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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 22922 | . . . 4 ⊢ 𝐵 = ∪ 𝐴 |
| 3 | 2 | restid 17478 | . . 3 ⊢ (𝐴 ∈ (TopOn‘𝐵) → (𝐴 ↾t 𝐵) = 𝐴) |
| 4 | 1, 3 | ax-mp 5 | . 2 ⊢ (𝐴 ↾t 𝐵) = 𝐴 |
| 5 | 4 | eqcomi 2746 | 1 ⊢ 𝐴 = (𝐴 ↾t 𝐵) |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 ∈ wcel 2108 ‘cfv 6561 (class class class)co 7431 ↾t crest 17465 TopOnctopon 22916 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-rest 17467 df-topon 22917 |
| This theorem is referenced by: cncfcn1 24937 cncfmpt2f 24941 cdivcncf 24947 cnrehmeo 24984 cnrehmeoOLD 24985 mulcncf 25480 cnlimc 25923 dvidlem 25950 dvcnp2 25955 dvcnp2OLD 25956 dvcn 25957 dvnres 25967 dvaddbr 25974 dvmulbr 25975 dvmulbrOLD 25976 dvcobr 25983 dvcobrOLD 25984 dvcjbr 25987 dvrec 25993 dvexp3 26016 dveflem 26017 dvlipcn 26033 lhop1lem 26052 ftc1cn 26084 dvply1 26325 dvtaylp 26412 taylthlem2 26416 taylthlem2OLD 26417 psercn 26470 pserdvlem2 26472 pserdv 26473 abelth 26485 logcn 26689 dvloglem 26690 dvlog 26693 dvlog2 26695 efopnlem2 26699 logtayl 26702 cxpcn 26787 cxpcnOLD 26788 cxpcn2 26789 cxpcn3 26791 resqrtcn 26792 sqrtcn 26793 dvatan 26978 ftalem3 27118 cxpcncf1 34610 knoppcnlem10 36503 knoppcnlem11 36504 dvtan 37677 ftc1cnnc 37699 dvasin 37711 dvacos 37712 cxpcncf2 45914 |
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