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Theorem indv 31170
Description: Value of the indicator function generator with domain 𝑂. (Contributed by Thierry Arnoux, 23-Aug-2017.)
Assertion
Ref Expression
indv (𝑂𝑉 → (𝟭‘𝑂) = (𝑎 ∈ 𝒫 𝑂 ↦ (𝑥𝑂 ↦ if(𝑥𝑎, 1, 0))))
Distinct variable groups:   𝑥,𝑎,𝑂   𝑉,𝑎
Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem indv
Dummy variable 𝑜 is distinct from all other variables.
StepHypRef Expression
1 df-ind 31169 . 2 𝟭 = (𝑜 ∈ V ↦ (𝑎 ∈ 𝒫 𝑜 ↦ (𝑥𝑜 ↦ if(𝑥𝑎, 1, 0))))
2 pweq 4538 . . 3 (𝑜 = 𝑂 → 𝒫 𝑜 = 𝒫 𝑂)
3 mpteq1 5145 . . 3 (𝑜 = 𝑂 → (𝑥𝑜 ↦ if(𝑥𝑎, 1, 0)) = (𝑥𝑂 ↦ if(𝑥𝑎, 1, 0)))
42, 3mpteq12dv 5142 . 2 (𝑜 = 𝑂 → (𝑎 ∈ 𝒫 𝑜 ↦ (𝑥𝑜 ↦ if(𝑥𝑎, 1, 0))) = (𝑎 ∈ 𝒫 𝑂 ↦ (𝑥𝑂 ↦ if(𝑥𝑎, 1, 0))))
5 elex 3510 . 2 (𝑂𝑉𝑂 ∈ V)
6 pwexg 5270 . . 3 (𝑂 ∈ V → 𝒫 𝑂 ∈ V)
7 mptexg 6975 . . 3 (𝒫 𝑂 ∈ V → (𝑎 ∈ 𝒫 𝑂 ↦ (𝑥𝑂 ↦ if(𝑥𝑎, 1, 0))) ∈ V)
85, 6, 73syl 18 . 2 (𝑂𝑉 → (𝑎 ∈ 𝒫 𝑂 ↦ (𝑥𝑂 ↦ if(𝑥𝑎, 1, 0))) ∈ V)
91, 4, 5, 8fvmptd3 6783 1 (𝑂𝑉 → (𝟭‘𝑂) = (𝑎 ∈ 𝒫 𝑂 ↦ (𝑥𝑂 ↦ if(𝑥𝑎, 1, 0))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1528  wcel 2105  Vcvv 3492  ifcif 4463  𝒫 cpw 4535  cmpt 5137  cfv 6348  0cc0 10525  1c1 10526  𝟭cind 31168
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ind 31169
This theorem is referenced by:  indval  31171  indf1o  31182
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