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Theorem indv 31345
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 31344 . 2 𝟭 = (𝑜 ∈ V ↦ (𝑎 ∈ 𝒫 𝑜 ↦ (𝑥𝑜 ↦ if(𝑥𝑎, 1, 0))))
2 pweq 4527 . . 3 (𝑜 = 𝑂 → 𝒫 𝑜 = 𝒫 𝑂)
3 mpteq1 5130 . . 3 (𝑜 = 𝑂 → (𝑥𝑜 ↦ if(𝑥𝑎, 1, 0)) = (𝑥𝑂 ↦ if(𝑥𝑎, 1, 0)))
42, 3mpteq12dv 5127 . 2 (𝑜 = 𝑂 → (𝑎 ∈ 𝒫 𝑜 ↦ (𝑥𝑜 ↦ if(𝑥𝑎, 1, 0))) = (𝑎 ∈ 𝒫 𝑂 ↦ (𝑥𝑂 ↦ if(𝑥𝑎, 1, 0))))
5 elex 3487 . 2 (𝑂𝑉𝑂 ∈ V)
6 pwexg 5256 . . 3 (𝑂 ∈ V → 𝒫 𝑂 ∈ V)
7 mptexg 6966 . . 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 6773 1 (𝑂𝑉 → (𝟭‘𝑂) = (𝑎 ∈ 𝒫 𝑂 ↦ (𝑥𝑂 ↦ if(𝑥𝑎, 1, 0))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2114  Vcvv 3469  ifcif 4439  𝒫 cpw 4511  cmpt 5122  cfv 6334  0cc0 10526  1c1 10527  𝟭cind 31343
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 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307
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 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-ind 31344
This theorem is referenced by:  indval  31346  indf1o  31357
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