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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 22938 | . . . 4 ⊢ 𝐵 = ∪ 𝐴 |
3 | 2 | restid 17480 | . . 3 ⊢ (𝐴 ∈ (TopOn‘𝐵) → (𝐴 ↾t 𝐵) = 𝐴) |
4 | 1, 3 | ax-mp 5 | . 2 ⊢ (𝐴 ↾t 𝐵) = 𝐴 |
5 | 4 | eqcomi 2744 | 1 ⊢ 𝐴 = (𝐴 ↾t 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2106 ‘cfv 6563 (class class class)co 7431 ↾t crest 17467 TopOnctopon 22932 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-rest 17469 df-topon 22933 |
This theorem is referenced by: cncfcn1 24951 cncfmpt2f 24955 cdivcncf 24961 cnrehmeo 24998 cnrehmeoOLD 24999 mulcncf 25494 cnlimc 25938 dvidlem 25965 dvcnp2 25970 dvcnp2OLD 25971 dvcn 25972 dvnres 25982 dvaddbr 25989 dvmulbr 25990 dvmulbrOLD 25991 dvcobr 25998 dvcobrOLD 25999 dvcjbr 26002 dvrec 26008 dvexp3 26031 dveflem 26032 dvlipcn 26048 lhop1lem 26067 ftc1cn 26099 dvply1 26340 dvtaylp 26427 taylthlem2 26431 taylthlem2OLD 26432 psercn 26485 pserdvlem2 26487 pserdv 26488 abelth 26500 logcn 26704 dvloglem 26705 dvlog 26708 dvlog2 26710 efopnlem2 26714 logtayl 26717 cxpcn 26802 cxpcnOLD 26803 cxpcn2 26804 cxpcn3 26806 resqrtcn 26807 sqrtcn 26808 dvatan 26993 ftalem3 27133 cxpcncf1 34589 knoppcnlem10 36485 knoppcnlem11 36486 dvtan 37657 ftc1cnnc 37679 dvasin 37691 dvacos 37692 cxpcncf2 45855 |
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