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| Mirrors > Home > MPE Home > Th. List > Mathboxes > indf | Structured version Visualization version GIF version | ||
| Description: An indicator function as a function with domain and codomain. (Contributed by Thierry Arnoux, 13-Aug-2017.) |
| Ref | Expression |
|---|---|
| indf | ⊢ ((𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂) → ((𝟭‘𝑂)‘𝐴):𝑂⟶{0, 1}) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | indval 32838 | . 2 ⊢ ((𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂) → ((𝟭‘𝑂)‘𝐴) = (𝑥 ∈ 𝑂 ↦ if(𝑥 ∈ 𝐴, 1, 0))) | |
| 2 | 1re 11261 | . . . . 5 ⊢ 1 ∈ ℝ | |
| 3 | 0re 11263 | . . . . 5 ⊢ 0 ∈ ℝ | |
| 4 | ifpr 4693 | . . . . 5 ⊢ ((1 ∈ ℝ ∧ 0 ∈ ℝ) → if(𝑥 ∈ 𝐴, 1, 0) ∈ {1, 0}) | |
| 5 | 2, 3, 4 | mp2an 692 | . . . 4 ⊢ if(𝑥 ∈ 𝐴, 1, 0) ∈ {1, 0} |
| 6 | prcom 4732 | . . . 4 ⊢ {1, 0} = {0, 1} | |
| 7 | 5, 6 | eleqtri 2839 | . . 3 ⊢ if(𝑥 ∈ 𝐴, 1, 0) ∈ {0, 1} |
| 8 | 7 | a1i 11 | . 2 ⊢ (((𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂) ∧ 𝑥 ∈ 𝑂) → if(𝑥 ∈ 𝐴, 1, 0) ∈ {0, 1}) |
| 9 | 1, 8 | fmpt3d 7136 | 1 ⊢ ((𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂) → ((𝟭‘𝑂)‘𝐴):𝑂⟶{0, 1}) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2108 ⊆ wss 3951 ifcif 4525 {cpr 4628 ⟶wf 6557 ‘cfv 6561 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-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-ind 32836 |
| This theorem is referenced by: indpi1 32845 indsum 32846 indsumin 32847 prodindf 32848 indpreima 32850 indf1ofs 32851 indsupp 32852 elrgspnsubrunlem1 33251 breprexpnat 34649 circlemethnat 34656 |
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