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Mirrors > Home > MPE Home > Th. List > toponrestid | Structured version Visualization version GIF version |
Description: Given a topology on a set, restricting it to that same set has no effect. (Contributed by Jim Kingdon, 6-Jul-2022.) |
Ref | Expression |
---|---|
toponrestid.t | ⊢ 𝐴 ∈ (TopOn‘𝐵) |
Ref | Expression |
---|---|
toponrestid | ⊢ 𝐴 = (𝐴 ↾t 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | toponrestid.t | . . 3 ⊢ 𝐴 ∈ (TopOn‘𝐵) | |
2 | 1 | toponunii 21785 | . . . 4 ⊢ 𝐵 = ∪ 𝐴 |
3 | 2 | restid 16910 | . . 3 ⊢ (𝐴 ∈ (TopOn‘𝐵) → (𝐴 ↾t 𝐵) = 𝐴) |
4 | 1, 3 | ax-mp 5 | . 2 ⊢ (𝐴 ↾t 𝐵) = 𝐴 |
5 | 4 | eqcomi 2743 | 1 ⊢ 𝐴 = (𝐴 ↾t 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1543 ∈ wcel 2110 ‘cfv 6369 (class class class)co 7202 ↾t crest 16897 TopOnctopon 21779 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-rep 5168 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-ral 3059 df-rex 3060 df-reu 3061 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-op 4538 df-uni 4810 df-iun 4896 df-br 5044 df-opab 5106 df-mpt 5125 df-id 5444 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-ov 7205 df-oprab 7206 df-mpo 7207 df-rest 16899 df-topon 21780 |
This theorem is referenced by: cncfcn1 23780 cncfmpt2f 23784 cdivcncf 23790 cnrehmeo 23822 cnlimc 24757 dvidlem 24784 dvcnp2 24789 dvcn 24790 dvnres 24800 dvaddbr 24807 dvmulbr 24808 dvcobr 24815 dvcjbr 24818 dvrec 24824 dvexp3 24847 dveflem 24848 dvlipcn 24863 lhop1lem 24882 ftc1cn 24912 dvply1 25149 dvtaylp 25234 taylthlem2 25238 psercn 25290 pserdvlem2 25292 pserdv 25293 abelth 25305 logcn 25507 dvloglem 25508 dvlog 25511 dvlog2 25513 efopnlem2 25517 logtayl 25520 cxpcn 25603 cxpcn2 25604 cxpcn3 25606 resqrtcn 25607 sqrtcn 25608 dvatan 25790 ftalem3 25929 cxpcncf1 32259 knoppcnlem10 34376 knoppcnlem11 34377 dvtan 35521 ftc1cnnc 35543 dvasin 35555 dvacos 35556 cxpcncf2 43069 |
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