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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 2733 | . . . 4 ⊢ (𝐴 ∩ 𝑆) = (𝐴 ∩ 𝑆) | |
2 | ineq1 4166 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ∩ 𝑆) = (𝐴 ∩ 𝑆)) | |
3 | 2 | rspceeqv 3596 | . . . 4 ⊢ ((𝐴 ∈ 𝐽 ∧ (𝐴 ∩ 𝑆) = (𝐴 ∩ 𝑆)) → ∃𝑥 ∈ 𝐽 (𝐴 ∩ 𝑆) = (𝑥 ∩ 𝑆)) |
4 | 1, 3 | mpan2 690 | . . 3 ⊢ (𝐴 ∈ 𝐽 → ∃𝑥 ∈ 𝐽 (𝐴 ∩ 𝑆) = (𝑥 ∩ 𝑆)) |
5 | elrest 17314 | . . 3 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → ((𝐴 ∩ 𝑆) ∈ (𝐽 ↾t 𝑆) ↔ ∃𝑥 ∈ 𝐽 (𝐴 ∩ 𝑆) = (𝑥 ∩ 𝑆))) | |
6 | 4, 5 | syl5ibr 246 | . 2 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (𝐴 ∈ 𝐽 → (𝐴 ∩ 𝑆) ∈ (𝐽 ↾t 𝑆))) |
7 | 6 | 3impia 1118 | 1 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊 ∧ 𝐴 ∈ 𝐽) → (𝐴 ∩ 𝑆) ∈ (𝐽 ↾t 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ∃wrex 3070 ∩ cin 3910 (class class class)co 7358 ↾t crest 17307 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-ov 7361 df-oprab 7362 df-mpo 7363 df-rest 17309 |
This theorem is referenced by: firest 17319 restbas 22525 tgrest 22526 resttopon 22528 restcld 22539 restfpw 22546 neitr 22547 restntr 22549 ordtrest 22569 cnrest 22652 lmss 22665 connsubclo 22791 restnlly 22849 islly2 22851 cldllycmp 22862 lly1stc 22863 kgenss 22910 xkococnlem 23026 xkoinjcn 23054 qtoprest 23084 trfbas2 23210 trfil1 23253 trfil2 23254 fgtr 23257 trfg 23258 uzrest 23264 trufil 23277 flimrest 23350 cnextcn 23434 trust 23597 restutop 23605 trcfilu 23662 cfiluweak 23663 xrsmopn 24191 zdis 24195 xrge0tsms 24213 cnheibor 24334 cfilres 24676 lhop2 25395 psercn 25801 xrlimcnp 26334 xrge0tsmsd 31948 ordtrestNEW 32559 pnfneige0 32589 lmxrge0 32590 rrhre 32659 cvmscld 33924 cvmopnlem 33929 cvmliftmolem1 33932 poimirlem30 36154 subspopn 36257 iocopn 43844 icoopn 43849 limcresiooub 43969 limcresioolb 43970 fourierdlem32 44466 fourierdlem33 44467 fourierdlem48 44481 fourierdlem49 44482 i0oii 47038 io1ii 47039 iscnrm3llem2 47069 |
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