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Mirrors > Home > MPE Home > Th. List > opncldf2 | Structured version Visualization version GIF version |
Description: The values of the open-closed bijection. (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 |
---|---|
opncldf2 | ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝐽) → (𝐹‘𝐴) = (𝑋 ∖ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opncldf.2 | . 2 ⊢ 𝐹 = (𝑢 ∈ 𝐽 ↦ (𝑋 ∖ 𝑢)) | |
2 | difeq2 4129 | . 2 ⊢ (𝑢 = 𝐴 → (𝑋 ∖ 𝑢) = (𝑋 ∖ 𝐴)) | |
3 | simpr 484 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝐽) → 𝐴 ∈ 𝐽) | |
4 | opncldf.1 | . . 3 ⊢ 𝑋 = ∪ 𝐽 | |
5 | 4 | opncld 23056 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝐽) → (𝑋 ∖ 𝐴) ∈ (Clsd‘𝐽)) |
6 | 1, 2, 3, 5 | fvmptd3 7038 | 1 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝐽) → (𝐹‘𝐴) = (𝑋 ∖ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1536 ∈ wcel 2105 ∖ cdif 3959 ∪ cuni 4911 ↦ cmpt 5230 ‘cfv 6562 Topctop 22914 Clsdccld 23039 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-iota 6515 df-fun 6564 df-fv 6570 df-top 22915 df-cld 23042 |
This theorem is referenced by: (None) |
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