Theorem cncls2i 23253

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) ∧ 𝑆 ⊆ 𝑌) → ((cls‘𝐽)‘(◡𝐹 “ 𝑆)) ⊆ (◡𝐹 “ ((cls‘𝐾)‘𝑆)))

Proof of Theorem cncls2i

Step	Hyp	Ref	Expression
1		cntop2 23224	. . . 4 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
2		cncls2i.1	. . . . 5 ⊢ 𝑌 = ∪ 𝐾
3	2	clscld 23030	. . . 4 ⊢ ((𝐾 ∈ Top ∧ 𝑆 ⊆ 𝑌) → ((cls‘𝐾)‘𝑆) ∈ (Clsd‘𝐾))
4	1, 3	sylan 586	. . 3 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆 ⊆ 𝑌) → ((cls‘𝐾)‘𝑆) ∈ (Clsd‘𝐾))
5		cnclima 23251	. . 3 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ ((cls‘𝐾)‘𝑆) ∈ (Clsd‘𝐾)) → (◡𝐹 “ ((cls‘𝐾)‘𝑆)) ∈ (Clsd‘𝐽))
6	4, 5	syldan 597	. 2 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆 ⊆ 𝑌) → (◡𝐹 “ ((cls‘𝐾)‘𝑆)) ∈ (Clsd‘𝐽))
7	2	sscls 23039	. . . 4 ⊢ ((𝐾 ∈ Top ∧ 𝑆 ⊆ 𝑌) → 𝑆 ⊆ ((cls‘𝐾)‘𝑆))
8	1, 7	sylan 586	. . 3 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆 ⊆ 𝑌) → 𝑆 ⊆ ((cls‘𝐾)‘𝑆))
9		imass2 6054	. . 3 ⊢ (𝑆 ⊆ ((cls‘𝐾)‘𝑆) → (◡𝐹 “ 𝑆) ⊆ (◡𝐹 “ ((cls‘𝐾)‘𝑆)))
10	8, 9	syl 17	. 2 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆 ⊆ 𝑌) → (◡𝐹 “ 𝑆) ⊆ (◡𝐹 “ ((cls‘𝐾)‘𝑆)))
11		eqid 2739	. . 3 ⊢ ∪ 𝐽 = ∪ 𝐽
12	11	clsss2 23055	. 2 ⊢ (((◡𝐹 “ ((cls‘𝐾)‘𝑆)) ∈ (Clsd‘𝐽) ∧ (◡𝐹 “ 𝑆) ⊆ (◡𝐹 “ ((cls‘𝐾)‘𝑆))) → ((cls‘𝐽)‘(◡𝐹 “ 𝑆)) ⊆ (◡𝐹 “ ((cls‘𝐾)‘𝑆)))
13	6, 10, 12	syl2anc 590	1 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆 ⊆ 𝑌) → ((cls‘𝐽)‘(◡𝐹 “ 𝑆)) ⊆ (◡𝐹 “ ((cls‘𝐾)‘𝑆)))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119   ⊆ wss 3883  ∪ cuni 4838  ^◡ccnv 5617   “ cima 5621  ‘cfv 6485  (class class class)co 7356  Topctop 22876  Clsdccld 22999  clsccl 23001   Cn ccn 23207

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-int 4878  df-iun 4923  df-iin 4924  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-ov 7359  df-oprab 7360  df-mpo 7361  df-map 8765  df-top 22877  df-topon 22894  df-cld 23002  df-cls 23004  df-cn 23210

This theorem is referenced by:  cnclsi  23255  cncls2  23256  imasncls  23675  hmeocls  23751  clssubg  24092