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Theorem cldval 17079
 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 16971 . . 3
3 pwexg 4375 . . 3
4 rabexg 4345 . . 3
52, 3, 43syl 19 . 2
6 unieq 4016 . . . . . 6
76, 1syl6eqr 2485 . . . . 5
87pweqd 3796 . . . 4
97difeq1d 3456 . . . . 5
10 eleq12 2497 . . . . 5
119, 10mpancom 651 . . . 4
128, 11rabeqbidv 2943 . . 3
13 df-cld 17075 . . 3
1412, 13fvmptg 5796 . 2
155, 14mpdan 650 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wceq 1652   wcel 1725  crab 2701  cvv 2948   cdif 3309  cpw 3791  cuni 4007  cfv 5446  ctop 16950  ccld 17072 This theorem is referenced by:  iscld  17083  mretopd  17148 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-iota 5410  df-fun 5448  df-fv 5454  df-top 16955  df-cld 17075
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