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Theorem cncls2i 20984
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 20955 . . . 4 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
2 cncls2i.1 . . . . 5 𝑌 = 𝐾
32clscld 20761 . . . 4 ((𝐾 ∈ Top ∧ 𝑆𝑌) → ((cls‘𝐾)‘𝑆) ∈ (Clsd‘𝐾))
41, 3sylan 488 . . 3 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆𝑌) → ((cls‘𝐾)‘𝑆) ∈ (Clsd‘𝐾))
5 cnclima 20982 . . 3 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ ((cls‘𝐾)‘𝑆) ∈ (Clsd‘𝐾)) → (𝐹 “ ((cls‘𝐾)‘𝑆)) ∈ (Clsd‘𝐽))
64, 5syldan 487 . 2 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆𝑌) → (𝐹 “ ((cls‘𝐾)‘𝑆)) ∈ (Clsd‘𝐽))
72sscls 20770 . . . 4 ((𝐾 ∈ Top ∧ 𝑆𝑌) → 𝑆 ⊆ ((cls‘𝐾)‘𝑆))
81, 7sylan 488 . . 3 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆𝑌) → 𝑆 ⊆ ((cls‘𝐾)‘𝑆))
9 imass2 5460 . . 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 2621 . . 3 𝐽 = 𝐽
1211clsss2 20786 . 2 (((𝐹 “ ((cls‘𝐾)‘𝑆)) ∈ (Clsd‘𝐽) ∧ (𝐹𝑆) ⊆ (𝐹 “ ((cls‘𝐾)‘𝑆))) → ((cls‘𝐽)‘(𝐹𝑆)) ⊆ (𝐹 “ ((cls‘𝐾)‘𝑆)))
136, 10, 12syl2anc 692 1 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆𝑌) → ((cls‘𝐽)‘(𝐹𝑆)) ⊆ (𝐹 “ ((cls‘𝐾)‘𝑆)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  wss 3555   cuni 4402  ccnv 5073  cima 5077  cfv 5847  (class class class)co 6604  Topctop 20617  Clsdccld 20730  clsccl 20732   Cn ccn 20938
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-iin 4488  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-map 7804  df-top 20621  df-topon 20623  df-cld 20733  df-cls 20735  df-cn 20941
This theorem is referenced by:  cnclsi  20986  cncls2  20987  imasncls  21405  hmeocls  21481  clssubg  21822
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