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Mirrors > Home > MPE Home > Th. List > cncls2i | Structured version Visualization version GIF version |
Description: Property of the preimage of a closure. (Contributed by Mario Carneiro, 25-Aug-2015.) |
Ref | Expression |
---|---|
cncls2i.1 | ⊢ 𝑌 = ∪ 𝐾 |
Ref | Expression |
---|---|
cncls2i | ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆 ⊆ 𝑌) → ((cls‘𝐽)‘(◡𝐹 “ 𝑆)) ⊆ (◡𝐹 “ ((cls‘𝐾)‘𝑆))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cntop2 23095 | . . . 4 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top) | |
2 | cncls2i.1 | . . . . 5 ⊢ 𝑌 = ∪ 𝐾 | |
3 | 2 | clscld 22901 | . . . 4 ⊢ ((𝐾 ∈ Top ∧ 𝑆 ⊆ 𝑌) → ((cls‘𝐾)‘𝑆) ∈ (Clsd‘𝐾)) |
4 | 1, 3 | sylan 579 | . . 3 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆 ⊆ 𝑌) → ((cls‘𝐾)‘𝑆) ∈ (Clsd‘𝐾)) |
5 | cnclima 23122 | . . 3 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ ((cls‘𝐾)‘𝑆) ∈ (Clsd‘𝐾)) → (◡𝐹 “ ((cls‘𝐾)‘𝑆)) ∈ (Clsd‘𝐽)) | |
6 | 4, 5 | syldan 590 | . 2 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆 ⊆ 𝑌) → (◡𝐹 “ ((cls‘𝐾)‘𝑆)) ∈ (Clsd‘𝐽)) |
7 | 2 | sscls 22910 | . . . 4 ⊢ ((𝐾 ∈ Top ∧ 𝑆 ⊆ 𝑌) → 𝑆 ⊆ ((cls‘𝐾)‘𝑆)) |
8 | 1, 7 | sylan 579 | . . 3 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆 ⊆ 𝑌) → 𝑆 ⊆ ((cls‘𝐾)‘𝑆)) |
9 | imass2 6094 | . . 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 2726 | . . 3 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
12 | 11 | clsss2 22926 | . 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 583 | 1 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆 ⊆ 𝑌) → ((cls‘𝐽)‘(◡𝐹 “ 𝑆)) ⊆ (◡𝐹 “ ((cls‘𝐾)‘𝑆))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ⊆ wss 3943 ∪ cuni 4902 ◡ccnv 5668 “ cima 5672 ‘cfv 6536 (class class class)co 7404 Topctop 22745 Clsdccld 22870 clsccl 22872 Cn ccn 23078 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-iin 4993 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-ov 7407 df-oprab 7408 df-mpo 7409 df-map 8821 df-top 22746 df-topon 22763 df-cld 22873 df-cls 22875 df-cn 23081 |
This theorem is referenced by: cnclsi 23126 cncls2 23127 imasncls 23546 hmeocls 23622 clssubg 23963 |
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