![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > resthauslem | Structured version Visualization version GIF version |
Description: Lemma for resthaus 23333 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.) |
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) |
Ref | Expression |
---|---|
resthauslem | ⊢ ((𝐽 ∈ 𝐴 ∧ 𝑆 ∈ 𝑉) → (𝐽 ↾t 𝑆) ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 481 | . 2 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝑆 ∈ 𝑉) → 𝐽 ∈ 𝐴) | |
2 | f1oi 6876 | . . 3 ⊢ ( I ↾ (𝑆 ∩ ∪ 𝐽)):(𝑆 ∩ ∪ 𝐽)–1-1-onto→(𝑆 ∩ ∪ 𝐽) | |
3 | f1of1 6837 | . . 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 4228 | . . . . 5 ⊢ (𝑆 ∩ ∪ 𝐽) ⊆ ∪ 𝐽 | |
6 | resabs1 6012 | . . . . 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 479 | . . . . . . 7 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝑆 ∈ 𝑉) → 𝐽 ∈ Top) |
10 | toptopon2 22881 | . . . . . . 7 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽)) | |
11 | 9, 10 | sylib 217 | . . . . . 6 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝑆 ∈ 𝑉) → 𝐽 ∈ (TopOn‘∪ 𝐽)) |
12 | idcn 23222 | . . . . . 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 2725 | . . . . . 6 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
15 | 14 | cnrest 23250 | . . . . 5 ⊢ ((( I ↾ ∪ 𝐽) ∈ (𝐽 Cn 𝐽) ∧ (𝑆 ∩ ∪ 𝐽) ⊆ ∪ 𝐽) → (( I ↾ ∪ 𝐽) ↾ (𝑆 ∩ ∪ 𝐽)) ∈ ((𝐽 ↾t (𝑆 ∩ ∪ 𝐽)) Cn 𝐽)) |
16 | 13, 5, 15 | sylancl 584 | . . . 4 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝑆 ∈ 𝑉) → (( I ↾ ∪ 𝐽) ↾ (𝑆 ∩ ∪ 𝐽)) ∈ ((𝐽 ↾t (𝑆 ∩ ∪ 𝐽)) Cn 𝐽)) |
17 | 7, 16 | eqeltrrid 2830 | . . 3 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝑆 ∈ 𝑉) → ( I ↾ (𝑆 ∩ ∪ 𝐽)) ∈ ((𝐽 ↾t (𝑆 ∩ ∪ 𝐽)) Cn 𝐽)) |
18 | 14 | restin 23131 | . . . 4 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝑆 ∈ 𝑉) → (𝐽 ↾t 𝑆) = (𝐽 ↾t (𝑆 ∩ ∪ 𝐽))) |
19 | 18 | oveq1d 7434 | . . 3 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝑆 ∈ 𝑉) → ((𝐽 ↾t 𝑆) Cn 𝐽) = ((𝐽 ↾t (𝑆 ∩ ∪ 𝐽)) Cn 𝐽)) |
20 | 17, 19 | eleqtrrd 2828 | . 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 1368 | 1 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝑆 ∈ 𝑉) → (𝐽 ↾t 𝑆) ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ∩ cin 3943 ⊆ wss 3944 ∪ cuni 4909 I cid 5575 ↾ cres 5680 –1-1→wf1 6546 –1-1-onto→wf1o 6548 ‘cfv 6549 (class class class)co 7419 ↾t crest 17421 Topctop 22856 TopOnctopon 22873 Cn ccn 23189 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-1st 7994 df-2nd 7995 df-map 8847 df-en 8965 df-fin 8968 df-fi 9441 df-rest 17423 df-topgen 17444 df-top 22857 df-topon 22874 df-bases 22910 df-cn 23192 |
This theorem is referenced by: restt0 23331 restt1 23332 resthaus 23333 |
Copyright terms: Public domain | W3C validator |