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Mathbox for Thierry Arnoux |
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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 33994 | . 2 ⊢ ((𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂) → ((𝟭‘𝑂)‘𝐴) = (𝑥 ∈ 𝑂 ↦ if(𝑥 ∈ 𝐴, 1, 0))) | |
2 | 1re 11259 | . . . . 5 ⊢ 1 ∈ ℝ | |
3 | 0re 11261 | . . . . 5 ⊢ 0 ∈ ℝ | |
4 | ifpr 4698 | . . . . 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 4737 | . . . 4 ⊢ {1, 0} = {0, 1} | |
7 | 5, 6 | eleqtri 2837 | . . 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 2106 ⊆ wss 3963 ifcif 4531 {cpr 4633 ⟶wf 6559 ‘cfv 6563 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-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-ind 33992 |
This theorem is referenced by: indpi1 34001 indsum 34002 indsumin 34003 prodindf 34004 indpreima 34006 indf1ofs 34007 breprexpnat 34628 circlemethnat 34635 |
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