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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 11262 | . . 3 ⊢ (𝑂 ∈ 𝑉 → 0 ∈ ℝ) | |
3 | 1red 11260 | . . 3 ⊢ (𝑂 ∈ 𝑉 → 1 ∈ ℝ) | |
4 | 0ne1 12335 | . . . 4 ⊢ 0 ≠ 1 | |
5 | 4 | a1i 11 | . . 3 ⊢ (𝑂 ∈ 𝑉 → 0 ≠ 1) |
6 | eqid 2735 | . . 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 9116 | . 2 ⊢ (𝑂 ∈ 𝑉 → (𝑎 ∈ 𝒫 𝑂 ↦ (𝑥 ∈ 𝑂 ↦ if(𝑥 ∈ 𝑎, 1, 0))):𝒫 𝑂–1-1-onto→({0, 1} ↑m 𝑂)) |
8 | indv 33993 | . . 3 ⊢ (𝑂 ∈ 𝑉 → (𝟭‘𝑂) = (𝑎 ∈ 𝒫 𝑂 ↦ (𝑥 ∈ 𝑂 ↦ if(𝑥 ∈ 𝑎, 1, 0)))) | |
9 | 8 | f1oeq1d 6844 | . 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 2106 ≠ wne 2938 ifcif 4531 𝒫 cpw 4605 {cpr 4633 ↦ cmpt 5231 –1-1-onto→wf1o 6562 ‘cfv 6563 (class class class)co 7431 ↑m cmap 8865 ℝcr 11152 0cc0 11153 1c1 11154 𝟭cind 33991 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-i2m1 11221 ax-1ne0 11222 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-map 8867 df-ind 33992 |
This theorem is referenced by: indf1ofs 34007 eulerpartgbij 34354 |
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