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Theorem cnclsi 23296
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 23264 . . . . 5 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top)
21adantr 480 . . . 4 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆𝑋) → 𝐽 ∈ Top)
3 cnvimass 6102 . . . . 5 (𝐹 “ (𝐹𝑆)) ⊆ dom 𝐹
4 cnclsi.1 . . . . . . 7 𝑋 = 𝐽
5 eqid 2735 . . . . . . 7 𝐾 = 𝐾
64, 5cnf 23270 . . . . . 6 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:𝑋 𝐾)
76adantr 480 . . . . 5 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆𝑋) → 𝐹:𝑋 𝐾)
83, 7fssdm 6756 . . . 4 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆𝑋) → (𝐹 “ (𝐹𝑆)) ⊆ 𝑋)
9 simpr 484 . . . . . . 7 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆𝑋) → 𝑆𝑋)
107fdmd 6747 . . . . . . 7 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆𝑋) → dom 𝐹 = 𝑋)
119, 10sseqtrrd 4037 . . . . . 6 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆𝑋) → 𝑆 ⊆ dom 𝐹)
12 sseqin2 4231 . . . . . 6 (𝑆 ⊆ dom 𝐹 ↔ (dom 𝐹𝑆) = 𝑆)
1311, 12sylib 218 . . . . 5 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆𝑋) → (dom 𝐹𝑆) = 𝑆)
14 dminss 6175 . . . . 5 (dom 𝐹𝑆) ⊆ (𝐹 “ (𝐹𝑆))
1513, 14eqsstrrdi 4051 . . . 4 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆𝑋) → 𝑆 ⊆ (𝐹 “ (𝐹𝑆)))
164clsss 23078 . . . 4 ((𝐽 ∈ Top ∧ (𝐹 “ (𝐹𝑆)) ⊆ 𝑋𝑆 ⊆ (𝐹 “ (𝐹𝑆))) → ((cls‘𝐽)‘𝑆) ⊆ ((cls‘𝐽)‘(𝐹 “ (𝐹𝑆))))
172, 8, 15, 16syl3anc 1370 . . 3 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆𝑋) → ((cls‘𝐽)‘𝑆) ⊆ ((cls‘𝐽)‘(𝐹 “ (𝐹𝑆))))
18 imassrn 6091 . . . . 5 (𝐹𝑆) ⊆ ran 𝐹
197frnd 6745 . . . . 5 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆𝑋) → ran 𝐹 𝐾)
2018, 19sstrid 4007 . . . 4 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆𝑋) → (𝐹𝑆) ⊆ 𝐾)
215cncls2i 23294 . . . 4 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐹𝑆) ⊆ 𝐾) → ((cls‘𝐽)‘(𝐹 “ (𝐹𝑆))) ⊆ (𝐹 “ ((cls‘𝐾)‘(𝐹𝑆))))
2220, 21syldan 591 . . 3 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆𝑋) → ((cls‘𝐽)‘(𝐹 “ (𝐹𝑆))) ⊆ (𝐹 “ ((cls‘𝐾)‘(𝐹𝑆))))
2317, 22sstrd 4006 . 2 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆𝑋) → ((cls‘𝐽)‘𝑆) ⊆ (𝐹 “ ((cls‘𝐾)‘(𝐹𝑆))))
247ffund 6741 . . 3 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆𝑋) → Fun 𝐹)
254clsss3 23083 . . . . 5 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((cls‘𝐽)‘𝑆) ⊆ 𝑋)
261, 25sylan 580 . . . 4 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆𝑋) → ((cls‘𝐽)‘𝑆) ⊆ 𝑋)
2726, 10sseqtrrd 4037 . . 3 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆𝑋) → ((cls‘𝐽)‘𝑆) ⊆ dom 𝐹)
28 funimass3 7074 . . 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 584 . 2 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆𝑋) → ((𝐹 “ ((cls‘𝐽)‘𝑆)) ⊆ ((cls‘𝐾)‘(𝐹𝑆)) ↔ ((cls‘𝐽)‘𝑆) ⊆ (𝐹 “ ((cls‘𝐾)‘(𝐹𝑆)))))
3023, 29mpbird 257 1 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆𝑋) → (𝐹 “ ((cls‘𝐽)‘𝑆)) ⊆ ((cls‘𝐾)‘(𝐹𝑆)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  cin 3962  wss 3963   cuni 4912  ccnv 5688  dom cdm 5689  ran crn 5690  cima 5692  Fun wfun 6557  wf 6559  cfv 6563  (class class class)co 7431  Topctop 22915  clsccl 23042   Cn ccn 23248
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8867  df-top 22916  df-topon 22933  df-cld 23043  df-cls 23045  df-cn 23251
This theorem is referenced by:  cncls  23298  hmeocls  23792  clsnsg  24134
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