Theorem smores2 8337

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 8330	. . . . . . 7 ⊢ (Smo 𝐹 ↔ (𝐹:dom 𝐹⟶On ∧ Ord dom 𝐹 ∧ ∀𝑥 ∈ dom 𝐹∀𝑦 ∈ 𝑥 (𝐹‘𝑦) ∈ (𝐹‘𝑥)))
2	1	simp1bi 1163	. . . . . 6 ⊢ (Smo 𝐹 → 𝐹:dom 𝐹⟶On)
3	2	ffund 6710	. . . . 5 ⊢ (Smo 𝐹 → Fun 𝐹)
4		funres 6578	. . . . . 6 ⊢ (Fun 𝐹 → Fun (𝐹 ↾ 𝐴))
5	4	funfnd 6567	. . . . 5 ⊢ (Fun 𝐹 → (𝐹 ↾ 𝐴) Fn dom (𝐹 ↾ 𝐴))
6	3, 5	syl 18	. . . 4 ⊢ (Smo 𝐹 → (𝐹 ↾ 𝐴) Fn dom (𝐹 ↾ 𝐴))
7		df-ima 5674	. . . . . 6 ⊢ (𝐹 “ 𝐴) = ran (𝐹 ↾ 𝐴)
8		imassrn 6073	. . . . . 6 ⊢ (𝐹 “ 𝐴) ⊆ ran 𝐹
9	7, 8	eqsstrri 3984	. . . . 5 ⊢ ran (𝐹 ↾ 𝐴) ⊆ ran 𝐹
10	2	frnd 6714	. . . . 5 ⊢ (Smo 𝐹 → ran 𝐹 ⊆ On)
11	9, 10	sstrid 3948	. . . 4 ⊢ (Smo 𝐹 → ran (𝐹 ↾ 𝐴) ⊆ On)
12		df-f 6540	. . . 4 ⊢ ((𝐹 ↾ 𝐴):dom (𝐹 ↾ 𝐴)⟶On ↔ ((𝐹 ↾ 𝐴) Fn dom (𝐹 ↾ 𝐴) ∧ ran (𝐹 ↾ 𝐴) ⊆ On))
13	6, 11, 12	sylanbrc 594	. . 3 ⊢ (Smo 𝐹 → (𝐹 ↾ 𝐴):dom (𝐹 ↾ 𝐴)⟶On)
14	13	adantr 485	. 2 ⊢ ((Smo 𝐹 ∧ Ord 𝐴) → (𝐹 ↾ 𝐴):dom (𝐹 ↾ 𝐴)⟶On)
15		smodm 8334	. . 3 ⊢ (Smo 𝐹 → Ord dom 𝐹)
16		ordin 6391	. . . . 5 ⊢ ((Ord 𝐴 ∧ Ord dom 𝐹) → Ord (𝐴 ∩ dom 𝐹))
17		dmres 6011	. . . . . 6 ⊢ dom (𝐹 ↾ 𝐴) = (𝐴 ∩ dom 𝐹)
18		ordeq 6367	. . . . . 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 237	. . . 4 ⊢ ((Ord 𝐴 ∧ Ord dom 𝐹) → Ord dom (𝐹 ↾ 𝐴))
21	20	ancoms 463	. . 3 ⊢ ((Ord dom 𝐹 ∧ Ord 𝐴) → Ord dom (𝐹 ↾ 𝐴))
22	15, 21	sylan 591	. 2 ⊢ ((Smo 𝐹 ∧ Ord 𝐴) → Ord dom (𝐹 ↾ 𝐴))
23		resss 6000	. . . . . 6 ⊢ (𝐹 ↾ 𝐴) ⊆ 𝐹
24		dmss 5892	. . . . . 6 ⊢ ((𝐹 ↾ 𝐴) ⊆ 𝐹 → dom (𝐹 ↾ 𝐴) ⊆ dom 𝐹)
25	23, 24	ax-mp 5	. . . . 5 ⊢ dom (𝐹 ↾ 𝐴) ⊆ dom 𝐹
26	1	simp3bi 1165	. . . . 5 ⊢ (Smo 𝐹 → ∀𝑥 ∈ dom 𝐹∀𝑦 ∈ 𝑥 (𝐹‘𝑦) ∈ (𝐹‘𝑥))
27		ssralv 4006	. . . . 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 69	. . . 4 ⊢ (Smo 𝐹 → ∀𝑥 ∈ dom (𝐹 ↾ 𝐴)∀𝑦 ∈ 𝑥 (𝐹‘𝑦) ∈ (𝐹‘𝑥))
29	28	adantr 485	. . 3 ⊢ ((Smo 𝐹 ∧ Ord 𝐴) → ∀𝑥 ∈ dom (𝐹 ↾ 𝐴)∀𝑦 ∈ 𝑥 (𝐹‘𝑦) ∈ (𝐹‘𝑥))
30		ordtr1 6405	. . . . . . . . . . 11 ⊢ (Ord dom (𝐹 ↾ 𝐴) → ((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ dom (𝐹 ↾ 𝐴)) → 𝑦 ∈ dom (𝐹 ↾ 𝐴)))
31	22, 30	syl 18	. . . . . . . . . 10 ⊢ ((Smo 𝐹 ∧ Ord 𝐴) → ((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ dom (𝐹 ↾ 𝐴)) → 𝑦 ∈ dom (𝐹 ↾ 𝐴)))
32		inss1 4189	. . . . . . . . . . . 12 ⊢ (𝐴 ∩ dom 𝐹) ⊆ 𝐴
33	17, 32	eqsstri 3983	. . . . . . . . . . 11 ⊢ dom (𝐹 ↾ 𝐴) ⊆ 𝐴
34	33	sseli 3933	. . . . . . . . . 10 ⊢ (𝑦 ∈ dom (𝐹 ↾ 𝐴) → 𝑦 ∈ 𝐴)
35	31, 34	syl6 36	. . . . . . . . 9 ⊢ ((Smo 𝐹 ∧ Ord 𝐴) → ((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ dom (𝐹 ↾ 𝐴)) → 𝑦 ∈ 𝐴))
36	35	expcomd 421	. . . . . . . 8 ⊢ ((Smo 𝐹 ∧ Ord 𝐴) → (𝑥 ∈ dom (𝐹 ↾ 𝐴) → (𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐴)))
37	36	imp31 422	. . . . . . 7 ⊢ ((((Smo 𝐹 ∧ Ord 𝐴) ∧ 𝑥 ∈ dom (𝐹 ↾ 𝐴)) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ 𝐴)
38	37	fvresd 6901	. . . . . 6 ⊢ ((((Smo 𝐹 ∧ Ord 𝐴) ∧ 𝑥 ∈ dom (𝐹 ↾ 𝐴)) ∧ 𝑦 ∈ 𝑥) → ((𝐹 ↾ 𝐴)‘𝑦) = (𝐹‘𝑦))
39	33	sseli 3933	. . . . . . . 8 ⊢ (𝑥 ∈ dom (𝐹 ↾ 𝐴) → 𝑥 ∈ 𝐴)
40	39	fvresd 6901	. . . . . . 7 ⊢ (𝑥 ∈ dom (𝐹 ↾ 𝐴) → ((𝐹 ↾ 𝐴)‘𝑥) = (𝐹‘𝑥))
41	40	ad2antlr 739	. . . . . 6 ⊢ ((((Smo 𝐹 ∧ Ord 𝐴) ∧ 𝑥 ∈ dom (𝐹 ↾ 𝐴)) ∧ 𝑦 ∈ 𝑥) → ((𝐹 ↾ 𝐴)‘𝑥) = (𝐹‘𝑥))
42	38, 41	eleq12d 2857	. . . . 5 ⊢ ((((Smo 𝐹 ∧ Ord 𝐴) ∧ 𝑥 ∈ dom (𝐹 ↾ 𝐴)) ∧ 𝑦 ∈ 𝑥) → (((𝐹 ↾ 𝐴)‘𝑦) ∈ ((𝐹 ↾ 𝐴)‘𝑥) ↔ (𝐹‘𝑦) ∈ (𝐹‘𝑥)))
43	42	ralbidva 3186	. . . 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 3186	. . 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 260	. 2 ⊢ ((Smo 𝐹 ∧ Ord 𝐴) → ∀𝑥 ∈ dom (𝐹 ↾ 𝐴)∀𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐴)‘𝑦) ∈ ((𝐹 ↾ 𝐴)‘𝑥))
46		dfsmo2 8330	. 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 1362	1 ⊢ ((Smo 𝐹 ∧ Ord 𝐴) → Smo (𝐹 ↾ 𝐴))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1570   ∈ wcel 2143  ∀wral 3079   ∩ cin 3904   ⊆ wss 3905  dom cdm 5661  ran crn 5662   ↾ cres 5663   “ cima 5664  Ord word 6359  Oncon0 6360  Fun wfun 6530   Fn wfn 6531  ⟶wf 6532  ‘cfv 6536  Smo wsmo 8328

This theorem was proved from 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  ax-sep 5257  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-tr 5219  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-ord 6363  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-fv 6544  df-smo 8329

This theorem is referenced by: (None)