Theorem resthauslem 23301

Description: Lemma for resthaus 23306 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 482	. 2 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝑆 ∈ 𝑉) → 𝐽 ∈ 𝐴)
2		f1oi 6856	. . 3 ⊢ ( I ↾ (𝑆 ∩ ∪ 𝐽)):(𝑆 ∩ ∪ 𝐽)–1-1-onto→(𝑆 ∩ ∪ 𝐽)
3		f1of1 6817	. . 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 4213	. . . . 5 ⊢ (𝑆 ∩ ∪ 𝐽) ⊆ ∪ 𝐽
6		resabs1 5993	. . . . 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 480	. . . . . . 7 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝑆 ∈ 𝑉) → 𝐽 ∈ Top)
10		toptopon2 22856	. . . . . . 7 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽))
11	9, 10	sylib 218	. . . . . 6 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝑆 ∈ 𝑉) → 𝐽 ∈ (TopOn‘∪ 𝐽))
12		idcn 23195	. . . . . 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 2735	. . . . . 6 ⊢ ∪ 𝐽 = ∪ 𝐽
15	14	cnrest 23223	. . . . 5 ⊢ ((( I ↾ ∪ 𝐽) ∈ (𝐽 Cn 𝐽) ∧ (𝑆 ∩ ∪ 𝐽) ⊆ ∪ 𝐽) → (( I ↾ ∪ 𝐽) ↾ (𝑆 ∩ ∪ 𝐽)) ∈ ((𝐽 ↾t (𝑆 ∩ ∪ 𝐽)) Cn 𝐽))
16	13, 5, 15	sylancl 586	. . . 4 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝑆 ∈ 𝑉) → (( I ↾ ∪ 𝐽) ↾ (𝑆 ∩ ∪ 𝐽)) ∈ ((𝐽 ↾t (𝑆 ∩ ∪ 𝐽)) Cn 𝐽))
17	7, 16	eqeltrrid 2839	. . 3 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝑆 ∈ 𝑉) → ( I ↾ (𝑆 ∩ ∪ 𝐽)) ∈ ((𝐽 ↾t (𝑆 ∩ ∪ 𝐽)) Cn 𝐽))
18	14	restin 23104	. . . 4 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝑆 ∈ 𝑉) → (𝐽 ↾t 𝑆) = (𝐽 ↾t (𝑆 ∩ ∪ 𝐽)))
19	18	oveq1d 7420	. . 3 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝑆 ∈ 𝑉) → ((𝐽 ↾t 𝑆) Cn 𝐽) = ((𝐽 ↾t (𝑆 ∩ ∪ 𝐽)) Cn 𝐽))
20	17, 19	eleqtrrd 2837	. 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 1373	1 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝑆 ∈ 𝑉) → (𝐽 ↾t 𝑆) ∈ 𝐴)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2108   ∩ cin 3925   ⊆ wss 3926  ∪ cuni 4883   I cid 5547   ↾ cres 5656  –1-1→wf1 6528  –1-1-onto→wf1o 6530  ‘cfv 6531  (class class class)co 7405   ↾_t crest 17434  Topctop 22831  TopOnctopon 22848   Cn ccn 23162

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-int 4923  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7862  df-1st 7988  df-2nd 7989  df-map 8842  df-en 8960  df-fin 8963  df-fi 9423  df-rest 17436  df-topgen 17457  df-top 22832  df-topon 22849  df-bases 22884  df-cn 23165

This theorem is referenced by:  restt0  23304  restt1  23305  resthaus  23306