Theorem smores2 8325

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 8318	. . . . . . 7 ⊢ (Smo 𝐹 ↔ (𝐹:dom 𝐹⟶On ∧ Ord dom 𝐹 ∧ ∀𝑥 ∈ dom 𝐹∀𝑦 ∈ 𝑥 (𝐹‘𝑦) ∈ (𝐹‘𝑥)))
2	1	simp1bi 1158	. . . . . 6 ⊢ (Smo 𝐹 → 𝐹:dom 𝐹⟶On)
3	2	ffund 6696	. . . . 5 ⊢ (Smo 𝐹 → Fun 𝐹)
4		funres 6563	. . . . . 6 ⊢ (Fun 𝐹 → Fun (𝐹 ↾ 𝐴))
5	4	funfnd 6552	. . . . 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 5660	. . . . . 6 ⊢ (𝐹 “ 𝐴) = ran (𝐹 ↾ 𝐴)
8		imassrn 6060	. . . . . 6 ⊢ (𝐹 “ 𝐴) ⊆ ran 𝐹
9	7, 8	eqsstrri 3983	. . . . 5 ⊢ ran (𝐹 ↾ 𝐴) ⊆ ran 𝐹
10	2	frnd 6700	. . . . 5 ⊢ (Smo 𝐹 → ran 𝐹 ⊆ On)
11	9, 10	sstrid 3947	. . . 4 ⊢ (Smo 𝐹 → ran (𝐹 ↾ 𝐴) ⊆ On)
12		df-f 6525	. . . 4 ⊢ ((𝐹 ↾ 𝐴):dom (𝐹 ↾ 𝐴)⟶On ↔ ((𝐹 ↾ 𝐴) Fn dom (𝐹 ↾ 𝐴) ∧ ran (𝐹 ↾ 𝐴) ⊆ On))
13	6, 11, 12	sylanbrc 592	. . 3 ⊢ (Smo 𝐹 → (𝐹 ↾ 𝐴):dom (𝐹 ↾ 𝐴)⟶On)
14	13	adantr 484	. 2 ⊢ ((Smo 𝐹 ∧ Ord 𝐴) → (𝐹 ↾ 𝐴):dom (𝐹 ↾ 𝐴)⟶On)
15		smodm 8322	. . 3 ⊢ (Smo 𝐹 → Ord dom 𝐹)
16		ordin 6376	. . . . 5 ⊢ ((Ord 𝐴 ∧ Ord dom 𝐹) → Ord (𝐴 ∩ dom 𝐹))
17		dmres 5998	. . . . . 6 ⊢ dom (𝐹 ↾ 𝐴) = (𝐴 ∩ dom 𝐹)
18		ordeq 6353	. . . . . 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 236	. . . 4 ⊢ ((Ord 𝐴 ∧ Ord dom 𝐹) → Ord dom (𝐹 ↾ 𝐴))
21	20	ancoms 462	. . 3 ⊢ ((Ord dom 𝐹 ∧ Ord 𝐴) → Ord dom (𝐹 ↾ 𝐴))
22	15, 21	sylan 589	. 2 ⊢ ((Smo 𝐹 ∧ Ord 𝐴) → Ord dom (𝐹 ↾ 𝐴))
23		resss 5987	. . . . . 6 ⊢ (𝐹 ↾ 𝐴) ⊆ 𝐹
24		dmss 5878	. . . . . 6 ⊢ ((𝐹 ↾ 𝐴) ⊆ 𝐹 → dom (𝐹 ↾ 𝐴) ⊆ dom 𝐹)
25	23, 24	ax-mp 5	. . . . 5 ⊢ dom (𝐹 ↾ 𝐴) ⊆ dom 𝐹
26	1	simp3bi 1160	. . . . 5 ⊢ (Smo 𝐹 → ∀𝑥 ∈ dom 𝐹∀𝑦 ∈ 𝑥 (𝐹‘𝑦) ∈ (𝐹‘𝑥))
27		ssralv 4005	. . . . 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 484	. . 3 ⊢ ((Smo 𝐹 ∧ Ord 𝐴) → ∀𝑥 ∈ dom (𝐹 ↾ 𝐴)∀𝑦 ∈ 𝑥 (𝐹‘𝑦) ∈ (𝐹‘𝑥))
30		ordtr1 6390	. . . . . . . . . . 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 4188	. . . . . . . . . . . 12 ⊢ (𝐴 ∩ dom 𝐹) ⊆ 𝐴
33	17, 32	eqsstri 3982	. . . . . . . . . . 11 ⊢ dom (𝐹 ↾ 𝐴) ⊆ 𝐴
34	33	sseli 3932	. . . . . . . . . 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 420	. . . . . . . 8 ⊢ ((Smo 𝐹 ∧ Ord 𝐴) → (𝑥 ∈ dom (𝐹 ↾ 𝐴) → (𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐴)))
37	36	imp31 421	. . . . . . 7 ⊢ ((((Smo 𝐹 ∧ Ord 𝐴) ∧ 𝑥 ∈ dom (𝐹 ↾ 𝐴)) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ 𝐴)
38	37	fvresd 6887	. . . . . 6 ⊢ ((((Smo 𝐹 ∧ Ord 𝐴) ∧ 𝑥 ∈ dom (𝐹 ↾ 𝐴)) ∧ 𝑦 ∈ 𝑥) → ((𝐹 ↾ 𝐴)‘𝑦) = (𝐹‘𝑦))
39	33	sseli 3932	. . . . . . . 8 ⊢ (𝑥 ∈ dom (𝐹 ↾ 𝐴) → 𝑥 ∈ 𝐴)
40	39	fvresd 6887	. . . . . . 7 ⊢ (𝑥 ∈ dom (𝐹 ↾ 𝐴) → ((𝐹 ↾ 𝐴)‘𝑥) = (𝐹‘𝑥))
41	40	ad2antlr 737	. . . . . 6 ⊢ ((((Smo 𝐹 ∧ Ord 𝐴) ∧ 𝑥 ∈ dom (𝐹 ↾ 𝐴)) ∧ 𝑦 ∈ 𝑥) → ((𝐹 ↾ 𝐴)‘𝑥) = (𝐹‘𝑥))
42	38, 41	eleq12d 2856	. . . . 5 ⊢ ((((Smo 𝐹 ∧ Ord 𝐴) ∧ 𝑥 ∈ dom (𝐹 ↾ 𝐴)) ∧ 𝑦 ∈ 𝑥) → (((𝐹 ↾ 𝐴)‘𝑦) ∈ ((𝐹 ↾ 𝐴)‘𝑥) ↔ (𝐹‘𝑦) ∈ (𝐹‘𝑥)))
43	42	ralbidva 3183	. . . 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 3183	. . 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 259	. 2 ⊢ ((Smo 𝐹 ∧ Ord 𝐴) → ∀𝑥 ∈ dom (𝐹 ↾ 𝐴)∀𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐴)‘𝑦) ∈ ((𝐹 ↾ 𝐴)‘𝑥))
46		dfsmo2 8318	. 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 1357	1 ⊢ ((Smo 𝐹 ∧ Ord 𝐴) → Smo (𝐹 ↾ 𝐴))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1560   ∈ wcel 2142  ∀wral 3076   ∩ cin 3903   ⊆ wss 3904  dom cdm 5647  ran crn 5648   ↾ cres 5649   “ cima 5650  Ord word 6345  Oncon0 6346  Fun wfun 6515   Fn wfn 6516  ⟶wf 6517  ‘cfv 6521  Smo wsmo 8316

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-11 2191  ax-ext 2734  ax-sep 5246  ax-pr 5390

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-tr 5208  df-po 5555  df-so 5556  df-fr 5600  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-ord 6349  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-fv 6529  df-smo 8317

This theorem is referenced by: (None)