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Mirrors > Home > MPE Home > Th. List > opncldf1 | Structured version Visualization version GIF version |
Description: A bijection useful for converting statements about open sets to statements about closed sets and vice versa. (Contributed by Jeff Hankins, 27-Aug-2009.) (Proof shortened by Mario Carneiro, 1-Sep-2015.) |
Ref | Expression |
---|---|
opncldf.1 | ⊢ 𝑋 = ∪ 𝐽 |
opncldf.2 | ⊢ 𝐹 = (𝑢 ∈ 𝐽 ↦ (𝑋 ∖ 𝑢)) |
Ref | Expression |
---|---|
opncldf1 | ⊢ (𝐽 ∈ Top → (𝐹:𝐽–1-1-onto→(Clsd‘𝐽) ∧ ◡𝐹 = (𝑥 ∈ (Clsd‘𝐽) ↦ (𝑋 ∖ 𝑥)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opncldf.2 | . 2 ⊢ 𝐹 = (𝑢 ∈ 𝐽 ↦ (𝑋 ∖ 𝑢)) | |
2 | opncldf.1 | . . 3 ⊢ 𝑋 = ∪ 𝐽 | |
3 | 2 | opncld 22400 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝑢 ∈ 𝐽) → (𝑋 ∖ 𝑢) ∈ (Clsd‘𝐽)) |
4 | 2 | cldopn 22398 | . . 3 ⊢ (𝑥 ∈ (Clsd‘𝐽) → (𝑋 ∖ 𝑥) ∈ 𝐽) |
5 | 4 | adantl 483 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ (Clsd‘𝐽)) → (𝑋 ∖ 𝑥) ∈ 𝐽) |
6 | 2 | cldss 22396 | . . . . . . 7 ⊢ (𝑥 ∈ (Clsd‘𝐽) → 𝑥 ⊆ 𝑋) |
7 | 6 | ad2antll 728 | . . . . . 6 ⊢ ((𝐽 ∈ Top ∧ (𝑢 ∈ 𝐽 ∧ 𝑥 ∈ (Clsd‘𝐽))) → 𝑥 ⊆ 𝑋) |
8 | dfss4 4223 | . . . . . 6 ⊢ (𝑥 ⊆ 𝑋 ↔ (𝑋 ∖ (𝑋 ∖ 𝑥)) = 𝑥) | |
9 | 7, 8 | sylib 217 | . . . . 5 ⊢ ((𝐽 ∈ Top ∧ (𝑢 ∈ 𝐽 ∧ 𝑥 ∈ (Clsd‘𝐽))) → (𝑋 ∖ (𝑋 ∖ 𝑥)) = 𝑥) |
10 | 9 | eqcomd 2743 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ (𝑢 ∈ 𝐽 ∧ 𝑥 ∈ (Clsd‘𝐽))) → 𝑥 = (𝑋 ∖ (𝑋 ∖ 𝑥))) |
11 | difeq2 4081 | . . . . 5 ⊢ (𝑢 = (𝑋 ∖ 𝑥) → (𝑋 ∖ 𝑢) = (𝑋 ∖ (𝑋 ∖ 𝑥))) | |
12 | 11 | eqeq2d 2748 | . . . 4 ⊢ (𝑢 = (𝑋 ∖ 𝑥) → (𝑥 = (𝑋 ∖ 𝑢) ↔ 𝑥 = (𝑋 ∖ (𝑋 ∖ 𝑥)))) |
13 | 10, 12 | syl5ibrcom 247 | . . 3 ⊢ ((𝐽 ∈ Top ∧ (𝑢 ∈ 𝐽 ∧ 𝑥 ∈ (Clsd‘𝐽))) → (𝑢 = (𝑋 ∖ 𝑥) → 𝑥 = (𝑋 ∖ 𝑢))) |
14 | 2 | eltopss 22272 | . . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑢 ∈ 𝐽) → 𝑢 ⊆ 𝑋) |
15 | 14 | adantrr 716 | . . . . . 6 ⊢ ((𝐽 ∈ Top ∧ (𝑢 ∈ 𝐽 ∧ 𝑥 ∈ (Clsd‘𝐽))) → 𝑢 ⊆ 𝑋) |
16 | dfss4 4223 | . . . . . 6 ⊢ (𝑢 ⊆ 𝑋 ↔ (𝑋 ∖ (𝑋 ∖ 𝑢)) = 𝑢) | |
17 | 15, 16 | sylib 217 | . . . . 5 ⊢ ((𝐽 ∈ Top ∧ (𝑢 ∈ 𝐽 ∧ 𝑥 ∈ (Clsd‘𝐽))) → (𝑋 ∖ (𝑋 ∖ 𝑢)) = 𝑢) |
18 | 17 | eqcomd 2743 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ (𝑢 ∈ 𝐽 ∧ 𝑥 ∈ (Clsd‘𝐽))) → 𝑢 = (𝑋 ∖ (𝑋 ∖ 𝑢))) |
19 | difeq2 4081 | . . . . 5 ⊢ (𝑥 = (𝑋 ∖ 𝑢) → (𝑋 ∖ 𝑥) = (𝑋 ∖ (𝑋 ∖ 𝑢))) | |
20 | 19 | eqeq2d 2748 | . . . 4 ⊢ (𝑥 = (𝑋 ∖ 𝑢) → (𝑢 = (𝑋 ∖ 𝑥) ↔ 𝑢 = (𝑋 ∖ (𝑋 ∖ 𝑢)))) |
21 | 18, 20 | syl5ibrcom 247 | . . 3 ⊢ ((𝐽 ∈ Top ∧ (𝑢 ∈ 𝐽 ∧ 𝑥 ∈ (Clsd‘𝐽))) → (𝑥 = (𝑋 ∖ 𝑢) → 𝑢 = (𝑋 ∖ 𝑥))) |
22 | 13, 21 | impbid 211 | . 2 ⊢ ((𝐽 ∈ Top ∧ (𝑢 ∈ 𝐽 ∧ 𝑥 ∈ (Clsd‘𝐽))) → (𝑢 = (𝑋 ∖ 𝑥) ↔ 𝑥 = (𝑋 ∖ 𝑢))) |
23 | 1, 3, 5, 22 | f1ocnv2d 7611 | 1 ⊢ (𝐽 ∈ Top → (𝐹:𝐽–1-1-onto→(Clsd‘𝐽) ∧ ◡𝐹 = (𝑥 ∈ (Clsd‘𝐽) ↦ (𝑋 ∖ 𝑥)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∖ cdif 3912 ⊆ wss 3915 ∪ cuni 4870 ↦ cmpt 5193 ◡ccnv 5637 –1-1-onto→wf1o 6500 ‘cfv 6501 Topctop 22258 Clsdccld 22383 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-br 5111 df-opab 5173 df-mpt 5194 df-id 5536 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-top 22259 df-cld 22386 |
This theorem is referenced by: opncldf3 22453 cmpfi 22775 |
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