Theorem smores2 7734

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 7727	. . . . . . 7 ⊢ (Smo 𝐹 ↔ (𝐹:dom 𝐹⟶On ∧ Ord dom 𝐹 ∧ ∀𝑥 ∈ dom 𝐹∀𝑦 ∈ 𝑥 (𝐹‘𝑦) ∈ (𝐹‘𝑥)))
2	1	simp1bi 1136	. . . . . 6 ⊢ (Smo 𝐹 → 𝐹:dom 𝐹⟶On)
3	2	ffund 6295	. . . . 5 ⊢ (Smo 𝐹 → Fun 𝐹)
4		funres 6177	. . . . . 6 ⊢ (Fun 𝐹 → Fun (𝐹 ↾ 𝐴))
5	4	funfnd 6166	. . . . 5 ⊢ (Fun 𝐹 → (𝐹 ↾ 𝐴) Fn dom (𝐹 ↾ 𝐴))
6	3, 5	syl 17	. . . 4 ⊢ (Smo 𝐹 → (𝐹 ↾ 𝐴) Fn dom (𝐹 ↾ 𝐴))
7		df-ima 5368	. . . . . 6 ⊢ (𝐹 “ 𝐴) = ran (𝐹 ↾ 𝐴)
8		imassrn 5731	. . . . . 6 ⊢ (𝐹 “ 𝐴) ⊆ ran 𝐹
9	7, 8	eqsstr3i 3855	. . . . 5 ⊢ ran (𝐹 ↾ 𝐴) ⊆ ran 𝐹
10	2	frnd 6298	. . . . 5 ⊢ (Smo 𝐹 → ran 𝐹 ⊆ On)
11	9, 10	syl5ss 3832	. . . 4 ⊢ (Smo 𝐹 → ran (𝐹 ↾ 𝐴) ⊆ On)
12		df-f 6139	. . . 4 ⊢ ((𝐹 ↾ 𝐴):dom (𝐹 ↾ 𝐴)⟶On ↔ ((𝐹 ↾ 𝐴) Fn dom (𝐹 ↾ 𝐴) ∧ ran (𝐹 ↾ 𝐴) ⊆ On))
13	6, 11, 12	sylanbrc 578	. . 3 ⊢ (Smo 𝐹 → (𝐹 ↾ 𝐴):dom (𝐹 ↾ 𝐴)⟶On)
14	13	adantr 474	. 2 ⊢ ((Smo 𝐹 ∧ Ord 𝐴) → (𝐹 ↾ 𝐴):dom (𝐹 ↾ 𝐴)⟶On)
15		smodm 7731	. . 3 ⊢ (Smo 𝐹 → Ord dom 𝐹)
16		ordin 6006	. . . . 5 ⊢ ((Ord 𝐴 ∧ Ord dom 𝐹) → Ord (𝐴 ∩ dom 𝐹))
17		dmres 5668	. . . . . 6 ⊢ dom (𝐹 ↾ 𝐴) = (𝐴 ∩ dom 𝐹)
18		ordeq 5983	. . . . . 6 ⊢ (dom (𝐹 ↾ 𝐴) = (𝐴 ∩ dom 𝐹) → (Ord dom (𝐹 ↾ 𝐴) ↔ Ord (𝐴 ∩ dom 𝐹)))
19	17, 18	ax-mp 5	. . . . 5 ⊢ (Ord dom (𝐹 ↾ 𝐴) ↔ Ord (𝐴 ∩ dom 𝐹))
20	16, 19	sylibr 226	. . . 4 ⊢ ((Ord 𝐴 ∧ Ord dom 𝐹) → Ord dom (𝐹 ↾ 𝐴))
21	20	ancoms 452	. . 3 ⊢ ((Ord dom 𝐹 ∧ Ord 𝐴) → Ord dom (𝐹 ↾ 𝐴))
22	15, 21	sylan 575	. 2 ⊢ ((Smo 𝐹 ∧ Ord 𝐴) → Ord dom (𝐹 ↾ 𝐴))
23		resss 5671	. . . . . 6 ⊢ (𝐹 ↾ 𝐴) ⊆ 𝐹
24		dmss 5568	. . . . . 6 ⊢ ((𝐹 ↾ 𝐴) ⊆ 𝐹 → dom (𝐹 ↾ 𝐴) ⊆ dom 𝐹)
25	23, 24	ax-mp 5	. . . . 5 ⊢ dom (𝐹 ↾ 𝐴) ⊆ dom 𝐹
26	1	simp3bi 1138	. . . . 5 ⊢ (Smo 𝐹 → ∀𝑥 ∈ dom 𝐹∀𝑦 ∈ 𝑥 (𝐹‘𝑦) ∈ (𝐹‘𝑥))
27		ssralv 3885	. . . . 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)))
28	25, 26, 27	mpsyl 68	. . . 4 ⊢ (Smo 𝐹 → ∀𝑥 ∈ dom (𝐹 ↾ 𝐴)∀𝑦 ∈ 𝑥 (𝐹‘𝑦) ∈ (𝐹‘𝑥))
29	28	adantr 474	. . 3 ⊢ ((Smo 𝐹 ∧ Ord 𝐴) → ∀𝑥 ∈ dom (𝐹 ↾ 𝐴)∀𝑦 ∈ 𝑥 (𝐹‘𝑦) ∈ (𝐹‘𝑥))
30		ordtr1 6019	. . . . . . . . . . 11 ⊢ (Ord dom (𝐹 ↾ 𝐴) → ((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ dom (𝐹 ↾ 𝐴)) → 𝑦 ∈ dom (𝐹 ↾ 𝐴)))
31	22, 30	syl 17	. . . . . . . . . 10 ⊢ ((Smo 𝐹 ∧ Ord 𝐴) → ((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ dom (𝐹 ↾ 𝐴)) → 𝑦 ∈ dom (𝐹 ↾ 𝐴)))
32		inss1 4053	. . . . . . . . . . . 12 ⊢ (𝐴 ∩ dom 𝐹) ⊆ 𝐴
33	17, 32	eqsstri 3854	. . . . . . . . . . 11 ⊢ dom (𝐹 ↾ 𝐴) ⊆ 𝐴
34	33	sseli 3817	. . . . . . . . . 10 ⊢ (𝑦 ∈ dom (𝐹 ↾ 𝐴) → 𝑦 ∈ 𝐴)
35	31, 34	syl6 35	. . . . . . . . 9 ⊢ ((Smo 𝐹 ∧ Ord 𝐴) → ((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ dom (𝐹 ↾ 𝐴)) → 𝑦 ∈ 𝐴))
36	35	expcomd 408	. . . . . . . 8 ⊢ ((Smo 𝐹 ∧ Ord 𝐴) → (𝑥 ∈ dom (𝐹 ↾ 𝐴) → (𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐴)))
37	36	imp31 410	. . . . . . 7 ⊢ ((((Smo 𝐹 ∧ Ord 𝐴) ∧ 𝑥 ∈ dom (𝐹 ↾ 𝐴)) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ 𝐴)
38	37	fvresd 6466	. . . . . 6 ⊢ ((((Smo 𝐹 ∧ Ord 𝐴) ∧ 𝑥 ∈ dom (𝐹 ↾ 𝐴)) ∧ 𝑦 ∈ 𝑥) → ((𝐹 ↾ 𝐴)‘𝑦) = (𝐹‘𝑦))
39	33	sseli 3817	. . . . . . . 8 ⊢ (𝑥 ∈ dom (𝐹 ↾ 𝐴) → 𝑥 ∈ 𝐴)
40	39	fvresd 6466	. . . . . . 7 ⊢ (𝑥 ∈ dom (𝐹 ↾ 𝐴) → ((𝐹 ↾ 𝐴)‘𝑥) = (𝐹‘𝑥))
41	40	ad2antlr 717	. . . . . 6 ⊢ ((((Smo 𝐹 ∧ Ord 𝐴) ∧ 𝑥 ∈ dom (𝐹 ↾ 𝐴)) ∧ 𝑦 ∈ 𝑥) → ((𝐹 ↾ 𝐴)‘𝑥) = (𝐹‘𝑥))
42	38, 41	eleq12d 2853	. . . . 5 ⊢ ((((Smo 𝐹 ∧ Ord 𝐴) ∧ 𝑥 ∈ dom (𝐹 ↾ 𝐴)) ∧ 𝑦 ∈ 𝑥) → (((𝐹 ↾ 𝐴)‘𝑦) ∈ ((𝐹 ↾ 𝐴)‘𝑥) ↔ (𝐹‘𝑦) ∈ (𝐹‘𝑥)))
43	42	ralbidva 3167	. . . 4 ⊢ (((Smo 𝐹 ∧ Ord 𝐴) ∧ 𝑥 ∈ dom (𝐹 ↾ 𝐴)) → (∀𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐴)‘𝑦) ∈ ((𝐹 ↾ 𝐴)‘𝑥) ↔ ∀𝑦 ∈ 𝑥 (𝐹‘𝑦) ∈ (𝐹‘𝑥)))
44	43	ralbidva 3167	. . 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)))
45	29, 44	mpbird 249	. 2 ⊢ ((Smo 𝐹 ∧ Ord 𝐴) → ∀𝑥 ∈ dom (𝐹 ↾ 𝐴)∀𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐴)‘𝑦) ∈ ((𝐹 ↾ 𝐴)‘𝑥))
46		dfsmo2 7727	. 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)))
47	14, 22, 45, 46	syl3anbrc 1400	1 ⊢ ((Smo 𝐹 ∧ Ord 𝐴) → Smo (𝐹 ↾ 𝐴))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   = wceq 1601   ∈ wcel 2107  ∀wral 3090   ∩ cin 3791   ⊆ wss 3792  dom cdm 5355  ran crn 5356   ↾ cres 5357   “ cima 5358  Ord word 5975  Oncon0 5976  Fun wfun 6129   Fn wfn 6130  ⟶wf 6131  ‘cfv 6135  Smo wsmo 7725

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pr 5138

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-br 4887  df-opab 4949  df-tr 4988  df-po 5274  df-so 5275  df-fr 5314  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-ord 5979  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-fv 6143  df-smo 7726

This theorem is referenced by: (None)