Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  cncls2i Structured version   Visualization version   GIF version

Theorem cncls2i 20884
 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 20855 . . . 4 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
2 cncls2i.1 . . . . 5 𝑌 = 𝐾
32clscld 20661 . . . 4 ((𝐾 ∈ Top ∧ 𝑆𝑌) → ((cls‘𝐾)‘𝑆) ∈ (Clsd‘𝐾))
41, 3sylan 487 . . 3 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆𝑌) → ((cls‘𝐾)‘𝑆) ∈ (Clsd‘𝐾))
5 cnclima 20882 . . 3 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ ((cls‘𝐾)‘𝑆) ∈ (Clsd‘𝐾)) → (𝐹 “ ((cls‘𝐾)‘𝑆)) ∈ (Clsd‘𝐽))
64, 5syldan 486 . 2 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆𝑌) → (𝐹 “ ((cls‘𝐾)‘𝑆)) ∈ (Clsd‘𝐽))
72sscls 20670 . . . 4 ((𝐾 ∈ Top ∧ 𝑆𝑌) → 𝑆 ⊆ ((cls‘𝐾)‘𝑆))
81, 7sylan 487 . . 3 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆𝑌) → 𝑆 ⊆ ((cls‘𝐾)‘𝑆))
9 imass2 5420 . . 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 2610 . . 3 𝐽 = 𝐽
1211clsss2 20686 . 2 (((𝐹 “ ((cls‘𝐾)‘𝑆)) ∈ (Clsd‘𝐽) ∧ (𝐹𝑆) ⊆ (𝐹 “ ((cls‘𝐾)‘𝑆))) → ((cls‘𝐽)‘(𝐹𝑆)) ⊆ (𝐹 “ ((cls‘𝐾)‘𝑆)))
136, 10, 12syl2anc 691 1 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆𝑌) → ((cls‘𝐽)‘(𝐹𝑆)) ⊆ (𝐹 “ ((cls‘𝐾)‘𝑆)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ⊆ wss 3540  ∪ cuni 4372  ◡ccnv 5037   “ cima 5041  ‘cfv 5804  (class class class)co 6549  Topctop 20517  Clsdccld 20630  clsccl 20632   Cn ccn 20838 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-map 7746  df-top 20521  df-topon 20523  df-cld 20633  df-cls 20635  df-cn 20841 This theorem is referenced by:  cnclsi  20886  cncls2  20887  imasncls  21305  hmeocls  21381  clssubg  21722
 Copyright terms: Public domain W3C validator