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Theorem toponrestid 22943
Description: Given a topology on a set, restricting it to that same set has no effect. (Contributed by Jim Kingdon, 6-Jul-2022.)
Hypothesis
Ref Expression
toponrestid.t 𝐴 ∈ (TopOn‘𝐵)
Assertion
Ref Expression
toponrestid 𝐴 = (𝐴t 𝐵)

Proof of Theorem toponrestid
StepHypRef Expression
1 toponrestid.t . . 3 𝐴 ∈ (TopOn‘𝐵)
21toponunii 22938 . . . 4 𝐵 = 𝐴
32restid 17480 . . 3 (𝐴 ∈ (TopOn‘𝐵) → (𝐴t 𝐵) = 𝐴)
41, 3ax-mp 5 . 2 (𝐴t 𝐵) = 𝐴
54eqcomi 2744 1 𝐴 = (𝐴t 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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