Theorem resthauslem 23392

Description: Lemma for resthaus 23397 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 485	. 2 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝑆 ∈ 𝑉) → 𝐽 ∈ 𝐴)
2		f1oi 6830	. . 3 ⊢ ( I ↾ (𝑆 ∩ ∪ 𝐽)):(𝑆 ∩ ∪ 𝐽)–1-1-onto→(𝑆 ∩ ∪ 𝐽)
3		f1of1 6790	. . 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 4180	. . . . 5 ⊢ (𝑆 ∩ ∪ 𝐽) ⊆ ∪ 𝐽
6		resabs1 5981	. . . . 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 483	. . . . . . 7 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝑆 ∈ 𝑉) → 𝐽 ∈ Top)
10		toptopon2 22947	. . . . . . 7 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽))
11	9, 10	sylib 220	. . . . . 6 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝑆 ∈ 𝑉) → 𝐽 ∈ (TopOn‘∪ 𝐽))
12		idcn 23286	. . . . . 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 2752	. . . . . 6 ⊢ ∪ 𝐽 = ∪ 𝐽
15	14	cnrest 23314	. . . . 5 ⊢ ((( I ↾ ∪ 𝐽) ∈ (𝐽 Cn 𝐽) ∧ (𝑆 ∩ ∪ 𝐽) ⊆ ∪ 𝐽) → (( I ↾ ∪ 𝐽) ↾ (𝑆 ∩ ∪ 𝐽)) ∈ ((𝐽 ↾t (𝑆 ∩ ∪ 𝐽)) Cn 𝐽))
16	13, 5, 15	sylancl 594	. . . 4 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝑆 ∈ 𝑉) → (( I ↾ ∪ 𝐽) ↾ (𝑆 ∩ ∪ 𝐽)) ∈ ((𝐽 ↾t (𝑆 ∩ ∪ 𝐽)) Cn 𝐽))
17	7, 16	eqeltrrid 2857	. . 3 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝑆 ∈ 𝑉) → ( I ↾ (𝑆 ∩ ∪ 𝐽)) ∈ ((𝐽 ↾t (𝑆 ∩ ∪ 𝐽)) Cn 𝐽))
18	14	restin 23195	. . . 4 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝑆 ∈ 𝑉) → (𝐽 ↾t 𝑆) = (𝐽 ↾t (𝑆 ∩ ∪ 𝐽)))
19	18	oveq1d 7396	. . 3 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝑆 ∈ 𝑉) → ((𝐽 ↾t 𝑆) Cn 𝐽) = ((𝐽 ↾t (𝑆 ∩ ∪ 𝐽)) Cn 𝐽))
20	17, 19	eleqtrrd 2855	. 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 1382	1 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝑆 ∈ 𝑉) → (𝐽 ↾t 𝑆) ∈ 𝐴)

Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1095   = wceq 1550   ∈ wcel 2132   ∩ cin 3894   ⊆ wss 3895  ∪ cuni 4855   I cid 5530   ↾ cres 5638  –1-1→wf1 6503  –1-1-onto→wf1o 6505  ‘cfv 6506  (class class class)co 7381   ↾_t crest 17421  Topctop 22922  TopOnctopon 22939   Cn ccn 23253

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1805  ax-4 1819  ax-5 1920  ax-6 1977  ax-7 2018  ax-8 2134  ax-9 2142  ax-10 2165  ax-11 2181  ax-12 2202  ax-ext 2724  ax-rep 5217  ax-sep 5236  ax-nul 5246  ax-pow 5312  ax-pr 5380  ax-un 7703

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3or 1096  df-3an 1097  df-tru 1553  df-fal 1563  df-ex 1790  df-nf 1794  df-sb 2081  df-mo 2556  df-eu 2586  df-clab 2731  df-cleq 2744  df-clel 2827  df-nfc 2901  df-ne 2948  df-ral 3067  df-rex 3077  df-reu 3358  df-rab 3405  df-v 3446  df-sbc 3736  df-csb 3844  df-dif 3898  df-un 3900  df-in 3902  df-ss 3912  df-pss 3915  df-nul 4277  df-if 4471  df-pw 4547  df-sn 4573  df-pr 4575  df-op 4579  df-uni 4856  df-int 4896  df-iun 4941  df-br 5091  df-opab 5153  df-mpt 5172  df-tr 5198  df-id 5531  df-eprel 5536  df-po 5544  df-so 5545  df-fr 5589  df-we 5591  df-xp 5642  df-rel 5643  df-cnv 5644  df-co 5645  df-dm 5646  df-rn 5647  df-res 5648  df-ima 5649  df-ord 6334  df-on 6335  df-lim 6336  df-suc 6337  df-iota 6462  df-fun 6508  df-fn 6509  df-f 6510  df-f1 6511  df-fo 6512  df-f1o 6513  df-fv 6514  df-ov 7384  df-oprab 7385  df-mpo 7386  df-om 7832  df-1st 7955  df-2nd 7956  df-map 8794  df-en 8913  df-fin 8916  df-fi 9343  df-rest 17423  df-topgen 17444  df-top 22923  df-topon 22940  df-bases 22975  df-cn 23256

This theorem is referenced by:  restt0  23395  restt1  23396  resthaus  23397