Theorem smores2 8300

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 8293	. . . . . . 7 ⊢ (Smo 𝐹 ↔ (𝐹:dom 𝐹⟶On ∧ Ord dom 𝐹 ∧ ∀𝑥 ∈ dom 𝐹∀𝑦 ∈ 𝑥 (𝐹‘𝑦) ∈ (𝐹‘𝑥)))
2	1	simp1bi 1145	. . . . . 6 ⊢ (Smo 𝐹 → 𝐹:dom 𝐹⟶On)
3	2	ffund 6674	. . . . 5 ⊢ (Smo 𝐹 → Fun 𝐹)
4		funres 6542	. . . . . 6 ⊢ (Fun 𝐹 → Fun (𝐹 ↾ 𝐴))
5	4	funfnd 6531	. . . . 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 5644	. . . . . 6 ⊢ (𝐹 “ 𝐴) = ran (𝐹 ↾ 𝐴)
8		imassrn 6031	. . . . . 6 ⊢ (𝐹 “ 𝐴) ⊆ ran 𝐹
9	7, 8	eqsstrri 3991	. . . . 5 ⊢ ran (𝐹 ↾ 𝐴) ⊆ ran 𝐹
10	2	frnd 6678	. . . . 5 ⊢ (Smo 𝐹 → ran 𝐹 ⊆ On)
11	9, 10	sstrid 3955	. . . 4 ⊢ (Smo 𝐹 → ran (𝐹 ↾ 𝐴) ⊆ On)
12		df-f 6503	. . . 4 ⊢ ((𝐹 ↾ 𝐴):dom (𝐹 ↾ 𝐴)⟶On ↔ ((𝐹 ↾ 𝐴) Fn dom (𝐹 ↾ 𝐴) ∧ ran (𝐹 ↾ 𝐴) ⊆ On))
13	6, 11, 12	sylanbrc 583	. . 3 ⊢ (Smo 𝐹 → (𝐹 ↾ 𝐴):dom (𝐹 ↾ 𝐴)⟶On)
14	13	adantr 480	. 2 ⊢ ((Smo 𝐹 ∧ Ord 𝐴) → (𝐹 ↾ 𝐴):dom (𝐹 ↾ 𝐴)⟶On)
15		smodm 8297	. . 3 ⊢ (Smo 𝐹 → Ord dom 𝐹)
16		ordin 6350	. . . . 5 ⊢ ((Ord 𝐴 ∧ Ord dom 𝐹) → Ord (𝐴 ∩ dom 𝐹))
17		dmres 5972	. . . . . 6 ⊢ dom (𝐹 ↾ 𝐴) = (𝐴 ∩ dom 𝐹)
18		ordeq 6327	. . . . . 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 234	. . . 4 ⊢ ((Ord 𝐴 ∧ Ord dom 𝐹) → Ord dom (𝐹 ↾ 𝐴))
21	20	ancoms 458	. . 3 ⊢ ((Ord dom 𝐹 ∧ Ord 𝐴) → Ord dom (𝐹 ↾ 𝐴))
22	15, 21	sylan 580	. 2 ⊢ ((Smo 𝐹 ∧ Ord 𝐴) → Ord dom (𝐹 ↾ 𝐴))
23		resss 5961	. . . . . 6 ⊢ (𝐹 ↾ 𝐴) ⊆ 𝐹
24		dmss 5856	. . . . . 6 ⊢ ((𝐹 ↾ 𝐴) ⊆ 𝐹 → dom (𝐹 ↾ 𝐴) ⊆ dom 𝐹)
25	23, 24	ax-mp 5	. . . . 5 ⊢ dom (𝐹 ↾ 𝐴) ⊆ dom 𝐹
26	1	simp3bi 1147	. . . . 5 ⊢ (Smo 𝐹 → ∀𝑥 ∈ dom 𝐹∀𝑦 ∈ 𝑥 (𝐹‘𝑦) ∈ (𝐹‘𝑥))
27		ssralv 4012	. . . . 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 480	. . 3 ⊢ ((Smo 𝐹 ∧ Ord 𝐴) → ∀𝑥 ∈ dom (𝐹 ↾ 𝐴)∀𝑦 ∈ 𝑥 (𝐹‘𝑦) ∈ (𝐹‘𝑥))
30		ordtr1 6364	. . . . . . . . . . 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 4196	. . . . . . . . . . . 12 ⊢ (𝐴 ∩ dom 𝐹) ⊆ 𝐴
33	17, 32	eqsstri 3990	. . . . . . . . . . 11 ⊢ dom (𝐹 ↾ 𝐴) ⊆ 𝐴
34	33	sseli 3939	. . . . . . . . . 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 416	. . . . . . . 8 ⊢ ((Smo 𝐹 ∧ Ord 𝐴) → (𝑥 ∈ dom (𝐹 ↾ 𝐴) → (𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐴)))
37	36	imp31 417	. . . . . . 7 ⊢ ((((Smo 𝐹 ∧ Ord 𝐴) ∧ 𝑥 ∈ dom (𝐹 ↾ 𝐴)) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ 𝐴)
38	37	fvresd 6860	. . . . . 6 ⊢ ((((Smo 𝐹 ∧ Ord 𝐴) ∧ 𝑥 ∈ dom (𝐹 ↾ 𝐴)) ∧ 𝑦 ∈ 𝑥) → ((𝐹 ↾ 𝐴)‘𝑦) = (𝐹‘𝑦))
39	33	sseli 3939	. . . . . . . 8 ⊢ (𝑥 ∈ dom (𝐹 ↾ 𝐴) → 𝑥 ∈ 𝐴)
40	39	fvresd 6860	. . . . . . 7 ⊢ (𝑥 ∈ dom (𝐹 ↾ 𝐴) → ((𝐹 ↾ 𝐴)‘𝑥) = (𝐹‘𝑥))
41	40	ad2antlr 727	. . . . . 6 ⊢ ((((Smo 𝐹 ∧ Ord 𝐴) ∧ 𝑥 ∈ dom (𝐹 ↾ 𝐴)) ∧ 𝑦 ∈ 𝑥) → ((𝐹 ↾ 𝐴)‘𝑥) = (𝐹‘𝑥))
42	38, 41	eleq12d 2822	. . . . 5 ⊢ ((((Smo 𝐹 ∧ Ord 𝐴) ∧ 𝑥 ∈ dom (𝐹 ↾ 𝐴)) ∧ 𝑦 ∈ 𝑥) → (((𝐹 ↾ 𝐴)‘𝑦) ∈ ((𝐹 ↾ 𝐴)‘𝑥) ↔ (𝐹‘𝑦) ∈ (𝐹‘𝑥)))
43	42	ralbidva 3154	. . . 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 3154	. . 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 257	. 2 ⊢ ((Smo 𝐹 ∧ Ord 𝐴) → ∀𝑥 ∈ dom (𝐹 ↾ 𝐴)∀𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐴)‘𝑦) ∈ ((𝐹 ↾ 𝐴)‘𝑥))
46		dfsmo2 8293	. 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 1344	1 ⊢ ((Smo 𝐹 ∧ Ord 𝐴) → Smo (𝐹 ↾ 𝐴))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044   ∩ cin 3910   ⊆ wss 3911  dom cdm 5631  ran crn 5632   ↾ cres 5633   “ cima 5634  Ord word 6319  Oncon0 6320  Fun wfun 6493   Fn wfn 6494  ⟶wf 6495  ‘cfv 6499  Smo wsmo 8291

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-11 2158  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-tr 5210  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-ord 6323  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-fv 6507  df-smo 8292

This theorem is referenced by: (None)