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Mirrors > Home > MPE Home > Th. List > elrestr | Structured version Visualization version GIF version |
Description: Sufficient condition for being an open set in a subspace. (Contributed by Jeff Hankins, 11-Jul-2009.) (Revised by Mario Carneiro, 15-Dec-2013.) |
Ref | Expression |
---|---|
elrestr | ⊢ ((𝐽 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊 ∧ 𝐴 ∈ 𝐽) → (𝐴 ∩ 𝑆) ∈ (𝐽 ↾t 𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2732 | . . . 4 ⊢ (𝐴 ∩ 𝑆) = (𝐴 ∩ 𝑆) | |
2 | ineq1 4205 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ∩ 𝑆) = (𝐴 ∩ 𝑆)) | |
3 | 2 | rspceeqv 3633 | . . . 4 ⊢ ((𝐴 ∈ 𝐽 ∧ (𝐴 ∩ 𝑆) = (𝐴 ∩ 𝑆)) → ∃𝑥 ∈ 𝐽 (𝐴 ∩ 𝑆) = (𝑥 ∩ 𝑆)) |
4 | 1, 3 | mpan2 689 | . . 3 ⊢ (𝐴 ∈ 𝐽 → ∃𝑥 ∈ 𝐽 (𝐴 ∩ 𝑆) = (𝑥 ∩ 𝑆)) |
5 | elrest 17375 | . . 3 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → ((𝐴 ∩ 𝑆) ∈ (𝐽 ↾t 𝑆) ↔ ∃𝑥 ∈ 𝐽 (𝐴 ∩ 𝑆) = (𝑥 ∩ 𝑆))) | |
6 | 4, 5 | imbitrrid 245 | . 2 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (𝐴 ∈ 𝐽 → (𝐴 ∩ 𝑆) ∈ (𝐽 ↾t 𝑆))) |
7 | 6 | 3impia 1117 | 1 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊 ∧ 𝐴 ∈ 𝐽) → (𝐴 ∩ 𝑆) ∈ (𝐽 ↾t 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ∃wrex 3070 ∩ cin 3947 (class class class)co 7411 ↾t crest 17368 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pr 5427 ax-un 7727 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7414 df-oprab 7415 df-mpo 7416 df-rest 17370 |
This theorem is referenced by: firest 17380 restbas 22669 tgrest 22670 resttopon 22672 restcld 22683 restfpw 22690 neitr 22691 restntr 22693 ordtrest 22713 cnrest 22796 lmss 22809 connsubclo 22935 restnlly 22993 islly2 22995 cldllycmp 23006 lly1stc 23007 kgenss 23054 xkococnlem 23170 xkoinjcn 23198 qtoprest 23228 trfbas2 23354 trfil1 23397 trfil2 23398 fgtr 23401 trfg 23402 uzrest 23408 trufil 23421 flimrest 23494 cnextcn 23578 trust 23741 restutop 23749 trcfilu 23806 cfiluweak 23807 xrsmopn 24335 zdis 24339 xrge0tsms 24357 cnheibor 24478 cfilres 24820 lhop2 25539 psercn 25945 xrlimcnp 26480 xrge0tsmsd 32250 ordtrestNEW 32970 pnfneige0 33000 lmxrge0 33001 rrhre 33070 cvmscld 34333 cvmopnlem 34338 cvmliftmolem1 34341 poimirlem30 36604 subspopn 36706 iocopn 44312 icoopn 44317 limcresiooub 44437 limcresioolb 44438 fourierdlem32 44934 fourierdlem33 44935 fourierdlem48 44949 fourierdlem49 44950 i0oii 47630 io1ii 47631 iscnrm3llem2 47661 |
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