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 10715 | . . 3 ⊢ (𝑂 ∈ 𝑉 → 0 ∈ ℝ) | |
3 | 1red 10713 | . . 3 ⊢ (𝑂 ∈ 𝑉 → 1 ∈ ℝ) | |
4 | 0ne1 11780 | . . . 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 8664 | . 2 ⊢ (𝑂 ∈ 𝑉 → (𝑎 ∈ 𝒫 𝑂 ↦ (𝑥 ∈ 𝑂 ↦ if(𝑥 ∈ 𝑎, 1, 0))):𝒫 𝑂–1-1-onto→({0, 1} ↑m 𝑂)) |
8 | indv 31542 | . . 3 ⊢ (𝑂 ∈ 𝑉 → (𝟭‘𝑂) = (𝑎 ∈ 𝒫 𝑂 ↦ (𝑥 ∈ 𝑂 ↦ if(𝑥 ∈ 𝑎, 1, 0)))) | |
9 | 8 | f1oeq1d 6607 | . 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 260 | 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 2113 ≠ wne 2934 ifcif 4411 𝒫 cpw 4485 {cpr 4515 ↦ cmpt 5107 –1-1-onto→wf1o 6332 ‘cfv 6333 (class class class)co 7164 ↑m cmap 8430 ℝcr 10607 0cc0 10608 1c1 10609 𝟭cind 31540 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-rep 5151 ax-sep 5164 ax-nul 5171 ax-pow 5229 ax-pr 5293 ax-un 7473 ax-1cn 10666 ax-icn 10667 ax-addcl 10668 ax-addrcl 10669 ax-mulcl 10670 ax-mulrcl 10671 ax-i2m1 10676 ax-1ne0 10677 ax-rnegex 10679 ax-rrecex 10680 ax-cnre 10681 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-reu 3060 df-rab 3062 df-v 3399 df-sbc 3680 df-csb 3789 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-nul 4210 df-if 4412 df-pw 4487 df-sn 4514 df-pr 4516 df-op 4520 df-uni 4794 df-iun 4880 df-br 5028 df-opab 5090 df-mpt 5108 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6291 df-fun 6335 df-fn 6336 df-f 6337 df-f1 6338 df-fo 6339 df-f1o 6340 df-fv 6341 df-ov 7167 df-oprab 7168 df-mpo 7169 df-map 8432 df-ind 31541 |
This theorem is referenced by: indf1ofs 31556 eulerpartgbij 31901 |
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