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Theorem cldval 12257
 Description: The set of closed sets of a topology. (Note that the set of open sets is just the topology itself, so we don't have a separate definition.) (Contributed by NM, 2-Oct-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)
Hypothesis
Ref Expression
cldval.1
Assertion
Ref Expression
cldval
Distinct variable groups:   ,   ,

Proof of Theorem cldval
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 cldval.1 . . . 4
21topopn 12164 . . 3
3 pwexg 4099 . . 3
4 rabexg 4066 . . 3
52, 3, 43syl 17 . 2
6 unieq 3740 . . . . . 6
76, 1syl6eqr 2188 . . . . 5
87pweqd 3510 . . . 4
97difeq1d 3188 . . . . 5
10 eleq12 2202 . . . . 5
119, 10mpancom 418 . . . 4
128, 11rabeqbidv 2676 . . 3
13 df-cld 12253 . . 3
1412, 13fvmptg 5490 . 2
155, 14mpdan 417 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 104   wceq 1331   wcel 1480  crab 2418  cvv 2681   cdif 3063  cpw 3505  cuni 3731  cfv 5118  ctop 12153  ccld 12250 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126 This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-rab 2423  df-v 2683  df-sbc 2905  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-iota 5083  df-fun 5120  df-fv 5126  df-top 12154  df-cld 12253 This theorem is referenced by:  iscld  12261
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