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| Mirrors > Home > MPE Home > Th. List > elrest | Structured version Visualization version GIF version | ||
| Description: The predicate "is an open set of a subspace topology". (Contributed by FL, 5-Jan-2009.) (Revised by Mario Carneiro, 15-Dec-2013.) |
| Ref | Expression |
|---|---|
| elrest | ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∈ (𝐽 ↾t 𝐵) ↔ ∃𝑥 ∈ 𝐽 𝐴 = (𝑥 ∩ 𝐵))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | restval 17471 | . . 3 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐽 ↾t 𝐵) = ran (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐵))) | |
| 2 | 1 | eleq2d 2827 | . 2 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∈ (𝐽 ↾t 𝐵) ↔ 𝐴 ∈ ran (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐵)))) |
| 3 | eqid 2737 | . . 3 ⊢ (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐵)) = (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐵)) | |
| 4 | vex 3484 | . . . 4 ⊢ 𝑥 ∈ V | |
| 5 | 4 | inex1 5317 | . . 3 ⊢ (𝑥 ∩ 𝐵) ∈ V |
| 6 | 3, 5 | elrnmpti 5973 | . 2 ⊢ (𝐴 ∈ ran (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐵)) ↔ ∃𝑥 ∈ 𝐽 𝐴 = (𝑥 ∩ 𝐵)) |
| 7 | 2, 6 | bitrdi 287 | 1 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∈ (𝐽 ↾t 𝐵) ↔ ∃𝑥 ∈ 𝐽 𝐴 = (𝑥 ∩ 𝐵))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∃wrex 3070 ∩ cin 3950 ↦ cmpt 5225 ran crn 5686 (class class class)co 7431 ↾t crest 17465 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-rest 17467 |
| This theorem is referenced by: elrestr 17473 restsspw 17476 firest 17477 restbas 23166 restsn 23178 restcld 23180 restopnb 23183 ssrest 23184 neitr 23188 restntr 23190 cnrest2 23294 cnpresti 23296 cnprest 23297 cnprest2 23298 lmss 23306 cmpsublem 23407 cmpsub 23408 connsuba 23428 1stcrest 23461 subislly 23489 cldllycmp 23503 txrest 23639 trfbas2 23851 trfbas 23852 trfil2 23895 flimrest 23991 fclsrest 24032 cnextcn 24075 tsmssubm 24151 trust 24238 restutop 24246 restutopopn 24247 trcfilu 24303 metrest 24537 xrtgioo 24828 xrge0tsms 24856 icoopnst 24969 iocopnst 24970 subopnmbl 25639 mbfimaopn2 25692 xrlimcnp 27011 xrge0tsmsd 33065 rspectopn 33866 bj-restsn 37083 bj-rest10 37089 bj-restn0 37091 bj-restpw 37093 bj-rest0 37094 bj-restb 37095 bj-restuni 37098 bj-restreg 37100 ptrest 37626 poimirlem29 37656 elrestd 45113 restuni3 45123 restsubel 45158 icccncfext 45902 subsaliuncl 46373 subsalsal 46374 salrestss 46376 sssmf 46753 incsmf 46757 decsmf 46782 smflimlem6 46791 smfco 46817 smfpimcc 46823 |
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