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Theorem indval 29403
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 29402 . . 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 2677 . . . . 5 (𝑎 = 𝐴 → (𝑥𝑎𝑥𝐴))
43ifbid 4058 . . . 4 (𝑎 = 𝐴 → if(𝑥𝑎, 1, 0) = if(𝑥𝐴, 1, 0))
54mpteq2dv 4673 . . 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 simpr 476 . . 3 ((𝑂𝑉𝐴𝑂) → 𝐴𝑂)
8 ssexg 4732 . . . . 5 ((𝐴𝑂𝑂𝑉) → 𝐴 ∈ V)
98ancoms 468 . . . 4 ((𝑂𝑉𝐴𝑂) → 𝐴 ∈ V)
10 elpwg 4116 . . . 4 (𝐴 ∈ V → (𝐴 ∈ 𝒫 𝑂𝐴𝑂))
119, 10syl 17 . . 3 ((𝑂𝑉𝐴𝑂) → (𝐴 ∈ 𝒫 𝑂𝐴𝑂))
127, 11mpbird 246 . 2 ((𝑂𝑉𝐴𝑂) → 𝐴 ∈ 𝒫 𝑂)
13 mptexg 6389 . . 3 (𝑂𝑉 → (𝑥𝑂 ↦ if(𝑥𝐴, 1, 0)) ∈ V)
1413adantr 480 . 2 ((𝑂𝑉𝐴𝑂) → (𝑥𝑂 ↦ if(𝑥𝐴, 1, 0)) ∈ V)
152, 6, 12, 14fvmptd 6197 1 ((𝑂𝑉𝐴𝑂) → ((𝟭‘𝑂)‘𝐴) = (𝑥𝑂 ↦ if(𝑥𝐴, 1, 0)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383   = wceq 1475  wcel 1977  Vcvv 3173  wss 3540  ifcif 4036  𝒫 cpw 4108  cmpt 4643  cfv 5804  0cc0 9815  1c1 9816  𝟭cind 29400
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ind 29401
This theorem is referenced by:  indval2  29404  indf  29405  indfval  29406
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