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Theorem iscld 21780
Description: The predicate "the class 𝑆 is a closed set". (Contributed by NM, 2-Oct-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)
Hypothesis
Ref Expression
iscld.1 𝑋 = 𝐽
Assertion
Ref Expression
iscld (𝐽 ∈ Top → (𝑆 ∈ (Clsd‘𝐽) ↔ (𝑆𝑋 ∧ (𝑋𝑆) ∈ 𝐽)))

Proof of Theorem iscld
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 iscld.1 . . . . 5 𝑋 = 𝐽
21cldval 21776 . . . 4 (𝐽 ∈ Top → (Clsd‘𝐽) = {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ∈ 𝐽})
32eleq2d 2818 . . 3 (𝐽 ∈ Top → (𝑆 ∈ (Clsd‘𝐽) ↔ 𝑆 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ∈ 𝐽}))
4 difeq2 4007 . . . . 5 (𝑥 = 𝑆 → (𝑋𝑥) = (𝑋𝑆))
54eleq1d 2817 . . . 4 (𝑥 = 𝑆 → ((𝑋𝑥) ∈ 𝐽 ↔ (𝑋𝑆) ∈ 𝐽))
65elrab 3588 . . 3 (𝑆 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ∈ 𝐽} ↔ (𝑆 ∈ 𝒫 𝑋 ∧ (𝑋𝑆) ∈ 𝐽))
73, 6bitrdi 290 . 2 (𝐽 ∈ Top → (𝑆 ∈ (Clsd‘𝐽) ↔ (𝑆 ∈ 𝒫 𝑋 ∧ (𝑋𝑆) ∈ 𝐽)))
81topopn 21659 . . . 4 (𝐽 ∈ Top → 𝑋𝐽)
9 elpw2g 5212 . . . 4 (𝑋𝐽 → (𝑆 ∈ 𝒫 𝑋𝑆𝑋))
108, 9syl 17 . . 3 (𝐽 ∈ Top → (𝑆 ∈ 𝒫 𝑋𝑆𝑋))
1110anbi1d 633 . 2 (𝐽 ∈ Top → ((𝑆 ∈ 𝒫 𝑋 ∧ (𝑋𝑆) ∈ 𝐽) ↔ (𝑆𝑋 ∧ (𝑋𝑆) ∈ 𝐽)))
127, 11bitrd 282 1 (𝐽 ∈ Top → (𝑆 ∈ (Clsd‘𝐽) ↔ (𝑆𝑋 ∧ (𝑋𝑆) ∈ 𝐽)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1542  wcel 2114  {crab 3057  cdif 3840  wss 3843  𝒫 cpw 4488   cuni 4796  cfv 6339  Topctop 21646  Clsdccld 21769
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3400  df-sbc 3681  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5429  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-iota 6297  df-fun 6341  df-fv 6347  df-top 21647  df-cld 21772
This theorem is referenced by:  iscld2  21781  cldss  21782  cldopn  21784  topcld  21788  discld  21842  indiscld  21844  restcld  21925  ordtcld1  21950  ordtcld2  21951  hauscmp  22160  txcld  22356  ptcld  22366  qtopcld  22466  opnsubg  22861  sszcld  23571  ist0cld  31357  stoweidlem57  43162
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