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Theorem cncls2i 23265
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 23236 . . . 4 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
2 cncls2i.1 . . . . 5 𝑌 = 𝐾
32clscld 23042 . . . 4 ((𝐾 ∈ Top ∧ 𝑆𝑌) → ((cls‘𝐾)‘𝑆) ∈ (Clsd‘𝐾))
41, 3sylan 578 . . 3 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆𝑌) → ((cls‘𝐾)‘𝑆) ∈ (Clsd‘𝐾))
5 cnclima 23263 . . 3 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ ((cls‘𝐾)‘𝑆) ∈ (Clsd‘𝐾)) → (𝐹 “ ((cls‘𝐾)‘𝑆)) ∈ (Clsd‘𝐽))
64, 5syldan 589 . 2 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆𝑌) → (𝐹 “ ((cls‘𝐾)‘𝑆)) ∈ (Clsd‘𝐽))
72sscls 23051 . . . 4 ((𝐾 ∈ Top ∧ 𝑆𝑌) → 𝑆 ⊆ ((cls‘𝐾)‘𝑆))
81, 7sylan 578 . . 3 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆𝑌) → 𝑆 ⊆ ((cls‘𝐾)‘𝑆))
9 imass2 6112 . . 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 2726 . . 3 𝐽 = 𝐽
1211clsss2 23067 . 2 (((𝐹 “ ((cls‘𝐾)‘𝑆)) ∈ (Clsd‘𝐽) ∧ (𝐹𝑆) ⊆ (𝐹 “ ((cls‘𝐾)‘𝑆))) → ((cls‘𝐽)‘(𝐹𝑆)) ⊆ (𝐹 “ ((cls‘𝐾)‘𝑆)))
136, 10, 12syl2anc 582 1 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆𝑌) → ((cls‘𝐽)‘(𝐹𝑆)) ⊆ (𝐹 “ ((cls‘𝐾)‘𝑆)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1534  wcel 2099  wss 3947   cuni 4913  ccnv 5681  cima 5685  cfv 6554  (class class class)co 7424  Topctop 22886  Clsdccld 23011  clsccl 23013   Cn ccn 23219
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5290  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-int 4955  df-iun 5003  df-iin 5004  df-br 5154  df-opab 5216  df-mpt 5237  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-ov 7427  df-oprab 7428  df-mpo 7429  df-map 8857  df-top 22887  df-topon 22904  df-cld 23014  df-cls 23016  df-cn 23222
This theorem is referenced by:  cnclsi  23267  cncls2  23268  imasncls  23687  hmeocls  23763  clssubg  24104
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