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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 21779 | . . . 4 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top) | |
2 | cncls2i.1 | . . . . 5 ⊢ 𝑌 = ∪ 𝐾 | |
3 | 2 | clscld 21585 | . . . 4 ⊢ ((𝐾 ∈ Top ∧ 𝑆 ⊆ 𝑌) → ((cls‘𝐾)‘𝑆) ∈ (Clsd‘𝐾)) |
4 | 1, 3 | sylan 580 | . . 3 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆 ⊆ 𝑌) → ((cls‘𝐾)‘𝑆) ∈ (Clsd‘𝐾)) |
5 | cnclima 21806 | . . 3 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ ((cls‘𝐾)‘𝑆) ∈ (Clsd‘𝐾)) → (◡𝐹 “ ((cls‘𝐾)‘𝑆)) ∈ (Clsd‘𝐽)) | |
6 | 4, 5 | syldan 591 | . 2 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆 ⊆ 𝑌) → (◡𝐹 “ ((cls‘𝐾)‘𝑆)) ∈ (Clsd‘𝐽)) |
7 | 2 | sscls 21594 | . . . 4 ⊢ ((𝐾 ∈ Top ∧ 𝑆 ⊆ 𝑌) → 𝑆 ⊆ ((cls‘𝐾)‘𝑆)) |
8 | 1, 7 | sylan 580 | . . 3 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆 ⊆ 𝑌) → 𝑆 ⊆ ((cls‘𝐾)‘𝑆)) |
9 | imass2 5959 | . . 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 2821 | . . 3 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
12 | 11 | clsss2 21610 | . 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 584 | 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 396 = wceq 1528 ∈ wcel 2105 ⊆ wss 3935 ∪ cuni 4832 ◡ccnv 5548 “ cima 5552 ‘cfv 6349 (class class class)co 7145 Topctop 21431 Clsdccld 21554 clsccl 21556 Cn ccn 21762 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4833 df-int 4870 df-iun 4914 df-iin 4915 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7148 df-oprab 7149 df-mpo 7150 df-map 8398 df-top 21432 df-topon 21449 df-cld 21557 df-cls 21559 df-cn 21765 |
This theorem is referenced by: cnclsi 21810 cncls2 21811 imasncls 22230 hmeocls 22306 clssubg 22646 |
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