Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > indval | Structured version Visualization version GIF version |
Description: Value of the indicator function generator for a set 𝐴 and a domain 𝑂. (Contributed by Thierry Arnoux, 2-Feb-2017.) |
Ref | Expression |
---|---|
indval | ⊢ ((𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂) → ((𝟭‘𝑂)‘𝐴) = (𝑥 ∈ 𝑂 ↦ if(𝑥 ∈ 𝐴, 1, 0))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | indv 31271 | . . 3 ⊢ (𝑂 ∈ 𝑉 → (𝟭‘𝑂) = (𝑎 ∈ 𝒫 𝑂 ↦ (𝑥 ∈ 𝑂 ↦ if(𝑥 ∈ 𝑎, 1, 0)))) | |
2 | 1 | adantr 483 | . 2 ⊢ ((𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂) → (𝟭‘𝑂) = (𝑎 ∈ 𝒫 𝑂 ↦ (𝑥 ∈ 𝑂 ↦ if(𝑥 ∈ 𝑎, 1, 0)))) |
3 | eleq2 2901 | . . . . 5 ⊢ (𝑎 = 𝐴 → (𝑥 ∈ 𝑎 ↔ 𝑥 ∈ 𝐴)) | |
4 | 3 | ifbid 4488 | . . . 4 ⊢ (𝑎 = 𝐴 → if(𝑥 ∈ 𝑎, 1, 0) = if(𝑥 ∈ 𝐴, 1, 0)) |
5 | 4 | mpteq2dv 5161 | . . 3 ⊢ (𝑎 = 𝐴 → (𝑥 ∈ 𝑂 ↦ if(𝑥 ∈ 𝑎, 1, 0)) = (𝑥 ∈ 𝑂 ↦ if(𝑥 ∈ 𝐴, 1, 0))) |
6 | 5 | adantl 484 | . 2 ⊢ (((𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂) ∧ 𝑎 = 𝐴) → (𝑥 ∈ 𝑂 ↦ if(𝑥 ∈ 𝑎, 1, 0)) = (𝑥 ∈ 𝑂 ↦ if(𝑥 ∈ 𝐴, 1, 0))) |
7 | ssexg 5226 | . . . 4 ⊢ ((𝐴 ⊆ 𝑂 ∧ 𝑂 ∈ 𝑉) → 𝐴 ∈ V) | |
8 | 7 | ancoms 461 | . . 3 ⊢ ((𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂) → 𝐴 ∈ V) |
9 | simpr 487 | . . 3 ⊢ ((𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂) → 𝐴 ⊆ 𝑂) | |
10 | 8, 9 | elpwd 4546 | . 2 ⊢ ((𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂) → 𝐴 ∈ 𝒫 𝑂) |
11 | mptexg 6983 | . . 3 ⊢ (𝑂 ∈ 𝑉 → (𝑥 ∈ 𝑂 ↦ if(𝑥 ∈ 𝐴, 1, 0)) ∈ V) | |
12 | 11 | adantr 483 | . 2 ⊢ ((𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂) → (𝑥 ∈ 𝑂 ↦ if(𝑥 ∈ 𝐴, 1, 0)) ∈ V) |
13 | 2, 6, 10, 12 | fvmptd 6774 | 1 ⊢ ((𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂) → ((𝟭‘𝑂)‘𝐴) = (𝑥 ∈ 𝑂 ↦ if(𝑥 ∈ 𝐴, 1, 0))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 Vcvv 3494 ⊆ wss 3935 ifcif 4466 𝒫 cpw 4538 ↦ cmpt 5145 ‘cfv 6354 0cc0 10536 1c1 10537 𝟭cind 31269 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-ind 31270 |
This theorem is referenced by: indval2 31273 indf 31274 indfval 31275 |
Copyright terms: Public domain | W3C validator |