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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 2610 | . . . 4 ⊢ (𝐴 ∩ 𝑆) = (𝐴 ∩ 𝑆) | |
2 | ineq1 3769 | . . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥 ∩ 𝑆) = (𝐴 ∩ 𝑆)) | |
3 | 2 | eqeq2d 2620 | . . . . 5 ⊢ (𝑥 = 𝐴 → ((𝐴 ∩ 𝑆) = (𝑥 ∩ 𝑆) ↔ (𝐴 ∩ 𝑆) = (𝐴 ∩ 𝑆))) |
4 | 3 | rspcev 3282 | . . . 4 ⊢ ((𝐴 ∈ 𝐽 ∧ (𝐴 ∩ 𝑆) = (𝐴 ∩ 𝑆)) → ∃𝑥 ∈ 𝐽 (𝐴 ∩ 𝑆) = (𝑥 ∩ 𝑆)) |
5 | 1, 4 | mpan2 703 | . . 3 ⊢ (𝐴 ∈ 𝐽 → ∃𝑥 ∈ 𝐽 (𝐴 ∩ 𝑆) = (𝑥 ∩ 𝑆)) |
6 | elrest 15911 | . . 3 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → ((𝐴 ∩ 𝑆) ∈ (𝐽 ↾t 𝑆) ↔ ∃𝑥 ∈ 𝐽 (𝐴 ∩ 𝑆) = (𝑥 ∩ 𝑆))) | |
7 | 5, 6 | syl5ibr 235 | . 2 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (𝐴 ∈ 𝐽 → (𝐴 ∩ 𝑆) ∈ (𝐽 ↾t 𝑆))) |
8 | 7 | 3impia 1253 | 1 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊 ∧ 𝐴 ∈ 𝐽) → (𝐴 ∩ 𝑆) ∈ (𝐽 ↾t 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ∃wrex 2897 ∩ cin 3539 (class class class)co 6549 ↾t crest 15904 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-rest 15906 |
This theorem is referenced by: firest 15916 restbas 20772 tgrest 20773 resttopon 20775 restcld 20786 restfpw 20793 neitr 20794 restntr 20796 ordtrest 20816 cnrest 20899 lmss 20912 consubclo 21037 restnlly 21095 islly2 21097 cldllycmp 21108 lly1stc 21109 kgenss 21156 xkococnlem 21272 xkoinjcn 21300 qtoprest 21330 trfbas2 21457 trfil1 21500 trfil2 21501 fgtr 21504 trfg 21505 uzrest 21511 trufil 21524 flimrest 21597 cnextcn 21681 trust 21843 restutop 21851 trcfilu 21908 cfiluweak 21909 xrsmopn 22423 zdis 22427 xrge0tsms 22445 cnheibor 22562 cfilres 22902 lhop2 23582 psercn 23984 xrlimcnp 24495 xrge0tsmsd 29116 ordtrestNEW 29295 pnfneige0 29325 lmxrge0 29326 rrhre 29393 cvmscld 30509 cvmopnlem 30514 cvmliftmolem1 30517 poimirlem30 32609 subspopn 32718 iocopn 38593 icoopn 38598 limcresiooub 38709 limcresioolb 38710 fourierdlem32 39032 fourierdlem33 39033 fourierdlem48 39047 fourierdlem49 39048 |
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