Theorem smores 5930

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 4961	. . . . . . . 8 ⊢ (Fun 𝐴 → Fun (𝐴 ↾ 𝐵))
2		funfn 4951	. . . . . . . 8 ⊢ (Fun 𝐴 ↔ 𝐴 Fn dom 𝐴)
3		funfn 4951	. . . . . . . 8 ⊢ (Fun (𝐴 ↾ 𝐵) ↔ (𝐴 ↾ 𝐵) Fn dom (𝐴 ↾ 𝐵))
4	1, 2, 3	3imtr3i 198	. . . . . . 7 ⊢ (𝐴 Fn dom 𝐴 → (𝐴 ↾ 𝐵) Fn dom (𝐴 ↾ 𝐵))
5		resss 4653	. . . . . . . . 9 ⊢ (𝐴 ↾ 𝐵) ⊆ 𝐴
6		rnss 4582	. . . . . . . . 9 ⊢ ((𝐴 ↾ 𝐵) ⊆ 𝐴 → ran (𝐴 ↾ 𝐵) ⊆ ran 𝐴)
7	5, 6	ax-mp 7	. . . . . . . 8 ⊢ ran (𝐴 ↾ 𝐵) ⊆ ran 𝐴
8		sstr 3007	. . . . . . . 8 ⊢ ((ran (𝐴 ↾ 𝐵) ⊆ ran 𝐴 ∧ ran 𝐴 ⊆ On) → ran (𝐴 ↾ 𝐵) ⊆ On)
9	7, 8	mpan 414	. . . . . . 7 ⊢ (ran 𝐴 ⊆ On → ran (𝐴 ↾ 𝐵) ⊆ On)
10	4, 9	anim12i 331	. . . . . 6 ⊢ ((𝐴 Fn dom 𝐴 ∧ ran 𝐴 ⊆ On) → ((𝐴 ↾ 𝐵) Fn dom (𝐴 ↾ 𝐵) ∧ ran (𝐴 ↾ 𝐵) ⊆ On))
11		df-f 4926	. . . . . 6 ⊢ (𝐴:dom 𝐴⟶On ↔ (𝐴 Fn dom 𝐴 ∧ ran 𝐴 ⊆ On))
12		df-f 4926	. . . . . 6 ⊢ ((𝐴 ↾ 𝐵):dom (𝐴 ↾ 𝐵)⟶On ↔ ((𝐴 ↾ 𝐵) Fn dom (𝐴 ↾ 𝐵) ∧ ran (𝐴 ↾ 𝐵) ⊆ On))
13	10, 11, 12	3imtr4i 199	. . . . 5 ⊢ (𝐴:dom 𝐴⟶On → (𝐴 ↾ 𝐵):dom (𝐴 ↾ 𝐵)⟶On)
14	13	a1i 9	. . . 4 ⊢ (𝐵 ∈ dom 𝐴 → (𝐴:dom 𝐴⟶On → (𝐴 ↾ 𝐵):dom (𝐴 ↾ 𝐵)⟶On))
15		ordelord 4136	. . . . . . 7 ⊢ ((Ord dom 𝐴 ∧ 𝐵 ∈ dom 𝐴) → Ord 𝐵)
16	15	expcom 114	. . . . . 6 ⊢ (𝐵 ∈ dom 𝐴 → (Ord dom 𝐴 → Ord 𝐵))
17		ordin 4140	. . . . . . 7 ⊢ ((Ord 𝐵 ∧ Ord dom 𝐴) → Ord (𝐵 ∩ dom 𝐴))
18	17	ex 113	. . . . . 6 ⊢ (Ord 𝐵 → (Ord dom 𝐴 → Ord (𝐵 ∩ dom 𝐴)))
19	16, 18	syli 37	. . . . 5 ⊢ (𝐵 ∈ dom 𝐴 → (Ord dom 𝐴 → Ord (𝐵 ∩ dom 𝐴)))
20		dmres 4650	. . . . . 6 ⊢ dom (𝐴 ↾ 𝐵) = (𝐵 ∩ dom 𝐴)
21		ordeq 4127	. . . . . 6 ⊢ (dom (𝐴 ↾ 𝐵) = (𝐵 ∩ dom 𝐴) → (Ord dom (𝐴 ↾ 𝐵) ↔ Ord (𝐵 ∩ dom 𝐴)))
22	20, 21	ax-mp 7	. . . . 5 ⊢ (Ord dom (𝐴 ↾ 𝐵) ↔ Ord (𝐵 ∩ dom 𝐴))
23	19, 22	syl6ibr 160	. . . 4 ⊢ (𝐵 ∈ dom 𝐴 → (Ord dom 𝐴 → Ord dom (𝐴 ↾ 𝐵)))
24		dmss 4552	. . . . . . . . 9 ⊢ ((𝐴 ↾ 𝐵) ⊆ 𝐴 → dom (𝐴 ↾ 𝐵) ⊆ dom 𝐴)
25	5, 24	ax-mp 7	. . . . . . . 8 ⊢ dom (𝐴 ↾ 𝐵) ⊆ dom 𝐴
26		ssralv 3058	. . . . . . . 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))))
27	25, 26	ax-mp 7	. . . . . . 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)))
28		ssralv 3058	. . . . . . . . 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))))
29	25, 28	ax-mp 7	. . . . . . . 8 ⊢ (∀𝑦 ∈ dom 𝐴(𝑥 ∈ 𝑦 → (𝐴‘𝑥) ∈ (𝐴‘𝑦)) → ∀𝑦 ∈ dom (𝐴 ↾ 𝐵)(𝑥 ∈ 𝑦 → (𝐴‘𝑥) ∈ (𝐴‘𝑦)))
30	29	ralimi 2426	. . . . . . 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)))
31	27, 30	syl 14	. . . . . 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)))
32		inss1 3186	. . . . . . . . . . . . 13 ⊢ (𝐵 ∩ dom 𝐴) ⊆ 𝐵
33	20, 32	eqsstri 3029	. . . . . . . . . . . 12 ⊢ dom (𝐴 ↾ 𝐵) ⊆ 𝐵
34		simpl 107	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ dom (𝐴 ↾ 𝐵) ∧ 𝑦 ∈ dom (𝐴 ↾ 𝐵)) → 𝑥 ∈ dom (𝐴 ↾ 𝐵))
35	33, 34	sseldi 2997	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ dom (𝐴 ↾ 𝐵) ∧ 𝑦 ∈ dom (𝐴 ↾ 𝐵)) → 𝑥 ∈ 𝐵)
36		fvres 5219	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐵 → ((𝐴 ↾ 𝐵)‘𝑥) = (𝐴‘𝑥))
37	35, 36	syl 14	. . . . . . . . . 10 ⊢ ((𝑥 ∈ dom (𝐴 ↾ 𝐵) ∧ 𝑦 ∈ dom (𝐴 ↾ 𝐵)) → ((𝐴 ↾ 𝐵)‘𝑥) = (𝐴‘𝑥))
38		simpr 108	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ dom (𝐴 ↾ 𝐵) ∧ 𝑦 ∈ dom (𝐴 ↾ 𝐵)) → 𝑦 ∈ dom (𝐴 ↾ 𝐵))
39	33, 38	sseldi 2997	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ dom (𝐴 ↾ 𝐵) ∧ 𝑦 ∈ dom (𝐴 ↾ 𝐵)) → 𝑦 ∈ 𝐵)
40		fvres 5219	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝐵 → ((𝐴 ↾ 𝐵)‘𝑦) = (𝐴‘𝑦))
41	39, 40	syl 14	. . . . . . . . . 10 ⊢ ((𝑥 ∈ dom (𝐴 ↾ 𝐵) ∧ 𝑦 ∈ dom (𝐴 ↾ 𝐵)) → ((𝐴 ↾ 𝐵)‘𝑦) = (𝐴‘𝑦))
42	37, 41	eleq12d 2149	. . . . . . . . 9 ⊢ ((𝑥 ∈ dom (𝐴 ↾ 𝐵) ∧ 𝑦 ∈ dom (𝐴 ↾ 𝐵)) → (((𝐴 ↾ 𝐵)‘𝑥) ∈ ((𝐴 ↾ 𝐵)‘𝑦) ↔ (𝐴‘𝑥) ∈ (𝐴‘𝑦)))
43	42	imbi2d 228	. . . . . . . 8 ⊢ ((𝑥 ∈ dom (𝐴 ↾ 𝐵) ∧ 𝑦 ∈ dom (𝐴 ↾ 𝐵)) → ((𝑥 ∈ 𝑦 → ((𝐴 ↾ 𝐵)‘𝑥) ∈ ((𝐴 ↾ 𝐵)‘𝑦)) ↔ (𝑥 ∈ 𝑦 → (𝐴‘𝑥) ∈ (𝐴‘𝑦))))
44	43	ralbidva 2364	. . . . . . 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))))
45	44	ralbiia 2380	. . . . . 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)))
46	31, 45	sylibr 132	. . . . 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)))
47	46	a1i 9	. . . 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))))
48	14, 23, 47	3anim123d 1250	. . 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)))))
49		df-smo 5924	. . 3 ⊢ (Smo 𝐴 ↔ (𝐴:dom 𝐴⟶On ∧ Ord dom 𝐴 ∧ ∀𝑥 ∈ dom 𝐴∀𝑦 ∈ dom 𝐴(𝑥 ∈ 𝑦 → (𝐴‘𝑥) ∈ (𝐴‘𝑦))))
50		df-smo 5924	. . 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))))
51	48, 49, 50	3imtr4g 203	. 2 ⊢ (𝐵 ∈ dom 𝐴 → (Smo 𝐴 → Smo (𝐴 ↾ 𝐵)))
52	51	impcom 123	1 ⊢ ((Smo 𝐴 ∧ 𝐵 ∈ dom 𝐴) → Smo (𝐴 ↾ 𝐵))

Syntax hints:   → wi 4   ∧ wa 102   ↔ wb 103   ∧ w3a 919   = wceq 1284   ∈ wcel 1433  ∀wral 2348   ∩ cin 2972   ⊆ wss 2973  Ord word 4117  Oncon0 4118  dom cdm 4363  ran crn 4364   ↾ cres 4365  Fun wfun 4916   Fn wfn 4917  ⟶wf 4918  ‘cfv 4922  Smo wsmo 5923

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-tr 3876  df-iord 4121  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-fv 4930  df-smo 5924

This theorem is referenced by:  smores3  5931