Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > indf1o | Structured version Visualization version GIF version |
Description: The bijection between a power set and the set of indicator functions. (Contributed by Thierry Arnoux, 14-Aug-2017.) |
Ref | Expression |
---|---|
indf1o | ⊢ (𝑂 ∈ 𝑉 → (𝟭‘𝑂):𝒫 𝑂–1-1-onto→({0, 1} ↑m 𝑂)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . . 3 ⊢ (𝑂 ∈ 𝑉 → 𝑂 ∈ 𝑉) | |
2 | 0red 10978 | . . 3 ⊢ (𝑂 ∈ 𝑉 → 0 ∈ ℝ) | |
3 | 1red 10976 | . . 3 ⊢ (𝑂 ∈ 𝑉 → 1 ∈ ℝ) | |
4 | 0ne1 12044 | . . . 4 ⊢ 0 ≠ 1 | |
5 | 4 | a1i 11 | . . 3 ⊢ (𝑂 ∈ 𝑉 → 0 ≠ 1) |
6 | eqid 2738 | . . 3 ⊢ (𝑎 ∈ 𝒫 𝑂 ↦ (𝑥 ∈ 𝑂 ↦ if(𝑥 ∈ 𝑎, 1, 0))) = (𝑎 ∈ 𝒫 𝑂 ↦ (𝑥 ∈ 𝑂 ↦ if(𝑥 ∈ 𝑎, 1, 0))) | |
7 | 1, 2, 3, 5, 6 | pw2f1o 8864 | . 2 ⊢ (𝑂 ∈ 𝑉 → (𝑎 ∈ 𝒫 𝑂 ↦ (𝑥 ∈ 𝑂 ↦ if(𝑥 ∈ 𝑎, 1, 0))):𝒫 𝑂–1-1-onto→({0, 1} ↑m 𝑂)) |
8 | indv 31980 | . . 3 ⊢ (𝑂 ∈ 𝑉 → (𝟭‘𝑂) = (𝑎 ∈ 𝒫 𝑂 ↦ (𝑥 ∈ 𝑂 ↦ if(𝑥 ∈ 𝑎, 1, 0)))) | |
9 | 8 | f1oeq1d 6711 | . 2 ⊢ (𝑂 ∈ 𝑉 → ((𝟭‘𝑂):𝒫 𝑂–1-1-onto→({0, 1} ↑m 𝑂) ↔ (𝑎 ∈ 𝒫 𝑂 ↦ (𝑥 ∈ 𝑂 ↦ if(𝑥 ∈ 𝑎, 1, 0))):𝒫 𝑂–1-1-onto→({0, 1} ↑m 𝑂))) |
10 | 7, 9 | mpbird 256 | 1 ⊢ (𝑂 ∈ 𝑉 → (𝟭‘𝑂):𝒫 𝑂–1-1-onto→({0, 1} ↑m 𝑂)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 ≠ wne 2943 ifcif 4459 𝒫 cpw 4533 {cpr 4563 ↦ cmpt 5157 –1-1-onto→wf1o 6432 ‘cfv 6433 (class class class)co 7275 ↑m cmap 8615 ℝcr 10870 0cc0 10871 1c1 10872 𝟭cind 31978 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-i2m1 10939 ax-1ne0 10940 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-ov 7278 df-oprab 7279 df-mpo 7280 df-map 8617 df-ind 31979 |
This theorem is referenced by: indf1ofs 31994 eulerpartgbij 32339 |
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