo 𝐹 ↔ (𝐹:dom 𝐹⟶On ∧ Ord dom 𝐹 ∧ ∀𝑥 ∈ dom 𝐹∀𝑦 ∈ 𝑥 (𝐹‘𝑦) ∈ (𝐹‘𝑥)))
13	11, 12	imbitrrdi 162	1 ⊢ (𝐹:𝐴⟶𝐵 → ((𝐵 ⊆ On ∧ Ord 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 (𝐹‘𝑦) ∈ (𝐹‘𝑥)) → Smo 𝐹))

Syntax hints:   → wi 4   ↔ wb 105   ∧ w3a 981   = wceq 1373   ∈ wcel 2177  ∀wral 2485   ⊆ wss 3170  Ord word 4417  Oncon0 4418  dom cdm 4683  ⟶wf 5276  ‘cfv 5280  Smo wsmo 6384

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-v 2775  df-in 3176  df-ss 3183  df-uni 3857  df-tr 4151  df-iord 4421  df-fn 5283  df-f 5284  df-smo 6385

This theorem is referenced by: (None)