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 10632 | . . 3 ⊢ (𝑂 ∈ 𝑉 → 0 ∈ ℝ) | |
3 | 1red 10630 | . . 3 ⊢ (𝑂 ∈ 𝑉 → 1 ∈ ℝ) | |
4 | 0ne1 11696 | . . . 4 ⊢ 0 ≠ 1 | |
5 | 4 | a1i 11 | . . 3 ⊢ (𝑂 ∈ 𝑉 → 0 ≠ 1) |
6 | eqid 2818 | . . 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 8610 | . 2 ⊢ (𝑂 ∈ 𝑉 → (𝑎 ∈ 𝒫 𝑂 ↦ (𝑥 ∈ 𝑂 ↦ if(𝑥 ∈ 𝑎, 1, 0))):𝒫 𝑂–1-1-onto→({0, 1} ↑m 𝑂)) |
8 | indv 31170 | . . 3 ⊢ (𝑂 ∈ 𝑉 → (𝟭‘𝑂) = (𝑎 ∈ 𝒫 𝑂 ↦ (𝑥 ∈ 𝑂 ↦ if(𝑥 ∈ 𝑎, 1, 0)))) | |
9 | f1oeq1 6597 | . . 3 ⊢ ((𝟭‘𝑂) = (𝑎 ∈ 𝒫 𝑂 ↦ (𝑥 ∈ 𝑂 ↦ if(𝑥 ∈ 𝑎, 1, 0))) → ((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 | 8, 9 | syl 17 | . 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))) |
11 | 7, 10 | mpbird 258 | 1 ⊢ (𝑂 ∈ 𝑉 → (𝟭‘𝑂):𝒫 𝑂–1-1-onto→({0, 1} ↑m 𝑂)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 ifcif 4463 𝒫 cpw 4535 {cpr 4559 ↦ cmpt 5137 –1-1-onto→wf1o 6347 ‘cfv 6348 (class class class)co 7145 ↑m cmap 8395 ℝcr 10524 0cc0 10525 1c1 10526 𝟭cind 31168 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-i2m1 10593 ax-1ne0 10594 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-map 8397 df-ind 31169 |
This theorem is referenced by: indf1ofs 31184 eulerpartgbij 31529 |
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