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Mirrors > Home > ILE Home > Th. List > iscld | GIF version |
Description: The predicate "the class 𝑆 is a closed set". (Contributed by NM, 2-Oct-2006.) (Revised by Mario Carneiro, 11-Nov-2013.) |
Ref | Expression |
---|---|
iscld.1 | ⊢ 𝑋 = ∪ 𝐽 |
Ref | Expression |
---|---|
iscld | ⊢ (𝐽 ∈ Top → (𝑆 ∈ (Clsd‘𝐽) ↔ (𝑆 ⊆ 𝑋 ∧ (𝑋 ∖ 𝑆) ∈ 𝐽))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iscld.1 | . . . . 5 ⊢ 𝑋 = ∪ 𝐽 | |
2 | 1 | cldval 12646 | . . . 4 ⊢ (𝐽 ∈ Top → (Clsd‘𝐽) = {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ∈ 𝐽}) |
3 | 2 | eleq2d 2234 | . . 3 ⊢ (𝐽 ∈ Top → (𝑆 ∈ (Clsd‘𝐽) ↔ 𝑆 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ∈ 𝐽})) |
4 | difeq2 3229 | . . . . 5 ⊢ (𝑥 = 𝑆 → (𝑋 ∖ 𝑥) = (𝑋 ∖ 𝑆)) | |
5 | 4 | eleq1d 2233 | . . . 4 ⊢ (𝑥 = 𝑆 → ((𝑋 ∖ 𝑥) ∈ 𝐽 ↔ (𝑋 ∖ 𝑆) ∈ 𝐽)) |
6 | 5 | elrab 2877 | . . 3 ⊢ (𝑆 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ∈ 𝐽} ↔ (𝑆 ∈ 𝒫 𝑋 ∧ (𝑋 ∖ 𝑆) ∈ 𝐽)) |
7 | 3, 6 | bitrdi 195 | . 2 ⊢ (𝐽 ∈ Top → (𝑆 ∈ (Clsd‘𝐽) ↔ (𝑆 ∈ 𝒫 𝑋 ∧ (𝑋 ∖ 𝑆) ∈ 𝐽))) |
8 | 1 | topopn 12553 | . . . 4 ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽) |
9 | elpw2g 4129 | . . . 4 ⊢ (𝑋 ∈ 𝐽 → (𝑆 ∈ 𝒫 𝑋 ↔ 𝑆 ⊆ 𝑋)) | |
10 | 8, 9 | syl 14 | . . 3 ⊢ (𝐽 ∈ Top → (𝑆 ∈ 𝒫 𝑋 ↔ 𝑆 ⊆ 𝑋)) |
11 | 10 | anbi1d 461 | . 2 ⊢ (𝐽 ∈ Top → ((𝑆 ∈ 𝒫 𝑋 ∧ (𝑋 ∖ 𝑆) ∈ 𝐽) ↔ (𝑆 ⊆ 𝑋 ∧ (𝑋 ∖ 𝑆) ∈ 𝐽))) |
12 | 7, 11 | bitrd 187 | 1 ⊢ (𝐽 ∈ Top → (𝑆 ∈ (Clsd‘𝐽) ↔ (𝑆 ⊆ 𝑋 ∧ (𝑋 ∖ 𝑆) ∈ 𝐽))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1342 ∈ wcel 2135 {crab 2446 ∖ cdif 3108 ⊆ wss 3111 𝒫 cpw 3553 ∪ cuni 3783 ‘cfv 5182 Topctop 12542 Clsdccld 12639 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-rab 2451 df-v 2723 df-sbc 2947 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-iota 5147 df-fun 5184 df-fv 5190 df-top 12543 df-cld 12642 |
This theorem is referenced by: iscld2 12651 cldss 12652 cldopn 12654 topcld 12656 discld 12683 |
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