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Mirrors > Home > ILE Home > Th. List > restid2 | GIF version |
Description: The subspace topology over a subset of the base set is the original topology. (Contributed by Mario Carneiro, 13-Aug-2015.) |
Ref | Expression |
---|---|
restid2 | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐽 ⊆ 𝒫 𝐴) → (𝐽 ↾t 𝐴) = 𝐽) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pwexg 4209 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ V) | |
2 | 1 | adantr 276 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐽 ⊆ 𝒫 𝐴) → 𝒫 𝐴 ∈ V) |
3 | simpr 110 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐽 ⊆ 𝒫 𝐴) → 𝐽 ⊆ 𝒫 𝐴) | |
4 | 2, 3 | ssexd 4169 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐽 ⊆ 𝒫 𝐴) → 𝐽 ∈ V) |
5 | simpl 109 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐽 ⊆ 𝒫 𝐴) → 𝐴 ∈ 𝑉) | |
6 | restval 12856 | . . 3 ⊢ ((𝐽 ∈ V ∧ 𝐴 ∈ 𝑉) → (𝐽 ↾t 𝐴) = ran (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐴))) | |
7 | 4, 5, 6 | syl2anc 411 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐽 ⊆ 𝒫 𝐴) → (𝐽 ↾t 𝐴) = ran (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐴))) |
8 | 3 | sselda 3179 | . . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐽 ⊆ 𝒫 𝐴) ∧ 𝑥 ∈ 𝐽) → 𝑥 ∈ 𝒫 𝐴) |
9 | 8 | elpwid 3612 | . . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐽 ⊆ 𝒫 𝐴) ∧ 𝑥 ∈ 𝐽) → 𝑥 ⊆ 𝐴) |
10 | df-ss 3166 | . . . . . . 7 ⊢ (𝑥 ⊆ 𝐴 ↔ (𝑥 ∩ 𝐴) = 𝑥) | |
11 | 9, 10 | sylib 122 | . . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐽 ⊆ 𝒫 𝐴) ∧ 𝑥 ∈ 𝐽) → (𝑥 ∩ 𝐴) = 𝑥) |
12 | 11 | mpteq2dva 4119 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐽 ⊆ 𝒫 𝐴) → (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐴)) = (𝑥 ∈ 𝐽 ↦ 𝑥)) |
13 | mptresid 4996 | . . . . 5 ⊢ ( I ↾ 𝐽) = (𝑥 ∈ 𝐽 ↦ 𝑥) | |
14 | 12, 13 | eqtr4di 2244 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐽 ⊆ 𝒫 𝐴) → (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐴)) = ( I ↾ 𝐽)) |
15 | 14 | rneqd 4891 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐽 ⊆ 𝒫 𝐴) → ran (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐴)) = ran ( I ↾ 𝐽)) |
16 | rnresi 5022 | . . 3 ⊢ ran ( I ↾ 𝐽) = 𝐽 | |
17 | 15, 16 | eqtrdi 2242 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐽 ⊆ 𝒫 𝐴) → ran (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐴)) = 𝐽) |
18 | 7, 17 | eqtrd 2226 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐽 ⊆ 𝒫 𝐴) → (𝐽 ↾t 𝐴) = 𝐽) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1364 ∈ wcel 2164 Vcvv 2760 ∩ cin 3152 ⊆ wss 3153 𝒫 cpw 3601 ↦ cmpt 4090 I cid 4319 ran crn 4660 ↾ cres 4661 (class class class)co 5918 ↾t crest 12850 |
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 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-coll 4144 ax-sep 4147 ax-pow 4203 ax-pr 4238 ax-un 4464 ax-setind 4569 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-ral 2477 df-rex 2478 df-reu 2479 df-rab 2481 df-v 2762 df-sbc 2986 df-csb 3081 df-dif 3155 df-un 3157 df-in 3159 df-ss 3166 df-pw 3603 df-sn 3624 df-pr 3625 df-op 3627 df-uni 3836 df-iun 3914 df-br 4030 df-opab 4091 df-mpt 4092 df-id 4324 df-xp 4665 df-rel 4666 df-cnv 4667 df-co 4668 df-dm 4669 df-rn 4670 df-res 4671 df-ima 4672 df-iota 5215 df-fun 5256 df-fn 5257 df-f 5258 df-f1 5259 df-fo 5260 df-f1o 5261 df-fv 5262 df-ov 5921 df-oprab 5922 df-mpo 5923 df-rest 12852 |
This theorem is referenced by: restid 12861 topnidg 12863 |
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