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Mirrors > Home > ILE Home > Th. List > ssrest | GIF version |
Description: If 𝐾 is a finer topology than 𝐽, then the subspace topologies induced by 𝐴 maintain this relationship. (Contributed by Mario Carneiro, 21-Mar-2015.) (Revised by Mario Carneiro, 1-May-2015.) |
Ref | Expression |
---|---|
ssrest | ⊢ ((𝐾 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐾) → (𝐽 ↾t 𝐴) ⊆ (𝐾 ↾t 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 110 | . . . 4 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ (𝐽 ↾t 𝐴)) → 𝑥 ∈ (𝐽 ↾t 𝐴)) | |
2 | ssrexv 3220 | . . . . . 6 ⊢ (𝐽 ⊆ 𝐾 → (∃𝑦 ∈ 𝐽 𝑥 = (𝑦 ∩ 𝐴) → ∃𝑦 ∈ 𝐾 𝑥 = (𝑦 ∩ 𝐴))) | |
3 | 2 | ad2antlr 489 | . . . . 5 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ (𝐽 ↾t 𝐴)) → (∃𝑦 ∈ 𝐽 𝑥 = (𝑦 ∩ 𝐴) → ∃𝑦 ∈ 𝐾 𝑥 = (𝑦 ∩ 𝐴))) |
4 | df-rest 12635 | . . . . . . . 8 ⊢ ↾t = (𝑗 ∈ V, 𝑥 ∈ V ↦ ran (𝑦 ∈ 𝑗 ↦ (𝑦 ∩ 𝑥))) | |
5 | 4 | elmpocl 6063 | . . . . . . 7 ⊢ (𝑥 ∈ (𝐽 ↾t 𝐴) → (𝐽 ∈ V ∧ 𝐴 ∈ V)) |
6 | 5 | adantl 277 | . . . . . 6 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ (𝐽 ↾t 𝐴)) → (𝐽 ∈ V ∧ 𝐴 ∈ V)) |
7 | elrest 12640 | . . . . . 6 ⊢ ((𝐽 ∈ V ∧ 𝐴 ∈ V) → (𝑥 ∈ (𝐽 ↾t 𝐴) ↔ ∃𝑦 ∈ 𝐽 𝑥 = (𝑦 ∩ 𝐴))) | |
8 | 6, 7 | syl 14 | . . . . 5 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ (𝐽 ↾t 𝐴)) → (𝑥 ∈ (𝐽 ↾t 𝐴) ↔ ∃𝑦 ∈ 𝐽 𝑥 = (𝑦 ∩ 𝐴))) |
9 | simpll 527 | . . . . . 6 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ (𝐽 ↾t 𝐴)) → 𝐾 ∈ 𝑉) | |
10 | 6 | simprd 114 | . . . . . 6 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ (𝐽 ↾t 𝐴)) → 𝐴 ∈ V) |
11 | elrest 12640 | . . . . . 6 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝐴 ∈ V) → (𝑥 ∈ (𝐾 ↾t 𝐴) ↔ ∃𝑦 ∈ 𝐾 𝑥 = (𝑦 ∩ 𝐴))) | |
12 | 9, 10, 11 | syl2anc 411 | . . . . 5 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ (𝐽 ↾t 𝐴)) → (𝑥 ∈ (𝐾 ↾t 𝐴) ↔ ∃𝑦 ∈ 𝐾 𝑥 = (𝑦 ∩ 𝐴))) |
13 | 3, 8, 12 | 3imtr4d 203 | . . . 4 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ (𝐽 ↾t 𝐴)) → (𝑥 ∈ (𝐽 ↾t 𝐴) → 𝑥 ∈ (𝐾 ↾t 𝐴))) |
14 | 1, 13 | mpd 13 | . . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ (𝐽 ↾t 𝐴)) → 𝑥 ∈ (𝐾 ↾t 𝐴)) |
15 | 14 | ex 115 | . 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐾) → (𝑥 ∈ (𝐽 ↾t 𝐴) → 𝑥 ∈ (𝐾 ↾t 𝐴))) |
16 | 15 | ssrdv 3161 | 1 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐾) → (𝐽 ↾t 𝐴) ⊆ (𝐾 ↾t 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1353 ∈ wcel 2148 ∃wrex 2456 Vcvv 2737 ∩ cin 3128 ⊆ wss 3129 ↦ cmpt 4061 ran crn 4624 (class class class)co 5869 ↾t crest 12633 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4115 ax-sep 4118 ax-pow 4171 ax-pr 4206 ax-un 4430 ax-setind 4533 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-iun 3886 df-br 4001 df-opab 4062 df-mpt 4063 df-id 4290 df-xp 4629 df-rel 4630 df-cnv 4631 df-co 4632 df-dm 4633 df-rn 4634 df-res 4635 df-ima 4636 df-iota 5174 df-fun 5214 df-fn 5215 df-f 5216 df-f1 5217 df-fo 5218 df-f1o 5219 df-fv 5220 df-ov 5872 df-oprab 5873 df-mpo 5874 df-rest 12635 |
This theorem is referenced by: (None) |
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