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Theorem cncls2i 21808
Description: Property of the preimage of a closure. (Contributed by Mario Carneiro, 25-Aug-2015.)
Hypothesis
Ref Expression
cncls2i.1 𝑌 = 𝐾
Assertion
Ref Expression
cncls2i ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆𝑌) → ((cls‘𝐽)‘(𝐹𝑆)) ⊆ (𝐹 “ ((cls‘𝐾)‘𝑆)))

Proof of Theorem cncls2i
StepHypRef Expression
1 cntop2 21779 . . . 4 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
2 cncls2i.1 . . . . 5 𝑌 = 𝐾
32clscld 21585 . . . 4 ((𝐾 ∈ Top ∧ 𝑆𝑌) → ((cls‘𝐾)‘𝑆) ∈ (Clsd‘𝐾))
41, 3sylan 580 . . 3 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆𝑌) → ((cls‘𝐾)‘𝑆) ∈ (Clsd‘𝐾))
5 cnclima 21806 . . 3 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ ((cls‘𝐾)‘𝑆) ∈ (Clsd‘𝐾)) → (𝐹 “ ((cls‘𝐾)‘𝑆)) ∈ (Clsd‘𝐽))
64, 5syldan 591 . 2 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆𝑌) → (𝐹 “ ((cls‘𝐾)‘𝑆)) ∈ (Clsd‘𝐽))
72sscls 21594 . . . 4 ((𝐾 ∈ Top ∧ 𝑆𝑌) → 𝑆 ⊆ ((cls‘𝐾)‘𝑆))
81, 7sylan 580 . . 3 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆𝑌) → 𝑆 ⊆ ((cls‘𝐾)‘𝑆))
9 imass2 5959 . . 3 (𝑆 ⊆ ((cls‘𝐾)‘𝑆) → (𝐹𝑆) ⊆ (𝐹 “ ((cls‘𝐾)‘𝑆)))
108, 9syl 17 . 2 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆𝑌) → (𝐹𝑆) ⊆ (𝐹 “ ((cls‘𝐾)‘𝑆)))
11 eqid 2821 . . 3 𝐽 = 𝐽
1211clsss2 21610 . 2 (((𝐹 “ ((cls‘𝐾)‘𝑆)) ∈ (Clsd‘𝐽) ∧ (𝐹𝑆) ⊆ (𝐹 “ ((cls‘𝐾)‘𝑆))) → ((cls‘𝐽)‘(𝐹𝑆)) ⊆ (𝐹 “ ((cls‘𝐾)‘𝑆)))
136, 10, 12syl2anc 584 1 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆𝑌) → ((cls‘𝐽)‘(𝐹𝑆)) ⊆ (𝐹 “ ((cls‘𝐾)‘𝑆)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  wss 3935   cuni 4832  ccnv 5548  cima 5552  cfv 6349  (class class class)co 7145  Topctop 21431  Clsdccld 21554  clsccl 21556   Cn ccn 21762
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3497  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4833  df-int 4870  df-iun 4914  df-iin 4915  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-ov 7148  df-oprab 7149  df-mpo 7150  df-map 8398  df-top 21432  df-topon 21449  df-cld 21557  df-cls 21559  df-cn 21765
This theorem is referenced by:  cnclsi  21810  cncls2  21811  imasncls  22230  hmeocls  22306  clssubg  22646
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