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Theorem indval 33006
Description: Value of the indicator function generator for a set 𝐴 and a domain 𝑂. (Contributed by Thierry Arnoux, 2-Feb-2017.)
Assertion
Ref Expression
indval ((𝑂𝑉𝐴𝑂) → ((𝟭‘𝑂)‘𝐴) = (𝑥𝑂 ↦ if(𝑥𝐴, 1, 0)))
Distinct variable groups:   𝑥,𝑂   𝑥,𝐴
Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem indval
Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 indv 33005 . . 3 (𝑂𝑉 → (𝟭‘𝑂) = (𝑎 ∈ 𝒫 𝑂 ↦ (𝑥𝑂 ↦ if(𝑥𝑎, 1, 0))))
21adantr 481 . 2 ((𝑂𝑉𝐴𝑂) → (𝟭‘𝑂) = (𝑎 ∈ 𝒫 𝑂 ↦ (𝑥𝑂 ↦ if(𝑥𝑎, 1, 0))))
3 eleq2 2822 . . . . 5 (𝑎 = 𝐴 → (𝑥𝑎𝑥𝐴))
43ifbid 4551 . . . 4 (𝑎 = 𝐴 → if(𝑥𝑎, 1, 0) = if(𝑥𝐴, 1, 0))
54mpteq2dv 5250 . . 3 (𝑎 = 𝐴 → (𝑥𝑂 ↦ if(𝑥𝑎, 1, 0)) = (𝑥𝑂 ↦ if(𝑥𝐴, 1, 0)))
65adantl 482 . 2 (((𝑂𝑉𝐴𝑂) ∧ 𝑎 = 𝐴) → (𝑥𝑂 ↦ if(𝑥𝑎, 1, 0)) = (𝑥𝑂 ↦ if(𝑥𝐴, 1, 0)))
7 ssexg 5323 . . . 4 ((𝐴𝑂𝑂𝑉) → 𝐴 ∈ V)
87ancoms 459 . . 3 ((𝑂𝑉𝐴𝑂) → 𝐴 ∈ V)
9 simpr 485 . . 3 ((𝑂𝑉𝐴𝑂) → 𝐴𝑂)
108, 9elpwd 4608 . 2 ((𝑂𝑉𝐴𝑂) → 𝐴 ∈ 𝒫 𝑂)
11 mptexg 7222 . . 3 (𝑂𝑉 → (𝑥𝑂 ↦ if(𝑥𝐴, 1, 0)) ∈ V)
1211adantr 481 . 2 ((𝑂𝑉𝐴𝑂) → (𝑥𝑂 ↦ if(𝑥𝐴, 1, 0)) ∈ V)
132, 6, 10, 12fvmptd 7005 1 ((𝑂𝑉𝐴𝑂) → ((𝟭‘𝑂)‘𝐴) = (𝑥𝑂 ↦ if(𝑥𝐴, 1, 0)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  Vcvv 3474  wss 3948  ifcif 4528  𝒫 cpw 4602  cmpt 5231  cfv 6543  0cc0 11109  1c1 11110  𝟭cind 33003
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 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427
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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ind 33004
This theorem is referenced by:  indval2  33007  indf  33008  indfval  33009
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