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Mirrors > Home > ILE Home > Th. List > restid | GIF version |
Description: The subspace topology of the base set is the original topology. (Contributed by Jeff Hankins, 9-Jul-2009.) (Revised by Mario Carneiro, 13-Aug-2015.) |
Ref | Expression |
---|---|
restid.1 | ⊢ 𝑋 = ∪ 𝐽 |
Ref | Expression |
---|---|
restid | ⊢ (𝐽 ∈ 𝑉 → (𝐽 ↾t 𝑋) = 𝐽) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | restid.1 | . . 3 ⊢ 𝑋 = ∪ 𝐽 | |
2 | uniexg 4457 | . . 3 ⊢ (𝐽 ∈ 𝑉 → ∪ 𝐽 ∈ V) | |
3 | 1, 2 | eqeltrid 2276 | . 2 ⊢ (𝐽 ∈ 𝑉 → 𝑋 ∈ V) |
4 | 1 | eqimss2i 3227 | . . 3 ⊢ ∪ 𝐽 ⊆ 𝑋 |
5 | sspwuni 3986 | . . 3 ⊢ (𝐽 ⊆ 𝒫 𝑋 ↔ ∪ 𝐽 ⊆ 𝑋) | |
6 | 4, 5 | mpbir 146 | . 2 ⊢ 𝐽 ⊆ 𝒫 𝑋 |
7 | restid2 12753 | . 2 ⊢ ((𝑋 ∈ V ∧ 𝐽 ⊆ 𝒫 𝑋) → (𝐽 ↾t 𝑋) = 𝐽) | |
8 | 3, 6, 7 | sylancl 413 | 1 ⊢ (𝐽 ∈ 𝑉 → (𝐽 ↾t 𝑋) = 𝐽) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1364 ∈ wcel 2160 Vcvv 2752 ⊆ wss 3144 𝒫 cpw 3590 ∪ cuni 3824 (class class class)co 5896 ↾t crest 12744 |
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 2162 ax-14 2163 ax-ext 2171 ax-coll 4133 ax-sep 4136 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 |
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 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-id 4311 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-res 4656 df-ima 4657 df-iota 5196 df-fun 5237 df-fn 5238 df-f 5239 df-f1 5240 df-fo 5241 df-f1o 5242 df-fv 5243 df-ov 5899 df-oprab 5900 df-mpo 5901 df-rest 12746 |
This theorem is referenced by: toponrestid 13981 restin 14136 |
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