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Theorem clstop 13923
Description: The closure of a topology's underlying set is the entire set. (Contributed by NM, 5-Oct-2007.) (Proof shortened by Jim Kingdon, 11-Mar-2023.)
Hypothesis
Ref Expression
clscld.1 𝑋 = 𝐽
Assertion
Ref Expression
clstop (𝐽 ∈ Top → ((cls‘𝐽)‘𝑋) = 𝑋)

Proof of Theorem clstop
StepHypRef Expression
1 clscld.1 . . 3 𝑋 = 𝐽
21topcld 13905 . 2 (𝐽 ∈ Top → 𝑋 ∈ (Clsd‘𝐽))
3 cldcls 13910 . 2 (𝑋 ∈ (Clsd‘𝐽) → ((cls‘𝐽)‘𝑋) = 𝑋)
42, 3syl 14 1 (𝐽 ∈ Top → ((cls‘𝐽)‘𝑋) = 𝑋)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1363  wcel 2158   cuni 3821  cfv 5228  Topctop 13793  Clsdccld 13888  clsccl 13890
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-reu 2472  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-top 13794  df-cld 13891  df-cls 13893
This theorem is referenced by: (None)
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