Theorem smores 8285

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 6530	. . . . . . . 8 ⊢ (Fun 𝐴 → Fun (𝐴 ↾ 𝐵))
2		funfn 6518	. . . . . . . 8 ⊢ (Fun 𝐴 ↔ 𝐴 Fn dom 𝐴)
3		funfn 6518	. . . . . . . 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 5959	. . . . . . . . 9 ⊢ (𝐴 ↾ 𝐵) ⊆ 𝐴
6	5	rnssi 5888	. . . . . . . 8 ⊢ ran (𝐴 ↾ 𝐵) ⊆ ran 𝐴
7		sstr 3924	. . . . . . . 8 ⊢ ((ran (𝐴 ↾ 𝐵) ⊆ ran 𝐴 ∧ ran 𝐴 ⊆ On) → ran (𝐴 ↾ 𝐵) ⊆ On)
8	6, 7	mpan 697	. . . . . . 7 ⊢ (ran 𝐴 ⊆ On → ran (𝐴 ↾ 𝐵) ⊆ On)
9	4, 8	anim12i 620	. . . . . 6 ⊢ ((𝐴 Fn dom 𝐴 ∧ ran 𝐴 ⊆ On) → ((𝐴 ↾ 𝐵) Fn dom (𝐴 ↾ 𝐵) ∧ ran (𝐴 ↾ 𝐵) ⊆ On))
10		df-f 6492	. . . . . 6 ⊢ (𝐴:dom 𝐴⟶On ↔ (𝐴 Fn dom 𝐴 ∧ ran 𝐴 ⊆ On))
11		df-f 6492	. . . . . 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 6335	. . . . . . 7 ⊢ ((Ord dom 𝐴 ∧ 𝐵 ∈ dom 𝐴) → Ord 𝐵)
15	14	expcom 415	. . . . . 6 ⊢ (𝐵 ∈ dom 𝐴 → (Ord dom 𝐴 → Ord 𝐵))
16		ordin 6343	. . . . . . 7 ⊢ ((Ord 𝐵 ∧ Ord dom 𝐴) → Ord (𝐵 ∩ dom 𝐴))
17	16	ex 414	. . . . . 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 5970	. . . . . 6 ⊢ dom (𝐴 ↾ 𝐵) = (𝐵 ∩ dom 𝐴)
20		ordeq 6320	. . . . . 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 5850	. . . . . . . . 9 ⊢ ((𝐴 ↾ 𝐵) ⊆ 𝐴 → dom (𝐴 ↾ 𝐵) ⊆ dom 𝐴)
24	5, 23	ax-mp 5	. . . . . . . 8 ⊢ dom (𝐴 ↾ 𝐵) ⊆ dom 𝐴
25		ssralv 3985	. . . . . . . 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 3985	. . . . . . . . 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 3078	. . . . . . 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 4167	. . . . . . . . . . . . 13 ⊢ (𝐵 ∩ dom 𝐴) ⊆ 𝐵
32	19, 31	eqsstri 3962	. . . . . . . . . . . 12 ⊢ dom (𝐴 ↾ 𝐵) ⊆ 𝐵
33		simpl 484	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ dom (𝐴 ↾ 𝐵) ∧ 𝑦 ∈ dom (𝐴 ↾ 𝐵)) → 𝑥 ∈ dom (𝐴 ↾ 𝐵))
34	32, 33	sselid 3914	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ dom (𝐴 ↾ 𝐵) ∧ 𝑦 ∈ dom (𝐴 ↾ 𝐵)) → 𝑥 ∈ 𝐵)
35	34	fvresd 6850	. . . . . . . . . 10 ⊢ ((𝑥 ∈ dom (𝐴 ↾ 𝐵) ∧ 𝑦 ∈ dom (𝐴 ↾ 𝐵)) → ((𝐴 ↾ 𝐵)‘𝑥) = (𝐴‘𝑥))
36		simpr 486	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ dom (𝐴 ↾ 𝐵) ∧ 𝑦 ∈ dom (𝐴 ↾ 𝐵)) → 𝑦 ∈ dom (𝐴 ↾ 𝐵))
37	32, 36	sselid 3914	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ dom (𝐴 ↾ 𝐵) ∧ 𝑦 ∈ dom (𝐴 ↾ 𝐵)) → 𝑦 ∈ 𝐵)
38	37	fvresd 6850	. . . . . . . . . 10 ⊢ ((𝑥 ∈ dom (𝐴 ↾ 𝐵) ∧ 𝑦 ∈ dom (𝐴 ↾ 𝐵)) → ((𝐴 ↾ 𝐵)‘𝑦) = (𝐴‘𝑦))
39	35, 38	eleq12d 2835	. . . . . . . . 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 3162	. . . . . . 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 3085	. . . . . 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 1452	. . 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 8279	. . 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 8279	. . 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 409	1 ⊢ ((Smo 𝐴 ∧ 𝐵 ∈ dom 𝐴) → Smo (𝐴 ↾ 𝐵))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 397   ∧ w3a 1093   = wceq 1548   ∈ wcel 2121  ∀wral 3055   ∩ cin 3883   ⊆ wss 3884  dom cdm 5620  ran crn 5621   ↾ cres 5622  Ord word 6312  Oncon0 6313  Fun wfun 6482   Fn wfn 6483  ⟶wf 6484  ‘cfv 6488  Smo wsmo 8278

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713  ax-sep 5220  ax-pr 5364

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-ne 2937  df-ral 3056  df-rex 3066  df-rab 3394  df-v 3435  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4264  df-if 4457  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4841  df-br 5075  df-opab 5137  df-tr 5182  df-eprel 5520  df-po 5528  df-so 5529  df-fr 5573  df-we 5575  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ord 6316  df-iota 6444  df-fun 6490  df-fn 6491  df-f 6492  df-fv 6496  df-smo 8279

This theorem is referenced by:  smores3  8286  alephsing  10194