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Theorem indval 31500
 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 31499 . . 3 (𝑂𝑉 → (𝟭‘𝑂) = (𝑎 ∈ 𝒫 𝑂 ↦ (𝑥𝑂 ↦ if(𝑥𝑎, 1, 0))))
21adantr 484 . 2 ((𝑂𝑉𝐴𝑂) → (𝟭‘𝑂) = (𝑎 ∈ 𝒫 𝑂 ↦ (𝑥𝑂 ↦ if(𝑥𝑎, 1, 0))))
3 eleq2 2840 . . . . 5 (𝑎 = 𝐴 → (𝑥𝑎𝑥𝐴))
43ifbid 4443 . . . 4 (𝑎 = 𝐴 → if(𝑥𝑎, 1, 0) = if(𝑥𝐴, 1, 0))
54mpteq2dv 5128 . . 3 (𝑎 = 𝐴 → (𝑥𝑂 ↦ if(𝑥𝑎, 1, 0)) = (𝑥𝑂 ↦ if(𝑥𝐴, 1, 0)))
65adantl 485 . 2 (((𝑂𝑉𝐴𝑂) ∧ 𝑎 = 𝐴) → (𝑥𝑂 ↦ if(𝑥𝑎, 1, 0)) = (𝑥𝑂 ↦ if(𝑥𝐴, 1, 0)))
7 ssexg 5193 . . . 4 ((𝐴𝑂𝑂𝑉) → 𝐴 ∈ V)
87ancoms 462 . . 3 ((𝑂𝑉𝐴𝑂) → 𝐴 ∈ V)
9 simpr 488 . . 3 ((𝑂𝑉𝐴𝑂) → 𝐴𝑂)
108, 9elpwd 4502 . 2 ((𝑂𝑉𝐴𝑂) → 𝐴 ∈ 𝒫 𝑂)
11 mptexg 6975 . . 3 (𝑂𝑉 → (𝑥𝑂 ↦ if(𝑥𝐴, 1, 0)) ∈ V)
1211adantr 484 . 2 ((𝑂𝑉𝐴𝑂) → (𝑥𝑂 ↦ if(𝑥𝐴, 1, 0)) ∈ V)
132, 6, 10, 12fvmptd 6766 1 ((𝑂𝑉𝐴𝑂) → ((𝟭‘𝑂)‘𝐴) = (𝑥𝑂 ↦ if(𝑥𝐴, 1, 0)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  Vcvv 3409   ⊆ wss 3858  ifcif 4420  𝒫 cpw 4494   ↦ cmpt 5112  ‘cfv 6335  0cc0 10575  1c1 10576  𝟭cind 31497 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 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5156  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-iun 4885  df-br 5033  df-opab 5095  df-mpt 5113  df-id 5430  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343  df-ind 31498 This theorem is referenced by:  indval2  31501  indf  31502  indfval  31503
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