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Mirrors > Home > ILE Home > Th. List > restsspw | GIF version |
Description: The subspace topology is a collection of subsets of the restriction set. (Contributed by Mario Carneiro, 13-Aug-2015.) |
Ref | Expression |
---|---|
restsspw | ⊢ (𝐽 ↾t 𝐴) ⊆ 𝒫 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rest 11962 | . . . . . . 7 ⊢ ↾t = (𝑗 ∈ V, 𝑥 ∈ V ↦ ran (𝑦 ∈ 𝑗 ↦ (𝑦 ∩ 𝑥))) | |
2 | 1 | elmpocl 5922 | . . . . . 6 ⊢ (𝑥 ∈ (𝐽 ↾t 𝐴) → (𝐽 ∈ V ∧ 𝐴 ∈ V)) |
3 | elrest 11967 | . . . . . 6 ⊢ ((𝐽 ∈ V ∧ 𝐴 ∈ V) → (𝑥 ∈ (𝐽 ↾t 𝐴) ↔ ∃𝑦 ∈ 𝐽 𝑥 = (𝑦 ∩ 𝐴))) | |
4 | 2, 3 | syl 14 | . . . . 5 ⊢ (𝑥 ∈ (𝐽 ↾t 𝐴) → (𝑥 ∈ (𝐽 ↾t 𝐴) ↔ ∃𝑦 ∈ 𝐽 𝑥 = (𝑦 ∩ 𝐴))) |
5 | 4 | ibi 175 | . . . 4 ⊢ (𝑥 ∈ (𝐽 ↾t 𝐴) → ∃𝑦 ∈ 𝐽 𝑥 = (𝑦 ∩ 𝐴)) |
6 | inss2 3263 | . . . . . 6 ⊢ (𝑦 ∩ 𝐴) ⊆ 𝐴 | |
7 | sseq1 3086 | . . . . . 6 ⊢ (𝑥 = (𝑦 ∩ 𝐴) → (𝑥 ⊆ 𝐴 ↔ (𝑦 ∩ 𝐴) ⊆ 𝐴)) | |
8 | 6, 7 | mpbiri 167 | . . . . 5 ⊢ (𝑥 = (𝑦 ∩ 𝐴) → 𝑥 ⊆ 𝐴) |
9 | 8 | rexlimivw 2519 | . . . 4 ⊢ (∃𝑦 ∈ 𝐽 𝑥 = (𝑦 ∩ 𝐴) → 𝑥 ⊆ 𝐴) |
10 | 5, 9 | syl 14 | . . 3 ⊢ (𝑥 ∈ (𝐽 ↾t 𝐴) → 𝑥 ⊆ 𝐴) |
11 | selpw 3483 | . . 3 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴) | |
12 | 10, 11 | sylibr 133 | . 2 ⊢ (𝑥 ∈ (𝐽 ↾t 𝐴) → 𝑥 ∈ 𝒫 𝐴) |
13 | 12 | ssriv 3067 | 1 ⊢ (𝐽 ↾t 𝐴) ⊆ 𝒫 𝐴 |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ↔ wb 104 = wceq 1314 ∈ wcel 1463 ∃wrex 2391 Vcvv 2657 ∩ cin 3036 ⊆ wss 3037 𝒫 cpw 3476 ↦ cmpt 3949 ran crn 4500 (class class class)co 5728 ↾t crest 11960 |
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 586 ax-in2 587 ax-io 681 ax-5 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-10 1466 ax-11 1467 ax-i12 1468 ax-bndl 1469 ax-4 1470 ax-13 1474 ax-14 1475 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-i5r 1498 ax-ext 2097 ax-coll 4003 ax-sep 4006 ax-pow 4058 ax-pr 4091 ax-un 4315 ax-setind 4412 |
This theorem depends on definitions: df-bi 116 df-3an 947 df-tru 1317 df-fal 1320 df-nf 1420 df-sb 1719 df-eu 1978 df-mo 1979 df-clab 2102 df-cleq 2108 df-clel 2111 df-nfc 2244 df-ne 2283 df-ral 2395 df-rex 2396 df-reu 2397 df-rab 2399 df-v 2659 df-sbc 2879 df-csb 2972 df-dif 3039 df-un 3041 df-in 3043 df-ss 3050 df-pw 3478 df-sn 3499 df-pr 3500 df-op 3502 df-uni 3703 df-iun 3781 df-br 3896 df-opab 3950 df-mpt 3951 df-id 4175 df-xp 4505 df-rel 4506 df-cnv 4507 df-co 4508 df-dm 4509 df-rn 4510 df-res 4511 df-ima 4512 df-iota 5046 df-fun 5083 df-fn 5084 df-f 5085 df-f1 5086 df-fo 5087 df-f1o 5088 df-fv 5089 df-ov 5731 df-oprab 5732 df-mpo 5733 df-rest 11962 |
This theorem is referenced by: (None) |
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