Theorem smores 8317

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 6558	. . . . . . . 8 ⊢ (Fun 𝐴 → Fun (𝐴 ↾ 𝐵))
2		funfn 6546	. . . . . . . 8 ⊢ (Fun 𝐴 ↔ 𝐴 Fn dom 𝐴)
3		funfn 6546	. . . . . . . 8 ⊢ (Fun (𝐴 ↾ 𝐵) ↔ (𝐴 ↾ 𝐵) Fn dom (𝐴 ↾ 𝐵))
4	1, 2, 3	3imtr3i 293	. . . . . . 7 ⊢ (𝐴 Fn dom 𝐴 → (𝐴 ↾ 𝐵) Fn dom (𝐴 ↾ 𝐵))
5		resss 5983	. . . . . . . . 9 ⊢ (𝐴 ↾ 𝐵) ⊆ 𝐴
6	5	rnssi 5912	. . . . . . . 8 ⊢ ran (𝐴 ↾ 𝐵) ⊆ ran 𝐴
7		sstr 3942	. . . . . . . 8 ⊢ ((ran (𝐴 ↾ 𝐵) ⊆ ran 𝐴 ∧ ran 𝐴 ⊆ On) → ran (𝐴 ↾ 𝐵) ⊆ On)
8	6, 7	mpan 700	. . . . . . 7 ⊢ (ran 𝐴 ⊆ On → ran (𝐴 ↾ 𝐵) ⊆ On)
9	4, 8	anim12i 622	. . . . . 6 ⊢ ((𝐴 Fn dom 𝐴 ∧ ran 𝐴 ⊆ On) → ((𝐴 ↾ 𝐵) Fn dom (𝐴 ↾ 𝐵) ∧ ran (𝐴 ↾ 𝐵) ⊆ On))
10		df-f 6520	. . . . . 6 ⊢ (𝐴:dom 𝐴⟶On ↔ (𝐴 Fn dom 𝐴 ∧ ran 𝐴 ⊆ On))
11		df-f 6520	. . . . . 6 ⊢ ((𝐴 ↾ 𝐵):dom (𝐴 ↾ 𝐵)⟶On ↔ ((𝐴 ↾ 𝐵) Fn dom (𝐴 ↾ 𝐵) ∧ ran (𝐴 ↾ 𝐵) ⊆ On))
12	9, 10, 11	3imtr4i 294	. . . . 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 6363	. . . . . . 7 ⊢ ((Ord dom 𝐴 ∧ 𝐵 ∈ dom 𝐴) → Ord 𝐵)
15	14	expcom 417	. . . . . 6 ⊢ (𝐵 ∈ dom 𝐴 → (Ord dom 𝐴 → Ord 𝐵))
16		ordin 6371	. . . . . . 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 5994	. . . . . 6 ⊢ dom (𝐴 ↾ 𝐵) = (𝐵 ∩ dom 𝐴)
20		ordeq 6348	. . . . . 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	imbitrrdi 254	. . . 4 ⊢ (𝐵 ∈ dom 𝐴 → (Ord dom 𝐴 → Ord dom (𝐴 ↾ 𝐵)))
23		dmss 5874	. . . . . . . . 9 ⊢ ((𝐴 ↾ 𝐵) ⊆ 𝐴 → dom (𝐴 ↾ 𝐵) ⊆ dom 𝐴)
24	5, 23	ax-mp 5	. . . . . . . 8 ⊢ dom (𝐴 ↾ 𝐵) ⊆ dom 𝐴
25		ssralv 4003	. . . . . . . 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 4003	. . . . . . . . 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 3098	. . . . . . 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 4186	. . . . . . . . . . . . 13 ⊢ (𝐵 ∩ dom 𝐴) ⊆ 𝐵
32	19, 31	eqsstri 3980	. . . . . . . . . . . 12 ⊢ dom (𝐴 ↾ 𝐵) ⊆ 𝐵
33		simpl 486	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ dom (𝐴 ↾ 𝐵) ∧ 𝑦 ∈ dom (𝐴 ↾ 𝐵)) → 𝑥 ∈ dom (𝐴 ↾ 𝐵))
34	32, 33	sselid 3932	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ dom (𝐴 ↾ 𝐵) ∧ 𝑦 ∈ dom (𝐴 ↾ 𝐵)) → 𝑥 ∈ 𝐵)
35	34	fvresd 6882	. . . . . . . . . 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	sselid 3932	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ dom (𝐴 ↾ 𝐵) ∧ 𝑦 ∈ dom (𝐴 ↾ 𝐵)) → 𝑦 ∈ 𝐵)
38	37	fvresd 6882	. . . . . . . . . 10 ⊢ ((𝑥 ∈ dom (𝐴 ↾ 𝐵) ∧ 𝑦 ∈ dom (𝐴 ↾ 𝐵)) → ((𝐴 ↾ 𝐵)‘𝑦) = (𝐴‘𝑦))
39	35, 38	eleq12d 2855	. . . . . . . . 9 ⊢ ((𝑥 ∈ dom (𝐴 ↾ 𝐵) ∧ 𝑦 ∈ dom (𝐴 ↾ 𝐵)) → (((𝐴 ↾ 𝐵)‘𝑥) ∈ ((𝐴 ↾ 𝐵)‘𝑦) ↔ (𝐴‘𝑥) ∈ (𝐴‘𝑦)))
40	39	imbi2d 342	. . . . . . . 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 3182	. . . . . . 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 3105	. . . . . 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 236	. . . . 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 1463	. . 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 8311	. . 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 8311	. . 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 298	. 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 208   ∧ wa 399   ∧ w3a 1097   = wceq 1559   ∈ wcel 2141  ∀wral 3075   ∩ cin 3901   ⊆ wss 3902  dom cdm 5643  ran crn 5644   ↾ cres 5645  Ord word 6340  Oncon0 6341  Fun wfun 6510   Fn wfn 6511  ⟶wf 6512  ‘cfv 6516  Smo wsmo 8310

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5243  ax-pr 5387

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-opab 5160  df-tr 5205  df-eprel 5543  df-po 5551  df-so 5552  df-fr 5596  df-we 5598  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ord 6344  df-iota 6472  df-fun 6518  df-fn 6519  df-f 6520  df-fv 6524  df-smo 8311

This theorem is referenced by:  smores3  8318  alephsing  10227