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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 23041 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝑢 ∈ 𝐽) → (𝑋 ∖ 𝑢) ∈ (Clsd‘𝐽)) |
| 4 | 2 | cldopn 23039 | . . 3 ⊢ (𝑥 ∈ (Clsd‘𝐽) → (𝑋 ∖ 𝑥) ∈ 𝐽) |
| 5 | 4 | adantl 481 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ (Clsd‘𝐽)) → (𝑋 ∖ 𝑥) ∈ 𝐽) |
| 6 | 2 | cldss 23037 | . . . . . . 7 ⊢ (𝑥 ∈ (Clsd‘𝐽) → 𝑥 ⊆ 𝑋) |
| 7 | 6 | ad2antll 729 | . . . . . 6 ⊢ ((𝐽 ∈ Top ∧ (𝑢 ∈ 𝐽 ∧ 𝑥 ∈ (Clsd‘𝐽))) → 𝑥 ⊆ 𝑋) |
| 8 | dfss4 4269 | . . . . . 6 ⊢ (𝑥 ⊆ 𝑋 ↔ (𝑋 ∖ (𝑋 ∖ 𝑥)) = 𝑥) | |
| 9 | 7, 8 | sylib 218 | . . . . 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 4120 | . . . . 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 22913 | . . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑢 ∈ 𝐽) → 𝑢 ⊆ 𝑋) |
| 15 | 14 | adantrr 717 | . . . . . 6 ⊢ ((𝐽 ∈ Top ∧ (𝑢 ∈ 𝐽 ∧ 𝑥 ∈ (Clsd‘𝐽))) → 𝑢 ⊆ 𝑋) |
| 16 | dfss4 4269 | . . . . . 6 ⊢ (𝑢 ⊆ 𝑋 ↔ (𝑋 ∖ (𝑋 ∖ 𝑢)) = 𝑢) | |
| 17 | 15, 16 | sylib 218 | . . . . 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 4120 | . . . . 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 212 | . 2 ⊢ ((𝐽 ∈ Top ∧ (𝑢 ∈ 𝐽 ∧ 𝑥 ∈ (Clsd‘𝐽))) → (𝑢 = (𝑋 ∖ 𝑥) ↔ 𝑥 = (𝑋 ∖ 𝑢))) |
| 23 | 1, 3, 5, 22 | f1ocnv2d 7686 | 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 395 = wceq 1540 ∈ wcel 2108 ∖ cdif 3948 ⊆ wss 3951 ∪ cuni 4907 ↦ cmpt 5225 ◡ccnv 5684 –1-1-onto→wf1o 6560 ‘cfv 6561 Topctop 22899 Clsdccld 23024 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-top 22900 df-cld 23027 |
| This theorem is referenced by: opncldf3 23094 cmpfi 23416 |
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