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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 7717 | . . 3 ⊢ (𝐽 ∈ 𝑉 → ∪ 𝐽 ∈ V) | |
3 | 1, 2 | eqeltrid 2838 | . 2 ⊢ (𝐽 ∈ 𝑉 → 𝑋 ∈ V) |
4 | 1 | eqimss2i 4041 | . . 3 ⊢ ∪ 𝐽 ⊆ 𝑋 |
5 | sspwuni 5099 | . . 3 ⊢ (𝐽 ⊆ 𝒫 𝑋 ↔ ∪ 𝐽 ⊆ 𝑋) | |
6 | 4, 5 | mpbir 230 | . 2 ⊢ 𝐽 ⊆ 𝒫 𝑋 |
7 | restid2 17363 | . 2 ⊢ ((𝑋 ∈ V ∧ 𝐽 ⊆ 𝒫 𝑋) → (𝐽 ↾t 𝑋) = 𝐽) | |
8 | 3, 6, 7 | sylancl 587 | 1 ⊢ (𝐽 ∈ 𝑉 → (𝐽 ↾t 𝑋) = 𝐽) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 Vcvv 3475 ⊆ wss 3946 𝒫 cpw 4598 ∪ cuni 4904 (class class class)co 7396 ↾t crest 17353 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5359 ax-pr 5423 ax-un 7712 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4321 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4905 df-iun 4995 df-br 5145 df-opab 5207 df-mpt 5228 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6487 df-fun 6537 df-fn 6538 df-f 6539 df-f1 6540 df-fo 6541 df-f1o 6542 df-fv 6543 df-ov 7399 df-oprab 7400 df-mpo 7401 df-rest 17355 |
This theorem is referenced by: toponrestid 22392 restin 22639 cnrmnrm 22834 cmpkgen 23024 xkopt 23128 xkoinjcn 23160 ussid 23734 tuslem 23740 tuslemOLD 23741 cnperf 24305 retopconn 24314 abscncfALT 24409 cnmpopc 24413 recnperf 25391 lhop1lem 25499 cxpcn3 26223 retopsconn 34171 ivthALT 35125 binomcxplemdvbinom 42983 binomcxplemnotnn0 42986 fsumcncf 44467 ioccncflimc 44474 cncfuni 44475 icocncflimc 44478 cncfiooicclem1 44482 itgsubsticclem 44564 dirkercncflem2 44693 dirkercncflem4 44695 fourierdlem32 44728 fourierdlem33 44729 fourierdlem62 44757 fourierdlem93 44788 fourierdlem101 44796 |
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