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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 11264 | . . 3 ⊢ (𝑂 ∈ 𝑉 → 0 ∈ ℝ) | |
| 3 | 1red 11262 | . . 3 ⊢ (𝑂 ∈ 𝑉 → 1 ∈ ℝ) | |
| 4 | 0ne1 12337 | . . . 4 ⊢ 0 ≠ 1 | |
| 5 | 4 | a1i 11 | . . 3 ⊢ (𝑂 ∈ 𝑉 → 0 ≠ 1) |
| 6 | eqid 2737 | . . 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 9117 | . 2 ⊢ (𝑂 ∈ 𝑉 → (𝑎 ∈ 𝒫 𝑂 ↦ (𝑥 ∈ 𝑂 ↦ if(𝑥 ∈ 𝑎, 1, 0))):𝒫 𝑂–1-1-onto→({0, 1} ↑m 𝑂)) |
| 8 | indv 32837 | . . 3 ⊢ (𝑂 ∈ 𝑉 → (𝟭‘𝑂) = (𝑎 ∈ 𝒫 𝑂 ↦ (𝑥 ∈ 𝑂 ↦ if(𝑥 ∈ 𝑎, 1, 0)))) | |
| 9 | 8 | f1oeq1d 6843 | . 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 257 | 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 2108 ≠ wne 2940 ifcif 4525 𝒫 cpw 4600 {cpr 4628 ↦ cmpt 5225 –1-1-onto→wf1o 6560 ‘cfv 6561 (class class class)co 7431 ↑m cmap 8866 ℝcr 11154 0cc0 11155 1c1 11156 𝟭cind 32835 |
| 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-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-i2m1 11223 ax-1ne0 11224 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 |
| 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-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 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-iun 4993 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-res 5697 df-ima 5698 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-ov 7434 df-oprab 7435 df-mpo 7436 df-map 8868 df-ind 32836 |
| This theorem is referenced by: indf1ofs 32851 eulerpartgbij 34374 |
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