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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 21378 | . . . 4 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top) | |
2 | cncls2i.1 | . . . . 5 ⊢ 𝑌 = ∪ 𝐾 | |
3 | 2 | clscld 21184 | . . . 4 ⊢ ((𝐾 ∈ Top ∧ 𝑆 ⊆ 𝑌) → ((cls‘𝐾)‘𝑆) ∈ (Clsd‘𝐾)) |
4 | 1, 3 | sylan 576 | . . 3 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆 ⊆ 𝑌) → ((cls‘𝐾)‘𝑆) ∈ (Clsd‘𝐾)) |
5 | cnclima 21405 | . . 3 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ ((cls‘𝐾)‘𝑆) ∈ (Clsd‘𝐾)) → (◡𝐹 “ ((cls‘𝐾)‘𝑆)) ∈ (Clsd‘𝐽)) | |
6 | 4, 5 | syldan 586 | . 2 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆 ⊆ 𝑌) → (◡𝐹 “ ((cls‘𝐾)‘𝑆)) ∈ (Clsd‘𝐽)) |
7 | 2 | sscls 21193 | . . . 4 ⊢ ((𝐾 ∈ Top ∧ 𝑆 ⊆ 𝑌) → 𝑆 ⊆ ((cls‘𝐾)‘𝑆)) |
8 | 1, 7 | sylan 576 | . . 3 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆 ⊆ 𝑌) → 𝑆 ⊆ ((cls‘𝐾)‘𝑆)) |
9 | imass2 5722 | . . 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 2803 | . . 3 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
12 | 11 | clsss2 21209 | . 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 580 | 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 385 = wceq 1653 ∈ wcel 2157 ⊆ wss 3773 ∪ cuni 4632 ◡ccnv 5315 “ cima 5319 ‘cfv 6105 (class class class)co 6882 Topctop 21030 Clsdccld 21153 clsccl 21155 Cn ccn 21361 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2379 ax-ext 2781 ax-rep 4968 ax-sep 4979 ax-nul 4987 ax-pow 5039 ax-pr 5101 ax-un 7187 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2593 df-eu 2611 df-clab 2790 df-cleq 2796 df-clel 2799 df-nfc 2934 df-ne 2976 df-ral 3098 df-rex 3099 df-reu 3100 df-rab 3102 df-v 3391 df-sbc 3638 df-csb 3733 df-dif 3776 df-un 3778 df-in 3780 df-ss 3787 df-nul 4120 df-if 4282 df-pw 4355 df-sn 4373 df-pr 4375 df-op 4379 df-uni 4633 df-int 4672 df-iun 4716 df-iin 4717 df-br 4848 df-opab 4910 df-mpt 4927 df-id 5224 df-xp 5322 df-rel 5323 df-cnv 5324 df-co 5325 df-dm 5326 df-rn 5327 df-res 5328 df-ima 5329 df-iota 6068 df-fun 6107 df-fn 6108 df-f 6109 df-f1 6110 df-fo 6111 df-f1o 6112 df-fv 6113 df-ov 6885 df-oprab 6886 df-mpt2 6887 df-map 8101 df-top 21031 df-topon 21048 df-cld 21156 df-cls 21158 df-cn 21364 |
This theorem is referenced by: cnclsi 21409 cncls2 21410 imasncls 21828 hmeocls 21904 clssubg 22244 |
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