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Theorem indval 33541
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 33540 . . 3 (𝑂𝑉 → (𝟭‘𝑂) = (𝑎 ∈ 𝒫 𝑂 ↦ (𝑥𝑂 ↦ if(𝑥𝑎, 1, 0))))
21adantr 480 . 2 ((𝑂𝑉𝐴𝑂) → (𝟭‘𝑂) = (𝑎 ∈ 𝒫 𝑂 ↦ (𝑥𝑂 ↦ if(𝑥𝑎, 1, 0))))
3 eleq2 2816 . . . . 5 (𝑎 = 𝐴 → (𝑥𝑎𝑥𝐴))
43ifbid 4546 . . . 4 (𝑎 = 𝐴 → if(𝑥𝑎, 1, 0) = if(𝑥𝐴, 1, 0))
54mpteq2dv 5243 . . 3 (𝑎 = 𝐴 → (𝑥𝑂 ↦ if(𝑥𝑎, 1, 0)) = (𝑥𝑂 ↦ if(𝑥𝐴, 1, 0)))
65adantl 481 . 2 (((𝑂𝑉𝐴𝑂) ∧ 𝑎 = 𝐴) → (𝑥𝑂 ↦ if(𝑥𝑎, 1, 0)) = (𝑥𝑂 ↦ if(𝑥𝐴, 1, 0)))
7 ssexg 5316 . . . 4 ((𝐴𝑂𝑂𝑉) → 𝐴 ∈ V)
87ancoms 458 . . 3 ((𝑂𝑉𝐴𝑂) → 𝐴 ∈ V)
9 simpr 484 . . 3 ((𝑂𝑉𝐴𝑂) → 𝐴𝑂)
108, 9elpwd 4603 . 2 ((𝑂𝑉𝐴𝑂) → 𝐴 ∈ 𝒫 𝑂)
11 mptexg 7217 . . 3 (𝑂𝑉 → (𝑥𝑂 ↦ if(𝑥𝐴, 1, 0)) ∈ V)
1211adantr 480 . 2 ((𝑂𝑉𝐴𝑂) → (𝑥𝑂 ↦ if(𝑥𝐴, 1, 0)) ∈ V)
132, 6, 10, 12fvmptd 6998 1 ((𝑂𝑉𝐴𝑂) → ((𝟭‘𝑂)‘𝐴) = (𝑥𝑂 ↦ if(𝑥𝐴, 1, 0)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1533  wcel 2098  Vcvv 3468  wss 3943  ifcif 4523  𝒫 cpw 4597  cmpt 5224  cfv 6536  0cc0 11109  1c1 11110  𝟭cind 33538
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ind 33539
This theorem is referenced by:  indval2  33542  indf  33543  indfval  33544
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