Theorem smores 7972

Description: A strictly monotone function restricted to an ordinal remains strictly monotone. (Contributed by Andrew Salmon, 16-Nov-2011.) (Proof shortened by Mario Carneiro, 5-Dec-2016.)

Assertion

Ref	Expression
smores	⊢ ((Smo 𝐴 ∧ 𝐵 ∈ dom 𝐴) → Smo (𝐴 ↾ 𝐵))

Proof of Theorem smores

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		funres 6366	. . . . . . . 8 ⊢ (Fun 𝐴 → Fun (𝐴 ↾ 𝐵))
2		funfn 6354	. . . . . . . 8 ⊢ (Fun 𝐴 ↔ 𝐴 Fn dom 𝐴)
3		funfn 6354	. . . . . . . 8 ⊢ (Fun (𝐴 ↾ 𝐵) ↔ (𝐴 ↾ 𝐵) Fn dom (𝐴 ↾ 𝐵))
4	1, 2, 3	3imtr3i 294	. . . . . . 7 ⊢ (𝐴 Fn dom 𝐴 → (𝐴 ↾ 𝐵) Fn dom (𝐴 ↾ 𝐵))
5		resss 5843	. . . . . . . . 9 ⊢ (𝐴 ↾ 𝐵) ⊆ 𝐴
6	5	rnssi 5774	. . . . . . . 8 ⊢ ran (𝐴 ↾ 𝐵) ⊆ ran 𝐴
7		sstr 3923	. . . . . . . 8 ⊢ ((ran (𝐴 ↾ 𝐵) ⊆ ran 𝐴 ∧ ran 𝐴 ⊆ On) → ran (𝐴 ↾ 𝐵) ⊆ On)
8	6, 7	mpan 689	. . . . . . 7 ⊢ (ran 𝐴 ⊆ On → ran (𝐴 ↾ 𝐵) ⊆ On)
9	4, 8	anim12i 615	. . . . . 6 ⊢ ((𝐴 Fn dom 𝐴 ∧ ran 𝐴 ⊆ On) → ((𝐴 ↾ 𝐵) Fn dom (𝐴 ↾ 𝐵) ∧ ran (𝐴 ↾ 𝐵) ⊆ On))
10		df-f 6328	. . . . . 6 ⊢ (𝐴:dom 𝐴⟶On ↔ (𝐴 Fn dom 𝐴 ∧ ran 𝐴 ⊆ On))
11		df-f 6328	. . . . . 6 ⊢ ((𝐴 ↾ 𝐵):dom (𝐴 ↾ 𝐵)⟶On ↔ ((𝐴 ↾ 𝐵) Fn dom (𝐴 ↾ 𝐵) ∧ ran (𝐴 ↾ 𝐵) ⊆ On))
12	9, 10, 11	3imtr4i 295	. . . . 5 ⊢ (𝐴:dom 𝐴⟶On → (𝐴 ↾ 𝐵):dom (𝐴 ↾ 𝐵)⟶On)
13	12	a1i 11	. . . 4 ⊢ (𝐵 ∈ dom 𝐴 → (𝐴:dom 𝐴⟶On → (𝐴 ↾ 𝐵):dom (𝐴 ↾ 𝐵)⟶On))
14		ordelord 6181	. . . . . . 7 ⊢ ((Ord dom 𝐴 ∧ 𝐵 ∈ dom 𝐴) → Ord 𝐵)
15	14	expcom 417	. . . . . 6 ⊢ (𝐵 ∈ dom 𝐴 → (Ord dom 𝐴 → Ord 𝐵))
16		ordin 6189	. . . . . . 7 ⊢ ((Ord 𝐵 ∧ Ord dom 𝐴) → Ord (𝐵 ∩ dom 𝐴))
17	16	ex 416	. . . . . 6 ⊢ (Ord 𝐵 → (Ord dom 𝐴 → Ord (𝐵 ∩ dom 𝐴)))
18	15, 17	syli 39	. . . . 5 ⊢ (𝐵 ∈ dom 𝐴 → (Ord dom 𝐴 → Ord (𝐵 ∩ dom 𝐴)))
19		dmres 5840	. . . . . 6 ⊢ dom (𝐴 ↾ 𝐵) = (𝐵 ∩ dom 𝐴)
20		ordeq 6166	. . . . . 6 ⊢ (dom (𝐴 ↾ 𝐵) = (𝐵 ∩ dom 𝐴) → (Ord dom (𝐴 ↾ 𝐵) ↔ Ord (𝐵 ∩ dom 𝐴)))
21	19, 20	ax-mp 5	. . . . 5 ⊢ (Ord dom (𝐴 ↾ 𝐵) ↔ Ord (𝐵 ∩ dom 𝐴))
22	18, 21	syl6ibr 255	. . . 4 ⊢ (𝐵 ∈ dom 𝐴 → (Ord dom 𝐴 → Ord dom (𝐴 ↾ 𝐵)))
23		dmss 5735	. . . . . . . . 9 ⊢ ((𝐴 ↾ 𝐵) ⊆ 𝐴 → dom (𝐴 ↾ 𝐵) ⊆ dom 𝐴)
24	5, 23	ax-mp 5	. . . . . . . 8 ⊢ dom (𝐴 ↾ 𝐵) ⊆ dom 𝐴
25		ssralv 3981	. . . . . . . 8 ⊢ (dom (𝐴 ↾ 𝐵) ⊆ dom 𝐴 → (∀𝑥 ∈ dom 𝐴∀𝑦 ∈ dom 𝐴(𝑥 ∈ 𝑦 → (𝐴‘𝑥) ∈ (𝐴‘𝑦)) → ∀𝑥 ∈ dom (𝐴 ↾ 𝐵)∀𝑦 ∈ dom 𝐴(𝑥 ∈ 𝑦 → (𝐴‘𝑥) ∈ (𝐴‘𝑦))))
26	24, 25	ax-mp 5	. . . . . . 7 ⊢ (∀𝑥 ∈ dom 𝐴∀𝑦 ∈ dom 𝐴(𝑥 ∈ 𝑦 → (𝐴‘𝑥) ∈ (𝐴‘𝑦)) → ∀𝑥 ∈ dom (𝐴 ↾ 𝐵)∀𝑦 ∈ dom 𝐴(𝑥 ∈ 𝑦 → (𝐴‘𝑥) ∈ (𝐴‘𝑦)))
27		ssralv 3981	. . . . . . . . 9 ⊢ (dom (𝐴 ↾ 𝐵) ⊆ dom 𝐴 → (∀𝑦 ∈ dom 𝐴(𝑥 ∈ 𝑦 → (𝐴‘𝑥) ∈ (𝐴‘𝑦)) → ∀𝑦 ∈ dom (𝐴 ↾ 𝐵)(𝑥 ∈ 𝑦 → (𝐴‘𝑥) ∈ (𝐴‘𝑦))))
28	24, 27	ax-mp 5	. . . . . . . 8 ⊢ (∀𝑦 ∈ dom 𝐴(𝑥 ∈ 𝑦 → (𝐴‘𝑥) ∈ (𝐴‘𝑦)) → ∀𝑦 ∈ dom (𝐴 ↾ 𝐵)(𝑥 ∈ 𝑦 → (𝐴‘𝑥) ∈ (𝐴‘𝑦)))
29	28	ralimi 3128	. . . . . . 7 ⊢ (∀𝑥 ∈ dom (𝐴 ↾ 𝐵)∀𝑦 ∈ dom 𝐴(𝑥 ∈ 𝑦 → (𝐴‘𝑥) ∈ (𝐴‘𝑦)) → ∀𝑥 ∈ dom (𝐴 ↾ 𝐵)∀𝑦 ∈ dom (𝐴 ↾ 𝐵)(𝑥 ∈ 𝑦 → (𝐴‘𝑥) ∈ (𝐴‘𝑦)))
30	26, 29	syl 17	. . . . . 6 ⊢ (∀𝑥 ∈ dom 𝐴∀𝑦 ∈ dom 𝐴(𝑥 ∈ 𝑦 → (𝐴‘𝑥) ∈ (𝐴‘𝑦)) → ∀𝑥 ∈ dom (𝐴 ↾ 𝐵)∀𝑦 ∈ dom (𝐴 ↾ 𝐵)(𝑥 ∈ 𝑦 → (𝐴‘𝑥) ∈ (𝐴‘𝑦)))
31		inss1 4155	. . . . . . . . . . . . 13 ⊢ (𝐵 ∩ dom 𝐴) ⊆ 𝐵
32	19, 31	eqsstri 3949	. . . . . . . . . . . 12 ⊢ dom (𝐴 ↾ 𝐵) ⊆ 𝐵
33		simpl 486	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ dom (𝐴 ↾ 𝐵) ∧ 𝑦 ∈ dom (𝐴 ↾ 𝐵)) → 𝑥 ∈ dom (𝐴 ↾ 𝐵))
34	32, 33	sseldi 3913	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ dom (𝐴 ↾ 𝐵) ∧ 𝑦 ∈ dom (𝐴 ↾ 𝐵)) → 𝑥 ∈ 𝐵)
35	34	fvresd 6665	. . . . . . . . . 10 ⊢ ((𝑥 ∈ dom (𝐴 ↾ 𝐵) ∧ 𝑦 ∈ dom (𝐴 ↾ 𝐵)) → ((𝐴 ↾ 𝐵)‘𝑥) = (𝐴‘𝑥))
36		simpr 488	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ dom (𝐴 ↾ 𝐵) ∧ 𝑦 ∈ dom (𝐴 ↾ 𝐵)) → 𝑦 ∈ dom (𝐴 ↾ 𝐵))
37	32, 36	sseldi 3913	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ dom (𝐴 ↾ 𝐵) ∧ 𝑦 ∈ dom (𝐴 ↾ 𝐵)) → 𝑦 ∈ 𝐵)
38	37	fvresd 6665	. . . . . . . . . 10 ⊢ ((𝑥 ∈ dom (𝐴 ↾ 𝐵) ∧ 𝑦 ∈ dom (𝐴 ↾ 𝐵)) → ((𝐴 ↾ 𝐵)‘𝑦) = (𝐴‘𝑦))
39	35, 38	eleq12d 2884	. . . . . . . . 9 ⊢ ((𝑥 ∈ dom (𝐴 ↾ 𝐵) ∧ 𝑦 ∈ dom (𝐴 ↾ 𝐵)) → (((𝐴 ↾ 𝐵)‘𝑥) ∈ ((𝐴 ↾ 𝐵)‘𝑦) ↔ (𝐴‘𝑥) ∈ (𝐴‘𝑦)))
40	39	imbi2d 344	. . . . . . . 8 ⊢ ((𝑥 ∈ dom (𝐴 ↾ 𝐵) ∧ 𝑦 ∈ dom (𝐴 ↾ 𝐵)) → ((𝑥 ∈ 𝑦 → ((𝐴 ↾ 𝐵)‘𝑥) ∈ ((𝐴 ↾ 𝐵)‘𝑦)) ↔ (𝑥 ∈ 𝑦 → (𝐴‘𝑥) ∈ (𝐴‘𝑦))))
41	40	ralbidva 3161	. . . . . . 7 ⊢ (𝑥 ∈ dom (𝐴 ↾ 𝐵) → (∀𝑦 ∈ dom (𝐴 ↾ 𝐵)(𝑥 ∈ 𝑦 → ((𝐴 ↾ 𝐵)‘𝑥) ∈ ((𝐴 ↾ 𝐵)‘𝑦)) ↔ ∀𝑦 ∈ dom (𝐴 ↾ 𝐵)(𝑥 ∈ 𝑦 → (𝐴‘𝑥) ∈ (𝐴‘𝑦))))
42	41	ralbiia 3132	. . . . . 6 ⊢ (∀𝑥 ∈ dom (𝐴 ↾ 𝐵)∀𝑦 ∈ dom (𝐴 ↾ 𝐵)(𝑥 ∈ 𝑦 → ((𝐴 ↾ 𝐵)‘𝑥) ∈ ((𝐴 ↾ 𝐵)‘𝑦)) ↔ ∀𝑥 ∈ dom (𝐴 ↾ 𝐵)∀𝑦 ∈ dom (𝐴 ↾ 𝐵)(𝑥 ∈ 𝑦 → (𝐴‘𝑥) ∈ (𝐴‘𝑦)))
43	30, 42	sylibr 237	. . . . 5 ⊢ (∀𝑥 ∈ dom 𝐴∀𝑦 ∈ dom 𝐴(𝑥 ∈ 𝑦 → (𝐴‘𝑥) ∈ (𝐴‘𝑦)) → ∀𝑥 ∈ dom (𝐴 ↾ 𝐵)∀𝑦 ∈ dom (𝐴 ↾ 𝐵)(𝑥 ∈ 𝑦 → ((𝐴 ↾ 𝐵)‘𝑥) ∈ ((𝐴 ↾ 𝐵)‘𝑦)))
44	43	a1i 11	. . . 4 ⊢ (𝐵 ∈ dom 𝐴 → (∀𝑥 ∈ dom 𝐴∀𝑦 ∈ dom 𝐴(𝑥 ∈ 𝑦 → (𝐴‘𝑥) ∈ (𝐴‘𝑦)) → ∀𝑥 ∈ dom (𝐴 ↾ 𝐵)∀𝑦 ∈ dom (𝐴 ↾ 𝐵)(𝑥 ∈ 𝑦 → ((𝐴 ↾ 𝐵)‘𝑥) ∈ ((𝐴 ↾ 𝐵)‘𝑦))))
45	13, 22, 44	3anim123d 1440	. . 3 ⊢ (𝐵 ∈ dom 𝐴 → ((𝐴:dom 𝐴⟶On ∧ Ord dom 𝐴 ∧ ∀𝑥 ∈ dom 𝐴∀𝑦 ∈ dom 𝐴(𝑥 ∈ 𝑦 → (𝐴‘𝑥) ∈ (𝐴‘𝑦))) → ((𝐴 ↾ 𝐵):dom (𝐴 ↾ 𝐵)⟶On ∧ Ord dom (𝐴 ↾ 𝐵) ∧ ∀𝑥 ∈ dom (𝐴 ↾ 𝐵)∀𝑦 ∈ dom (𝐴 ↾ 𝐵)(𝑥 ∈ 𝑦 → ((𝐴 ↾ 𝐵)‘𝑥) ∈ ((𝐴 ↾ 𝐵)‘𝑦)))))
46		df-smo 7966	. . 3 ⊢ (Smo 𝐴 ↔ (𝐴:dom 𝐴⟶On ∧ Ord dom 𝐴 ∧ ∀𝑥 ∈ dom 𝐴∀𝑦 ∈ dom 𝐴(𝑥 ∈ 𝑦 → (𝐴‘𝑥) ∈ (𝐴‘𝑦))))
47		df-smo 7966	. . 3 ⊢ (Smo (𝐴 ↾ 𝐵) ↔ ((𝐴 ↾ 𝐵):dom (𝐴 ↾ 𝐵)⟶On ∧ Ord dom (𝐴 ↾ 𝐵) ∧ ∀𝑥 ∈ dom (𝐴 ↾ 𝐵)∀𝑦 ∈ dom (𝐴 ↾ 𝐵)(𝑥 ∈ 𝑦 → ((𝐴 ↾ 𝐵)‘𝑥) ∈ ((𝐴 ↾ 𝐵)‘𝑦))))
48	45, 46, 47	3imtr4g 299	. 2 ⊢ (𝐵 ∈ dom 𝐴 → (Smo 𝐴 → Smo (𝐴 ↾ 𝐵)))
49	48	impcom 411	1 ⊢ ((Smo 𝐴 ∧ 𝐵 ∈ dom 𝐴) → Smo (𝐴 ↾ 𝐵))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∀wral 3106   ∩ cin 3880   ⊆ wss 3881  dom cdm 5519  ran crn 5520   ↾ cres 5521  Ord word 6158  Oncon0 6159  Fun wfun 6318   Fn wfn 6319  ⟶wf 6320  ‘cfv 6324  Smo wsmo 7965

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-tr 5137  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ord 6162  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-fv 6332  df-smo 7966

This theorem is referenced by:  smores3  7973  alephsing  9687