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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 33503 | . 2 ⊢ ((𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂) → ((𝟭‘𝑂)‘𝐴) = (𝑥 ∈ 𝑂 ↦ if(𝑥 ∈ 𝐴, 1, 0))) | |
2 | 1re 11212 | . . . . 5 ⊢ 1 ∈ ℝ | |
3 | 0re 11214 | . . . . 5 ⊢ 0 ∈ ℝ | |
4 | ifpr 4688 | . . . . 5 ⊢ ((1 ∈ ℝ ∧ 0 ∈ ℝ) → if(𝑥 ∈ 𝐴, 1, 0) ∈ {1, 0}) | |
5 | 2, 3, 4 | mp2an 689 | . . . 4 ⊢ if(𝑥 ∈ 𝐴, 1, 0) ∈ {1, 0} |
6 | prcom 4729 | . . . 4 ⊢ {1, 0} = {0, 1} | |
7 | 5, 6 | eleqtri 2823 | . . 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 7108 | 1 ⊢ ((𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂) → ((𝟭‘𝑂)‘𝐴):𝑂⟶{0, 1}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2098 ⊆ wss 3941 ifcif 4521 {cpr 4623 ⟶wf 6530 ‘cfv 6534 ℝcr 11106 0cc0 11107 1c1 11108 𝟭cind 33500 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-i2m1 11175 ax-1ne0 11176 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-id 5565 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-ov 7405 df-ind 33501 |
This theorem is referenced by: indpi1 33510 indsum 33511 indsumin 33512 prodindf 33513 indpreima 33515 indf1ofs 33516 breprexpnat 34137 circlemethnat 34144 |
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