Theorem smores2 7993

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 7986	. . . . . . 7 ⊢ (Smo 𝐹 ↔ (𝐹:dom 𝐹⟶On ∧ Ord dom 𝐹 ∧ ∀𝑥 ∈ dom 𝐹∀𝑦 ∈ 𝑥 (𝐹‘𝑦) ∈ (𝐹‘𝑥)))
2	1	simp1bi 1141	. . . . . 6 ⊢ (Smo 𝐹 → 𝐹:dom 𝐹⟶On)
3	2	ffund 6520	. . . . 5 ⊢ (Smo 𝐹 → Fun 𝐹)
4		funres 6399	. . . . . 6 ⊢ (Fun 𝐹 → Fun (𝐹 ↾ 𝐴))
5	4	funfnd 6388	. . . . 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 5570	. . . . . 6 ⊢ (𝐹 “ 𝐴) = ran (𝐹 ↾ 𝐴)
8		imassrn 5942	. . . . . 6 ⊢ (𝐹 “ 𝐴) ⊆ ran 𝐹
9	7, 8	eqsstrri 4004	. . . . 5 ⊢ ran (𝐹 ↾ 𝐴) ⊆ ran 𝐹
10	2	frnd 6523	. . . . 5 ⊢ (Smo 𝐹 → ran 𝐹 ⊆ On)
11	9, 10	sstrid 3980	. . . 4 ⊢ (Smo 𝐹 → ran (𝐹 ↾ 𝐴) ⊆ On)
12		df-f 6361	. . . 4 ⊢ ((𝐹 ↾ 𝐴):dom (𝐹 ↾ 𝐴)⟶On ↔ ((𝐹 ↾ 𝐴) Fn dom (𝐹 ↾ 𝐴) ∧ ran (𝐹 ↾ 𝐴) ⊆ On))
13	6, 11, 12	sylanbrc 585	. . 3 ⊢ (Smo 𝐹 → (𝐹 ↾ 𝐴):dom (𝐹 ↾ 𝐴)⟶On)
14	13	adantr 483	. 2 ⊢ ((Smo 𝐹 ∧ Ord 𝐴) → (𝐹 ↾ 𝐴):dom (𝐹 ↾ 𝐴)⟶On)
15		smodm 7990	. . 3 ⊢ (Smo 𝐹 → Ord dom 𝐹)
16		ordin 6223	. . . . 5 ⊢ ((Ord 𝐴 ∧ Ord dom 𝐹) → Ord (𝐴 ∩ dom 𝐹))
17		dmres 5877	. . . . . 6 ⊢ dom (𝐹 ↾ 𝐴) = (𝐴 ∩ dom 𝐹)
18		ordeq 6200	. . . . . 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 461	. . 3 ⊢ ((Ord dom 𝐹 ∧ Ord 𝐴) → Ord dom (𝐹 ↾ 𝐴))
22	15, 21	sylan 582	. 2 ⊢ ((Smo 𝐹 ∧ Ord 𝐴) → Ord dom (𝐹 ↾ 𝐴))
23		resss 5880	. . . . . 6 ⊢ (𝐹 ↾ 𝐴) ⊆ 𝐹
24		dmss 5773	. . . . . 6 ⊢ ((𝐹 ↾ 𝐴) ⊆ 𝐹 → dom (𝐹 ↾ 𝐴) ⊆ dom 𝐹)
25	23, 24	ax-mp 5	. . . . 5 ⊢ dom (𝐹 ↾ 𝐴) ⊆ dom 𝐹
26	1	simp3bi 1143	. . . . 5 ⊢ (Smo 𝐹 → ∀𝑥 ∈ dom 𝐹∀𝑦 ∈ 𝑥 (𝐹‘𝑦) ∈ (𝐹‘𝑥))
27		ssralv 4035	. . . . 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 483	. . 3 ⊢ ((Smo 𝐹 ∧ Ord 𝐴) → ∀𝑥 ∈ dom (𝐹 ↾ 𝐴)∀𝑦 ∈ 𝑥 (𝐹‘𝑦) ∈ (𝐹‘𝑥))
30		ordtr1 6236	. . . . . . . . . . 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 4207	. . . . . . . . . . . 12 ⊢ (𝐴 ∩ dom 𝐹) ⊆ 𝐴
33	17, 32	eqsstri 4003	. . . . . . . . . . 11 ⊢ dom (𝐹 ↾ 𝐴) ⊆ 𝐴
34	33	sseli 3965	. . . . . . . . . 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 419	. . . . . . . 8 ⊢ ((Smo 𝐹 ∧ Ord 𝐴) → (𝑥 ∈ dom (𝐹 ↾ 𝐴) → (𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐴)))
37	36	imp31 420	. . . . . . 7 ⊢ ((((Smo 𝐹 ∧ Ord 𝐴) ∧ 𝑥 ∈ dom (𝐹 ↾ 𝐴)) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ 𝐴)
38	37	fvresd 6692	. . . . . 6 ⊢ ((((Smo 𝐹 ∧ Ord 𝐴) ∧ 𝑥 ∈ dom (𝐹 ↾ 𝐴)) ∧ 𝑦 ∈ 𝑥) → ((𝐹 ↾ 𝐴)‘𝑦) = (𝐹‘𝑦))
39	33	sseli 3965	. . . . . . . 8 ⊢ (𝑥 ∈ dom (𝐹 ↾ 𝐴) → 𝑥 ∈ 𝐴)
40	39	fvresd 6692	. . . . . . 7 ⊢ (𝑥 ∈ dom (𝐹 ↾ 𝐴) → ((𝐹 ↾ 𝐴)‘𝑥) = (𝐹‘𝑥))
41	40	ad2antlr 725	. . . . . 6 ⊢ ((((Smo 𝐹 ∧ Ord 𝐴) ∧ 𝑥 ∈ dom (𝐹 ↾ 𝐴)) ∧ 𝑦 ∈ 𝑥) → ((𝐹 ↾ 𝐴)‘𝑥) = (𝐹‘𝑥))
42	38, 41	eleq12d 2909	. . . . 5 ⊢ ((((Smo 𝐹 ∧ Ord 𝐴) ∧ 𝑥 ∈ dom (𝐹 ↾ 𝐴)) ∧ 𝑦 ∈ 𝑥) → (((𝐹 ↾ 𝐴)‘𝑦) ∈ ((𝐹 ↾ 𝐴)‘𝑥) ↔ (𝐹‘𝑦) ∈ (𝐹‘𝑥)))
43	42	ralbidva 3198	. . . 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 3198	. . 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 7986	. 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 1339	1 ⊢ ((Smo 𝐹 ∧ Ord 𝐴) → Smo (𝐹 ↾ 𝐴))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114  ∀wral 3140   ∩ cin 3937   ⊆ wss 3938  dom cdm 5557  ran crn 5558   ↾ cres 5559   “ cima 5560  Ord word 6192  Oncon0 6193  Fun wfun 6351   Fn wfn 6352  ⟶wf 6353  ‘cfv 6357  Smo wsmo 7984

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pr 5332

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-tr 5175  df-po 5476  df-so 5477  df-fr 5516  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-ord 6196  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-fv 6365  df-smo 7985

This theorem is referenced by: (None)