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Theorem cnclsi 21124
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 21092 . . . . 5 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top)
21adantr 480 . . . 4 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆𝑋) → 𝐽 ∈ Top)
3 cnvimass 5520 . . . . 5 (𝐹 “ (𝐹𝑆)) ⊆ dom 𝐹
4 cnclsi.1 . . . . . . . 8 𝑋 = 𝐽
5 eqid 2651 . . . . . . . 8 𝐾 = 𝐾
64, 5cnf 21098 . . . . . . 7 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:𝑋 𝐾)
76adantr 480 . . . . . 6 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆𝑋) → 𝐹:𝑋 𝐾)
8 fdm 6089 . . . . . 6 (𝐹:𝑋 𝐾 → dom 𝐹 = 𝑋)
97, 8syl 17 . . . . 5 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆𝑋) → dom 𝐹 = 𝑋)
103, 9syl5sseq 3686 . . . 4 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆𝑋) → (𝐹 “ (𝐹𝑆)) ⊆ 𝑋)
11 simpr 476 . . . . . . 7 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆𝑋) → 𝑆𝑋)
1211, 9sseqtr4d 3675 . . . . . 6 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆𝑋) → 𝑆 ⊆ dom 𝐹)
13 sseqin2 3850 . . . . . 6 (𝑆 ⊆ dom 𝐹 ↔ (dom 𝐹𝑆) = 𝑆)
1412, 13sylib 208 . . . . 5 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆𝑋) → (dom 𝐹𝑆) = 𝑆)
15 dminss 5582 . . . . 5 (dom 𝐹𝑆) ⊆ (𝐹 “ (𝐹𝑆))
1614, 15syl6eqssr 3689 . . . 4 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆𝑋) → 𝑆 ⊆ (𝐹 “ (𝐹𝑆)))
174clsss 20906 . . . 4 ((𝐽 ∈ Top ∧ (𝐹 “ (𝐹𝑆)) ⊆ 𝑋𝑆 ⊆ (𝐹 “ (𝐹𝑆))) → ((cls‘𝐽)‘𝑆) ⊆ ((cls‘𝐽)‘(𝐹 “ (𝐹𝑆))))
182, 10, 16, 17syl3anc 1366 . . 3 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆𝑋) → ((cls‘𝐽)‘𝑆) ⊆ ((cls‘𝐽)‘(𝐹 “ (𝐹𝑆))))
19 imassrn 5512 . . . . 5 (𝐹𝑆) ⊆ ran 𝐹
20 frn 6091 . . . . . 6 (𝐹:𝑋 𝐾 → ran 𝐹 𝐾)
217, 20syl 17 . . . . 5 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆𝑋) → ran 𝐹 𝐾)
2219, 21syl5ss 3647 . . . 4 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆𝑋) → (𝐹𝑆) ⊆ 𝐾)
235cncls2i 21122 . . . 4 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐹𝑆) ⊆ 𝐾) → ((cls‘𝐽)‘(𝐹 “ (𝐹𝑆))) ⊆ (𝐹 “ ((cls‘𝐾)‘(𝐹𝑆))))
2422, 23syldan 486 . . 3 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆𝑋) → ((cls‘𝐽)‘(𝐹 “ (𝐹𝑆))) ⊆ (𝐹 “ ((cls‘𝐾)‘(𝐹𝑆))))
2518, 24sstrd 3646 . 2 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆𝑋) → ((cls‘𝐽)‘𝑆) ⊆ (𝐹 “ ((cls‘𝐾)‘(𝐹𝑆))))
26 ffun 6086 . . . 4 (𝐹:𝑋 𝐾 → Fun 𝐹)
277, 26syl 17 . . 3 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆𝑋) → Fun 𝐹)
284clsss3 20911 . . . . 5 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((cls‘𝐽)‘𝑆) ⊆ 𝑋)
291, 28sylan 487 . . . 4 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆𝑋) → ((cls‘𝐽)‘𝑆) ⊆ 𝑋)
3029, 9sseqtr4d 3675 . . 3 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆𝑋) → ((cls‘𝐽)‘𝑆) ⊆ dom 𝐹)
31 funimass3 6373 . . 3 ((Fun 𝐹 ∧ ((cls‘𝐽)‘𝑆) ⊆ dom 𝐹) → ((𝐹 “ ((cls‘𝐽)‘𝑆)) ⊆ ((cls‘𝐾)‘(𝐹𝑆)) ↔ ((cls‘𝐽)‘𝑆) ⊆ (𝐹 “ ((cls‘𝐾)‘(𝐹𝑆)))))
3227, 30, 31syl2anc 694 . 2 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆𝑋) → ((𝐹 “ ((cls‘𝐽)‘𝑆)) ⊆ ((cls‘𝐾)‘(𝐹𝑆)) ↔ ((cls‘𝐽)‘𝑆) ⊆ (𝐹 “ ((cls‘𝐾)‘(𝐹𝑆)))))
3325, 32mpbird 247 1 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆𝑋) → (𝐹 “ ((cls‘𝐽)‘𝑆)) ⊆ ((cls‘𝐾)‘(𝐹𝑆)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  cin 3606  wss 3607   cuni 4468  ccnv 5142  dom cdm 5143  ran crn 5144  cima 5146  Fun wfun 5920  wf 5922  cfv 5926  (class class class)co 6690  Topctop 20746  clsccl 20870   Cn ccn 21076
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-iin 4555  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-map 7901  df-top 20747  df-topon 20764  df-cld 20871  df-cls 20873  df-cn 21079
This theorem is referenced by:  cncls  21126  hmeocls  21619  clsnsg  21960
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