Theorem resthauslem 23346

Description: Lemma for resthaus 23351 and similar theorems. If the topological property 𝐴 is preserved under injective preimages, then property 𝐴 passes to subspaces. (Contributed by Mario Carneiro, 25-Aug-2015.)

Hypotheses

Ref	Expression
resthauslem.1	⊢ (𝐽 ∈ 𝐴 → 𝐽 ∈ Top)
resthauslem.2	⊢ ((𝐽 ∈ 𝐴 ∧ ( I ↾ (𝑆 ∩ ∪ 𝐽)):(𝑆 ∩ ∪ 𝐽)–1-1→(𝑆 ∩ ∪ 𝐽) ∧ ( I ↾ (𝑆 ∩ ∪ 𝐽)) ∈ ((𝐽 ↾t 𝑆) Cn 𝐽)) → (𝐽 ↾t 𝑆) ∈ 𝐴)

Assertion

Ref	Expression
resthauslem	⊢ ((𝐽 ∈ 𝐴 ∧ 𝑆 ∈ 𝑉) → (𝐽 ↾t 𝑆) ∈ 𝐴)

Proof of Theorem resthauslem

Step	Hyp	Ref	Expression
1		simpl 483	. 2 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝑆 ∈ 𝑉) → 𝐽 ∈ 𝐴)
2		f1oi 6805	. . 3 ⊢ ( I ↾ (𝑆 ∩ ∪ 𝐽)):(𝑆 ∩ ∪ 𝐽)–1-1-onto→(𝑆 ∩ ∪ 𝐽)
3		f1of1 6766	. . 3 ⊢ (( I ↾ (𝑆 ∩ ∪ 𝐽)):(𝑆 ∩ ∪ 𝐽)–1-1-onto→(𝑆 ∩ ∪ 𝐽) → ( I ↾ (𝑆 ∩ ∪ 𝐽)):(𝑆 ∩ ∪ 𝐽)–1-1→(𝑆 ∩ ∪ 𝐽))
4	2, 3	mp1i 13	. 2 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝑆 ∈ 𝑉) → ( I ↾ (𝑆 ∩ ∪ 𝐽)):(𝑆 ∩ ∪ 𝐽)–1-1→(𝑆 ∩ ∪ 𝐽))
5		inss2 4166	. . . . 5 ⊢ (𝑆 ∩ ∪ 𝐽) ⊆ ∪ 𝐽
6		resabs1 5958	. . . . 5 ⊢ ((𝑆 ∩ ∪ 𝐽) ⊆ ∪ 𝐽 → (( I ↾ ∪ 𝐽) ↾ (𝑆 ∩ ∪ 𝐽)) = ( I ↾ (𝑆 ∩ ∪ 𝐽)))
7	5, 6	ax-mp 5	. . . 4 ⊢ (( I ↾ ∪ 𝐽) ↾ (𝑆 ∩ ∪ 𝐽)) = ( I ↾ (𝑆 ∩ ∪ 𝐽))
8		resthauslem.1	. . . . . . . 8 ⊢ (𝐽 ∈ 𝐴 → 𝐽 ∈ Top)
9	8	adantr 481	. . . . . . 7 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝑆 ∈ 𝑉) → 𝐽 ∈ Top)
10		toptopon2 22901	. . . . . . 7 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽))
11	9, 10	sylib 219	. . . . . 6 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝑆 ∈ 𝑉) → 𝐽 ∈ (TopOn‘∪ 𝐽))
12		idcn 23240	. . . . . 6 ⊢ (𝐽 ∈ (TopOn‘∪ 𝐽) → ( I ↾ ∪ 𝐽) ∈ (𝐽 Cn 𝐽))
13	11, 12	syl 17	. . . . 5 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝑆 ∈ 𝑉) → ( I ↾ ∪ 𝐽) ∈ (𝐽 Cn 𝐽))
14		eqid 2739	. . . . . 6 ⊢ ∪ 𝐽 = ∪ 𝐽
15	14	cnrest 23268	. . . . 5 ⊢ ((( I ↾ ∪ 𝐽) ∈ (𝐽 Cn 𝐽) ∧ (𝑆 ∩ ∪ 𝐽) ⊆ ∪ 𝐽) → (( I ↾ ∪ 𝐽) ↾ (𝑆 ∩ ∪ 𝐽)) ∈ ((𝐽 ↾t (𝑆 ∩ ∪ 𝐽)) Cn 𝐽))
16	13, 5, 15	sylancl 592	. . . 4 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝑆 ∈ 𝑉) → (( I ↾ ∪ 𝐽) ↾ (𝑆 ∩ ∪ 𝐽)) ∈ ((𝐽 ↾t (𝑆 ∩ ∪ 𝐽)) Cn 𝐽))
17	7, 16	eqeltrrid 2844	. . 3 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝑆 ∈ 𝑉) → ( I ↾ (𝑆 ∩ ∪ 𝐽)) ∈ ((𝐽 ↾t (𝑆 ∩ ∪ 𝐽)) Cn 𝐽))
18	14	restin 23149	. . . 4 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝑆 ∈ 𝑉) → (𝐽 ↾t 𝑆) = (𝐽 ↾t (𝑆 ∩ ∪ 𝐽)))
19	18	oveq1d 7371	. . 3 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝑆 ∈ 𝑉) → ((𝐽 ↾t 𝑆) Cn 𝐽) = ((𝐽 ↾t (𝑆 ∩ ∪ 𝐽)) Cn 𝐽))
20	17, 19	eleqtrrd 2842	. 2 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝑆 ∈ 𝑉) → ( I ↾ (𝑆 ∩ ∪ 𝐽)) ∈ ((𝐽 ↾t 𝑆) Cn 𝐽))
21		resthauslem.2	. 2 ⊢ ((𝐽 ∈ 𝐴 ∧ ( I ↾ (𝑆 ∩ ∪ 𝐽)):(𝑆 ∩ ∪ 𝐽)–1-1→(𝑆 ∩ ∪ 𝐽) ∧ ( I ↾ (𝑆 ∩ ∪ 𝐽)) ∈ ((𝐽 ↾t 𝑆) Cn 𝐽)) → (𝐽 ↾t 𝑆) ∈ 𝐴)
22	1, 4, 20, 21	syl3anc 1379	1 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝑆 ∈ 𝑉) → (𝐽 ↾t 𝑆) ∈ 𝐴)

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119   ∩ cin 3882   ⊆ wss 3883  ∪ cuni 4838   I cid 5512   ↾ cres 5620  –1-1→wf1 6482  –1-1-onto→wf1o 6484  ‘cfv 6485  (class class class)co 7356   ↾_t crest 17374  Topctop 22876  TopOnctopon 22893   Cn ccn 23207

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-int 4878  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-tr 5180  df-id 5513  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-we 5573  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-ord 6313  df-on 6314  df-lim 6315  df-suc 6316  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-1st 7931  df-2nd 7932  df-map 8765  df-en 8884  df-fin 8887  df-fi 9314  df-rest 17376  df-topgen 17397  df-top 22877  df-topon 22894  df-bases 22929  df-cn 23210

This theorem is referenced by:  restt0  23349  restt1  23350  resthaus  23351