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Theorem cnclsi 21872
 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 21840 . . . . 5 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top)
21adantr 483 . . . 4 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆𝑋) → 𝐽 ∈ Top)
3 cnvimass 5942 . . . . 5 (𝐹 “ (𝐹𝑆)) ⊆ dom 𝐹
4 cnclsi.1 . . . . . . 7 𝑋 = 𝐽
5 eqid 2819 . . . . . . 7 𝐾 = 𝐾
64, 5cnf 21846 . . . . . 6 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:𝑋 𝐾)
76adantr 483 . . . . 5 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆𝑋) → 𝐹:𝑋 𝐾)
83, 7fssdm 6523 . . . 4 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆𝑋) → (𝐹 “ (𝐹𝑆)) ⊆ 𝑋)
9 simpr 487 . . . . . . 7 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆𝑋) → 𝑆𝑋)
107fdmd 6516 . . . . . . 7 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆𝑋) → dom 𝐹 = 𝑋)
119, 10sseqtrrd 4006 . . . . . 6 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆𝑋) → 𝑆 ⊆ dom 𝐹)
12 sseqin2 4190 . . . . . 6 (𝑆 ⊆ dom 𝐹 ↔ (dom 𝐹𝑆) = 𝑆)
1311, 12sylib 220 . . . . 5 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆𝑋) → (dom 𝐹𝑆) = 𝑆)
14 dminss 6003 . . . . 5 (dom 𝐹𝑆) ⊆ (𝐹 “ (𝐹𝑆))
1513, 14eqsstrrdi 4020 . . . 4 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆𝑋) → 𝑆 ⊆ (𝐹 “ (𝐹𝑆)))
164clsss 21654 . . . 4 ((𝐽 ∈ Top ∧ (𝐹 “ (𝐹𝑆)) ⊆ 𝑋𝑆 ⊆ (𝐹 “ (𝐹𝑆))) → ((cls‘𝐽)‘𝑆) ⊆ ((cls‘𝐽)‘(𝐹 “ (𝐹𝑆))))
172, 8, 15, 16syl3anc 1366 . . 3 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆𝑋) → ((cls‘𝐽)‘𝑆) ⊆ ((cls‘𝐽)‘(𝐹 “ (𝐹𝑆))))
18 imassrn 5933 . . . . 5 (𝐹𝑆) ⊆ ran 𝐹
197frnd 6514 . . . . 5 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆𝑋) → ran 𝐹 𝐾)
2018, 19sstrid 3976 . . . 4 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆𝑋) → (𝐹𝑆) ⊆ 𝐾)
215cncls2i 21870 . . . 4 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐹𝑆) ⊆ 𝐾) → ((cls‘𝐽)‘(𝐹 “ (𝐹𝑆))) ⊆ (𝐹 “ ((cls‘𝐾)‘(𝐹𝑆))))
2220, 21syldan 593 . . 3 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆𝑋) → ((cls‘𝐽)‘(𝐹 “ (𝐹𝑆))) ⊆ (𝐹 “ ((cls‘𝐾)‘(𝐹𝑆))))
2317, 22sstrd 3975 . 2 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆𝑋) → ((cls‘𝐽)‘𝑆) ⊆ (𝐹 “ ((cls‘𝐾)‘(𝐹𝑆))))
247ffund 6511 . . 3 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆𝑋) → Fun 𝐹)
254clsss3 21659 . . . . 5 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((cls‘𝐽)‘𝑆) ⊆ 𝑋)
261, 25sylan 582 . . . 4 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆𝑋) → ((cls‘𝐽)‘𝑆) ⊆ 𝑋)
2726, 10sseqtrrd 4006 . . 3 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆𝑋) → ((cls‘𝐽)‘𝑆) ⊆ dom 𝐹)
28 funimass3 6817 . . 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 586 . 2 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆𝑋) → ((𝐹 “ ((cls‘𝐽)‘𝑆)) ⊆ ((cls‘𝐾)‘(𝐹𝑆)) ↔ ((cls‘𝐽)‘𝑆) ⊆ (𝐹 “ ((cls‘𝐾)‘(𝐹𝑆)))))
3023, 29mpbird 259 1 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆𝑋) → (𝐹 “ ((cls‘𝐽)‘𝑆)) ⊆ ((cls‘𝐾)‘(𝐹𝑆)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1531   ∈ wcel 2108   ∩ cin 3933   ⊆ wss 3934  ∪ cuni 4830  ◡ccnv 5547  dom cdm 5548  ran crn 5549   “ cima 5551  Fun wfun 6342  ⟶wf 6344  ‘cfv 6348  (class class class)co 7148  Topctop 21493  clsccl 21618   Cn ccn 21824 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-reu 3143  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-int 4868  df-iun 4912  df-iin 4913  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7151  df-oprab 7152  df-mpo 7153  df-map 8400  df-top 21494  df-topon 21511  df-cld 21619  df-cls 21621  df-cn 21827 This theorem is referenced by:  cncls  21874  hmeocls  22368  clsnsg  22710
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