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Theorem indv 32502
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 32501 . 2 𝟭 = (𝑜 ∈ V ↦ (𝑎 ∈ 𝒫 𝑜 ↦ (𝑥𝑜 ↦ if(𝑥𝑎, 1, 0))))
2 pweq 4573 . . 3 (𝑜 = 𝑂 → 𝒫 𝑜 = 𝒫 𝑂)
3 mpteq1 5197 . . 3 (𝑜 = 𝑂 → (𝑥𝑜 ↦ if(𝑥𝑎, 1, 0)) = (𝑥𝑂 ↦ if(𝑥𝑎, 1, 0)))
42, 3mpteq12dv 5195 . 2 (𝑜 = 𝑂 → (𝑎 ∈ 𝒫 𝑜 ↦ (𝑥𝑜 ↦ if(𝑥𝑎, 1, 0))) = (𝑎 ∈ 𝒫 𝑂 ↦ (𝑥𝑂 ↦ if(𝑥𝑎, 1, 0))))
5 elex 3462 . 2 (𝑂𝑉𝑂 ∈ V)
6 pwexg 5332 . . 3 (𝑂 ∈ V → 𝒫 𝑂 ∈ V)
7 mptexg 7168 . . 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 6969 1 (𝑂𝑉 → (𝟭‘𝑂) = (𝑎 ∈ 𝒫 𝑂 ↦ (𝑥𝑂 ↦ if(𝑥𝑎, 1, 0))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541  wcel 2106  Vcvv 3444  ifcif 4485  𝒫 cpw 4559  cmpt 5187  cfv 6494  0cc0 11048  1c1 11049  𝟭cind 32500
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5241  ax-sep 5255  ax-nul 5262  ax-pow 5319  ax-pr 5383
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2888  df-ne 2943  df-ral 3064  df-rex 3073  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3739  df-csb 3855  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4865  df-iun 4955  df-br 5105  df-opab 5167  df-mpt 5188  df-id 5530  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6446  df-fun 6496  df-fn 6497  df-f 6498  df-f1 6499  df-fo 6500  df-f1o 6501  df-fv 6502  df-ind 32501
This theorem is referenced by:  indval  32503  indf1o  32514
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