![]() |
Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
|
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 31382 | . 2 ⊢ ((𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂) → ((𝟭‘𝑂)‘𝐴) = (𝑥 ∈ 𝑂 ↦ if(𝑥 ∈ 𝐴, 1, 0))) | |
2 | 1re 10630 | . . . . 5 ⊢ 1 ∈ ℝ | |
3 | 0re 10632 | . . . . 5 ⊢ 0 ∈ ℝ | |
4 | ifpr 4589 | . . . . 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 4628 | . . . 4 ⊢ {1, 0} = {0, 1} | |
7 | 5, 6 | eleqtri 2888 | . . 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 6857 | 1 ⊢ ((𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂) → ((𝟭‘𝑂)‘𝐴):𝑂⟶{0, 1}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∈ wcel 2111 ⊆ wss 3881 ifcif 4425 {cpr 4527 ⟶wf 6320 ‘cfv 6324 ℝcr 10525 0cc0 10526 1c1 10527 𝟭cind 31379 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-i2m1 10594 ax-1ne0 10595 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-ind 31380 |
This theorem is referenced by: indpi1 31389 indsum 31390 indsumin 31391 prodindf 31392 indpreima 31394 indf1ofs 31395 breprexpnat 32015 circlemethnat 32022 |
Copyright terms: Public domain | W3C validator |