Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > restid | Structured version Visualization version 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 7456 | . . 3 ⊢ (𝐽 ∈ 𝑉 → ∪ 𝐽 ∈ V) | |
3 | 1, 2 | eqeltrid 2914 | . 2 ⊢ (𝐽 ∈ 𝑉 → 𝑋 ∈ V) |
4 | 1 | eqimss2i 4023 | . . 3 ⊢ ∪ 𝐽 ⊆ 𝑋 |
5 | sspwuni 5013 | . . 3 ⊢ (𝐽 ⊆ 𝒫 𝑋 ↔ ∪ 𝐽 ⊆ 𝑋) | |
6 | 4, 5 | mpbir 232 | . 2 ⊢ 𝐽 ⊆ 𝒫 𝑋 |
7 | restid2 16692 | . 2 ⊢ ((𝑋 ∈ V ∧ 𝐽 ⊆ 𝒫 𝑋) → (𝐽 ↾t 𝑋) = 𝐽) | |
8 | 3, 6, 7 | sylancl 586 | 1 ⊢ (𝐽 ∈ 𝑉 → (𝐽 ↾t 𝑋) = 𝐽) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 Vcvv 3492 ⊆ wss 3933 𝒫 cpw 4535 ∪ cuni 4830 (class class class)co 7145 ↾t crest 16682 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-rest 16684 |
This theorem is referenced by: toponrestid 21457 restin 21702 cnrmnrm 21897 cmpkgen 22087 xkopt 22191 xkoinjcn 22223 ussid 22796 tuslem 22803 cnperf 23355 retopconn 23364 abscncfALT 23455 cnmpopc 23459 recnperf 24430 lhop1lem 24537 cxpcn3 25256 retopsconn 32393 ivthALT 33580 binomcxplemdvbinom 40562 binomcxplemnotnn0 40565 fsumcncf 42037 ioccncflimc 42044 cncfuni 42045 icocncflimc 42048 cncfiooicclem1 42052 itgsubsticclem 42136 dirkercncflem2 42266 dirkercncflem4 42268 fourierdlem32 42301 fourierdlem33 42302 fourierdlem62 42330 fourierdlem93 42361 fourierdlem101 42369 |
Copyright terms: Public domain | W3C validator |