Theorem smores2 5850

Description: A strictly monotone ordinal function restricted to an ordinal is still monotone. (Contributed by Mario Carneiro, 15-Mar-2013.)

Assertion

Ref	Expression
smores2	⊢ ((Smo 𝐹 ∧ Ord A) → Smo (𝐹 ↾ A))

Proof of Theorem smores2

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfsmo2 5843	. . . . . . 7 ⊢ (Smo 𝐹 ↔ (𝐹:dom 𝐹⟶On ∧ Ord dom 𝐹 ∧ ∀x ∈ dom 𝐹∀y ∈ x (𝐹‘y) ∈ (𝐹‘x)))
2	1	simp1bi 918	. . . . . 6 ⊢ (Smo 𝐹 → 𝐹:dom 𝐹⟶On)
3		ffun 4991	. . . . . 6 ⊢ (𝐹:dom 𝐹⟶On → Fun 𝐹)
4	2, 3	syl 14	. . . . 5 ⊢ (Smo 𝐹 → Fun 𝐹)
5		funres 4884	. . . . . 6 ⊢ (Fun 𝐹 → Fun (𝐹 ↾ A))
6		funfn 4874	. . . . . 6 ⊢ (Fun (𝐹 ↾ A) ↔ (𝐹 ↾ A) Fn dom (𝐹 ↾ A))
7	5, 6	sylib 127	. . . . 5 ⊢ (Fun 𝐹 → (𝐹 ↾ A) Fn dom (𝐹 ↾ A))
8	4, 7	syl 14	. . . 4 ⊢ (Smo 𝐹 → (𝐹 ↾ A) Fn dom (𝐹 ↾ A))
9		df-ima 4301	. . . . . 6 ⊢ (𝐹 “ A) = ran (𝐹 ↾ A)
10		imassrn 4622	. . . . . 6 ⊢ (𝐹 “ A) ⊆ ran 𝐹
11	9, 10	eqsstr3i 2970	. . . . 5 ⊢ ran (𝐹 ↾ A) ⊆ ran 𝐹
12		frn 4995	. . . . . 6 ⊢ (𝐹:dom 𝐹⟶On → ran 𝐹 ⊆ On)
13	2, 12	syl 14	. . . . 5 ⊢ (Smo 𝐹 → ran 𝐹 ⊆ On)
14	11, 13	syl5ss 2950	. . . 4 ⊢ (Smo 𝐹 → ran (𝐹 ↾ A) ⊆ On)
15		df-f 4849	. . . 4 ⊢ ((𝐹 ↾ A):dom (𝐹 ↾ A)⟶On ↔ ((𝐹 ↾ A) Fn dom (𝐹 ↾ A) ∧ ran (𝐹 ↾ A) ⊆ On))
16	8, 14, 15	sylanbrc 394	. . 3 ⊢ (Smo 𝐹 → (𝐹 ↾ A):dom (𝐹 ↾ A)⟶On)
17	16	adantr 261	. 2 ⊢ ((Smo 𝐹 ∧ Ord A) → (𝐹 ↾ A):dom (𝐹 ↾ A)⟶On)
18		smodm 5847	. . 3 ⊢ (Smo 𝐹 → Ord dom 𝐹)
19		ordin 4088	. . . . 5 ⊢ ((Ord A ∧ Ord dom 𝐹) → Ord (A ∩ dom 𝐹))
20		dmres 4575	. . . . . 6 ⊢ dom (𝐹 ↾ A) = (A ∩ dom 𝐹)
21		ordeq 4075	. . . . . 6 ⊢ (dom (𝐹 ↾ A) = (A ∩ dom 𝐹) → (Ord dom (𝐹 ↾ A) ↔ Ord (A ∩ dom 𝐹)))
22	20, 21	ax-mp 7	. . . . 5 ⊢ (Ord dom (𝐹 ↾ A) ↔ Ord (A ∩ dom 𝐹))
23	19, 22	sylibr 137	. . . 4 ⊢ ((Ord A ∧ Ord dom 𝐹) → Ord dom (𝐹 ↾ A))
24	23	ancoms 255	. . 3 ⊢ ((Ord dom 𝐹 ∧ Ord A) → Ord dom (𝐹 ↾ A))
25	18, 24	sylan 267	. 2 ⊢ ((Smo 𝐹 ∧ Ord A) → Ord dom (𝐹 ↾ A))
26		resss 4578	. . . . . 6 ⊢ (𝐹 ↾ A) ⊆ 𝐹
27		dmss 4477	. . . . . 6 ⊢ ((𝐹 ↾ A) ⊆ 𝐹 → dom (𝐹 ↾ A) ⊆ dom 𝐹)
28	26, 27	ax-mp 7	. . . . 5 ⊢ dom (𝐹 ↾ A) ⊆ dom 𝐹
29	1	simp3bi 920	. . . . 5 ⊢ (Smo 𝐹 → ∀x ∈ dom 𝐹∀y ∈ x (𝐹‘y) ∈ (𝐹‘x))
30		ssralv 2998	. . . . 5 ⊢ (dom (𝐹 ↾ A) ⊆ dom 𝐹 → (∀x ∈ dom 𝐹∀y ∈ x (𝐹‘y) ∈ (𝐹‘x) → ∀x ∈ dom (𝐹 ↾ A)∀y ∈ x (𝐹‘y) ∈ (𝐹‘x)))
31	28, 29, 30	mpsyl 59	. . . 4 ⊢ (Smo 𝐹 → ∀x ∈ dom (𝐹 ↾ A)∀y ∈ x (𝐹‘y) ∈ (𝐹‘x))
32	31	adantr 261	. . 3 ⊢ ((Smo 𝐹 ∧ Ord A) → ∀x ∈ dom (𝐹 ↾ A)∀y ∈ x (𝐹‘y) ∈ (𝐹‘x))
33		ordtr1 4091	. . . . . . . . . . 11 ⊢ (Ord dom (𝐹 ↾ A) → ((y ∈ x ∧ x ∈ dom (𝐹 ↾ A)) → y ∈ dom (𝐹 ↾ A)))
34	25, 33	syl 14	. . . . . . . . . 10 ⊢ ((Smo 𝐹 ∧ Ord A) → ((y ∈ x ∧ x ∈ dom (𝐹 ↾ A)) → y ∈ dom (𝐹 ↾ A)))
35		inss1 3151	. . . . . . . . . . . 12 ⊢ (A ∩ dom 𝐹) ⊆ A
36	20, 35	eqsstri 2969	. . . . . . . . . . 11 ⊢ dom (𝐹 ↾ A) ⊆ A
37	36	sseli 2935	. . . . . . . . . 10 ⊢ (y ∈ dom (𝐹 ↾ A) → y ∈ A)
38	34, 37	syl6 29	. . . . . . . . 9 ⊢ ((Smo 𝐹 ∧ Ord A) → ((y ∈ x ∧ x ∈ dom (𝐹 ↾ A)) → y ∈ A))
39	38	expcomd 1327	. . . . . . . 8 ⊢ ((Smo 𝐹 ∧ Ord A) → (x ∈ dom (𝐹 ↾ A) → (y ∈ x → y ∈ A)))
40	39	imp31 243	. . . . . . 7 ⊢ ((((Smo 𝐹 ∧ Ord A) ∧ x ∈ dom (𝐹 ↾ A)) ∧ y ∈ x) → y ∈ A)
41		fvres 5141	. . . . . . 7 ⊢ (y ∈ A → ((𝐹 ↾ A)‘y) = (𝐹‘y))
42	40, 41	syl 14	. . . . . 6 ⊢ ((((Smo 𝐹 ∧ Ord A) ∧ x ∈ dom (𝐹 ↾ A)) ∧ y ∈ x) → ((𝐹 ↾ A)‘y) = (𝐹‘y))
43	36	sseli 2935	. . . . . . . 8 ⊢ (x ∈ dom (𝐹 ↾ A) → x ∈ A)
44		fvres 5141	. . . . . . . 8 ⊢ (x ∈ A → ((𝐹 ↾ A)‘x) = (𝐹‘x))
45	43, 44	syl 14	. . . . . . 7 ⊢ (x ∈ dom (𝐹 ↾ A) → ((𝐹 ↾ A)‘x) = (𝐹‘x))
46	45	ad2antlr 458	. . . . . 6 ⊢ ((((Smo 𝐹 ∧ Ord A) ∧ x ∈ dom (𝐹 ↾ A)) ∧ y ∈ x) → ((𝐹 ↾ A)‘x) = (𝐹‘x))
47	42, 46	eleq12d 2105	. . . . 5 ⊢ ((((Smo 𝐹 ∧ Ord A) ∧ x ∈ dom (𝐹 ↾ A)) ∧ y ∈ x) → (((𝐹 ↾ A)‘y) ∈ ((𝐹 ↾ A)‘x) ↔ (𝐹‘y) ∈ (𝐹‘x)))
48	47	ralbidva 2316	. . . 4 ⊢ (((Smo 𝐹 ∧ Ord A) ∧ x ∈ dom (𝐹 ↾ A)) → (∀y ∈ x ((𝐹 ↾ A)‘y) ∈ ((𝐹 ↾ A)‘x) ↔ ∀y ∈ x (𝐹‘y) ∈ (𝐹‘x)))
49	48	ralbidva 2316	. . 3 ⊢ ((Smo 𝐹 ∧ Ord A) → (∀x ∈ dom (𝐹 ↾ A)∀y ∈ x ((𝐹 ↾ A)‘y) ∈ ((𝐹 ↾ A)‘x) ↔ ∀x ∈ dom (𝐹 ↾ A)∀y ∈ x (𝐹‘y) ∈ (𝐹‘x)))
50	32, 49	mpbird 156	. 2 ⊢ ((Smo 𝐹 ∧ Ord A) → ∀x ∈ dom (𝐹 ↾ A)∀y ∈ x ((𝐹 ↾ A)‘y) ∈ ((𝐹 ↾ A)‘x))
51		dfsmo2 5843	. 2 ⊢ (Smo (𝐹 ↾ A) ↔ ((𝐹 ↾ A):dom (𝐹 ↾ A)⟶On ∧ Ord dom (𝐹 ↾ A) ∧ ∀x ∈ dom (𝐹 ↾ A)∀y ∈ x ((𝐹 ↾ A)‘y) ∈ ((𝐹 ↾ A)‘x)))
52	17, 25, 50, 51	syl3anbrc 1087	1 ⊢ ((Smo 𝐹 ∧ Ord A) → Smo (𝐹 ↾ A))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1242   ∈ wcel 1390  ∀wral 2300   ∩ cin 2910   ⊆ wss 2911  Ord word 4065  Oncon0 4066  dom cdm 4288  ran crn 4289   ↾ cres 4290   “ cima 4291  Fun wfun 4839   Fn wfn 4840  ⟶wf 4841  ‘cfv 4845  Smo wsmo 5841

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-tr 3846  df-iord 4069  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-fv 4853  df-smo 5842

This theorem is referenced by: (None)