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Mirrors > Home > ILE Home > Th. List > elrest | 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 11908 | . . 3 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐽 ↾t 𝐵) = ran (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐵))) | |
2 | 1 | eleq2d 2169 | . 2 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∈ (𝐽 ↾t 𝐵) ↔ 𝐴 ∈ ran (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐵)))) |
3 | eqid 2100 | . . 3 ⊢ (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐵)) = (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐵)) | |
4 | vex 2644 | . . . 4 ⊢ 𝑥 ∈ V | |
5 | 4 | inex1 4002 | . . 3 ⊢ (𝑥 ∩ 𝐵) ∈ V |
6 | 3, 5 | elrnmpti 4730 | . 2 ⊢ (𝐴 ∈ ran (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐵)) ↔ ∃𝑥 ∈ 𝐽 𝐴 = (𝑥 ∩ 𝐵)) |
7 | 2, 6 | syl6bb 195 | 1 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∈ (𝐽 ↾t 𝐵) ↔ ∃𝑥 ∈ 𝐽 𝐴 = (𝑥 ∩ 𝐵))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1299 ∈ wcel 1448 ∃wrex 2376 ∩ cin 3020 ↦ cmpt 3929 ran crn 4478 (class class class)co 5706 ↾t crest 11902 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 584 ax-in2 585 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-13 1459 ax-14 1460 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 ax-coll 3983 ax-sep 3986 ax-pow 4038 ax-pr 4069 ax-un 4293 ax-setind 4390 |
This theorem depends on definitions: df-bi 116 df-3an 932 df-tru 1302 df-fal 1305 df-nf 1405 df-sb 1704 df-eu 1963 df-mo 1964 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-ne 2268 df-ral 2380 df-rex 2381 df-reu 2382 df-rab 2384 df-v 2643 df-sbc 2863 df-csb 2956 df-dif 3023 df-un 3025 df-in 3027 df-ss 3034 df-pw 3459 df-sn 3480 df-pr 3481 df-op 3483 df-uni 3684 df-iun 3762 df-br 3876 df-opab 3930 df-mpt 3931 df-id 4153 df-xp 4483 df-rel 4484 df-cnv 4485 df-co 4486 df-dm 4487 df-rn 4488 df-res 4489 df-ima 4490 df-iota 5024 df-fun 5061 df-fn 5062 df-f 5063 df-f1 5064 df-fo 5065 df-f1o 5066 df-fv 5067 df-ov 5709 df-oprab 5710 df-mpo 5711 df-rest 11904 |
This theorem is referenced by: elrestr 11910 restsspw 11912 restbasg 12119 restsn 12131 restopnb 12132 ssrest 12133 cnrest2 12186 cnptopresti 12188 cnptoprest 12189 cnptoprest2 12190 lmss 12196 txrest 12226 metrest 12434 |
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