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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 33626 | . 2 ⊢ ((𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂) → ((𝟭‘𝑂)‘𝐴) = (𝑥 ∈ 𝑂 ↦ if(𝑥 ∈ 𝐴, 1, 0))) | |
2 | 1re 11238 | . . . . 5 ⊢ 1 ∈ ℝ | |
3 | 0re 11240 | . . . . 5 ⊢ 0 ∈ ℝ | |
4 | ifpr 4691 | . . . . 5 ⊢ ((1 ∈ ℝ ∧ 0 ∈ ℝ) → if(𝑥 ∈ 𝐴, 1, 0) ∈ {1, 0}) | |
5 | 2, 3, 4 | mp2an 691 | . . . 4 ⊢ if(𝑥 ∈ 𝐴, 1, 0) ∈ {1, 0} |
6 | prcom 4732 | . . . 4 ⊢ {1, 0} = {0, 1} | |
7 | 5, 6 | eleqtri 2827 | . . 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 7120 | 1 ⊢ ((𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂) → ((𝟭‘𝑂)‘𝐴):𝑂⟶{0, 1}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2099 ⊆ wss 3945 ifcif 4524 {cpr 4626 ⟶wf 6538 ‘cfv 6542 ℝcr 11131 0cc0 11132 1c1 11133 𝟭cind 33623 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-i2m1 11200 ax-1ne0 11201 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-ral 3058 df-rex 3067 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-ov 7417 df-ind 33624 |
This theorem is referenced by: indpi1 33633 indsum 33634 indsumin 33635 prodindf 33636 indpreima 33638 indf1ofs 33639 breprexpnat 34260 circlemethnat 34267 |
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