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Theorem cnclsi 21409
Description: Property of the image of a closure. (Contributed by Mario Carneiro, 25-Aug-2015.)
Hypothesis
Ref Expression
cnclsi.1 𝑋 = 𝐽
Assertion
Ref Expression
cnclsi ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆𝑋) → (𝐹 “ ((cls‘𝐽)‘𝑆)) ⊆ ((cls‘𝐾)‘(𝐹𝑆)))

Proof of Theorem cnclsi
StepHypRef Expression
1 cntop1 21377 . . . . 5 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top)
21adantr 473 . . . 4 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆𝑋) → 𝐽 ∈ Top)
3 cnvimass 5706 . . . . 5 (𝐹 “ (𝐹𝑆)) ⊆ dom 𝐹
4 cnclsi.1 . . . . . . 7 𝑋 = 𝐽
5 eqid 2803 . . . . . . 7 𝐾 = 𝐾
64, 5cnf 21383 . . . . . 6 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:𝑋 𝐾)
76adantr 473 . . . . 5 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆𝑋) → 𝐹:𝑋 𝐾)
83, 7fssdm 6276 . . . 4 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆𝑋) → (𝐹 “ (𝐹𝑆)) ⊆ 𝑋)
9 simpr 478 . . . . . . 7 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆𝑋) → 𝑆𝑋)
107fdmd 6269 . . . . . . 7 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆𝑋) → dom 𝐹 = 𝑋)
119, 10sseqtr4d 3842 . . . . . 6 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆𝑋) → 𝑆 ⊆ dom 𝐹)
12 sseqin2 4019 . . . . . 6 (𝑆 ⊆ dom 𝐹 ↔ (dom 𝐹𝑆) = 𝑆)
1311, 12sylib 210 . . . . 5 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆𝑋) → (dom 𝐹𝑆) = 𝑆)
14 dminss 5768 . . . . 5 (dom 𝐹𝑆) ⊆ (𝐹 “ (𝐹𝑆))
1513, 14syl6eqssr 3856 . . . 4 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆𝑋) → 𝑆 ⊆ (𝐹 “ (𝐹𝑆)))
164clsss 21191 . . . 4 ((𝐽 ∈ Top ∧ (𝐹 “ (𝐹𝑆)) ⊆ 𝑋𝑆 ⊆ (𝐹 “ (𝐹𝑆))) → ((cls‘𝐽)‘𝑆) ⊆ ((cls‘𝐽)‘(𝐹 “ (𝐹𝑆))))
172, 8, 15, 16syl3anc 1491 . . 3 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆𝑋) → ((cls‘𝐽)‘𝑆) ⊆ ((cls‘𝐽)‘(𝐹 “ (𝐹𝑆))))
18 imassrn 5698 . . . . 5 (𝐹𝑆) ⊆ ran 𝐹
197frnd 6267 . . . . 5 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆𝑋) → ran 𝐹 𝐾)
2018, 19syl5ss 3813 . . . 4 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆𝑋) → (𝐹𝑆) ⊆ 𝐾)
215cncls2i 21407 . . . 4 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐹𝑆) ⊆ 𝐾) → ((cls‘𝐽)‘(𝐹 “ (𝐹𝑆))) ⊆ (𝐹 “ ((cls‘𝐾)‘(𝐹𝑆))))
2220, 21syldan 586 . . 3 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆𝑋) → ((cls‘𝐽)‘(𝐹 “ (𝐹𝑆))) ⊆ (𝐹 “ ((cls‘𝐾)‘(𝐹𝑆))))
2317, 22sstrd 3812 . 2 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆𝑋) → ((cls‘𝐽)‘𝑆) ⊆ (𝐹 “ ((cls‘𝐾)‘(𝐹𝑆))))
247ffund 6264 . . 3 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆𝑋) → Fun 𝐹)
254clsss3 21196 . . . . 5 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((cls‘𝐽)‘𝑆) ⊆ 𝑋)
261, 25sylan 576 . . . 4 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆𝑋) → ((cls‘𝐽)‘𝑆) ⊆ 𝑋)
2726, 10sseqtr4d 3842 . . 3 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆𝑋) → ((cls‘𝐽)‘𝑆) ⊆ dom 𝐹)
28 funimass3 6563 . . 3 ((Fun 𝐹 ∧ ((cls‘𝐽)‘𝑆) ⊆ dom 𝐹) → ((𝐹 “ ((cls‘𝐽)‘𝑆)) ⊆ ((cls‘𝐾)‘(𝐹𝑆)) ↔ ((cls‘𝐽)‘𝑆) ⊆ (𝐹 “ ((cls‘𝐾)‘(𝐹𝑆)))))
2924, 27, 28syl2anc 580 . 2 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆𝑋) → ((𝐹 “ ((cls‘𝐽)‘𝑆)) ⊆ ((cls‘𝐾)‘(𝐹𝑆)) ↔ ((cls‘𝐽)‘𝑆) ⊆ (𝐹 “ ((cls‘𝐾)‘(𝐹𝑆)))))
3023, 29mpbird 249 1 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆𝑋) → (𝐹 “ ((cls‘𝐽)‘𝑆)) ⊆ ((cls‘𝐾)‘(𝐹𝑆)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385   = wceq 1653  wcel 2157  cin 3772  wss 3773   cuni 4632  ccnv 5315  dom cdm 5316  ran crn 5317  cima 5319  Fun wfun 6099  wf 6101  cfv 6105  (class class class)co 6882  Topctop 21030  clsccl 21155   Cn ccn 21361
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2379  ax-ext 2781  ax-rep 4968  ax-sep 4979  ax-nul 4987  ax-pow 5039  ax-pr 5101  ax-un 7187
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2593  df-eu 2611  df-clab 2790  df-cleq 2796  df-clel 2799  df-nfc 2934  df-ne 2976  df-ral 3098  df-rex 3099  df-reu 3100  df-rab 3102  df-v 3391  df-sbc 3638  df-csb 3733  df-dif 3776  df-un 3778  df-in 3780  df-ss 3787  df-nul 4120  df-if 4282  df-pw 4355  df-sn 4373  df-pr 4375  df-op 4379  df-uni 4633  df-int 4672  df-iun 4716  df-iin 4717  df-br 4848  df-opab 4910  df-mpt 4927  df-id 5224  df-xp 5322  df-rel 5323  df-cnv 5324  df-co 5325  df-dm 5326  df-rn 5327  df-res 5328  df-ima 5329  df-iota 6068  df-fun 6107  df-fn 6108  df-f 6109  df-f1 6110  df-fo 6111  df-f1o 6112  df-fv 6113  df-ov 6885  df-oprab 6886  df-mpt2 6887  df-map 8101  df-top 21031  df-topon 21048  df-cld 21156  df-cls 21158  df-cn 21364
This theorem is referenced by:  cncls  21411  hmeocls  21904  clsnsg  22245
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