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Theorem indval2 29404
 Description: Alternate value of the indicator function generator. (Contributed by Thierry Arnoux, 2-Feb-2017.)
Assertion
Ref Expression
indval2 ((𝑂𝑉𝐴𝑂) → ((𝟭‘𝑂)‘𝐴) = ((𝐴 × {1}) ∪ ((𝑂𝐴) × {0})))

Proof of Theorem indval2
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 dfmpt3 5927 . . . 4 (𝑥𝑂 ↦ if(𝑥𝐴, 1, 0)) = 𝑥𝑂 ({𝑥} × {if(𝑥𝐴, 1, 0)})
2 indval 29403 . . . 4 ((𝑂𝑉𝐴𝑂) → ((𝟭‘𝑂)‘𝐴) = (𝑥𝑂 ↦ if(𝑥𝐴, 1, 0)))
3 undif 4001 . . . . . . 7 (𝐴𝑂 ↔ (𝐴 ∪ (𝑂𝐴)) = 𝑂)
43biimpi 205 . . . . . 6 (𝐴𝑂 → (𝐴 ∪ (𝑂𝐴)) = 𝑂)
54adantl 481 . . . . 5 ((𝑂𝑉𝐴𝑂) → (𝐴 ∪ (𝑂𝐴)) = 𝑂)
65iuneq1d 4481 . . . 4 ((𝑂𝑉𝐴𝑂) → 𝑥 ∈ (𝐴 ∪ (𝑂𝐴))({𝑥} × {if(𝑥𝐴, 1, 0)}) = 𝑥𝑂 ({𝑥} × {if(𝑥𝐴, 1, 0)}))
71, 2, 63eqtr4a 2670 . . 3 ((𝑂𝑉𝐴𝑂) → ((𝟭‘𝑂)‘𝐴) = 𝑥 ∈ (𝐴 ∪ (𝑂𝐴))({𝑥} × {if(𝑥𝐴, 1, 0)}))
8 iunxun 4541 . . 3 𝑥 ∈ (𝐴 ∪ (𝑂𝐴))({𝑥} × {if(𝑥𝐴, 1, 0)}) = ( 𝑥𝐴 ({𝑥} × {if(𝑥𝐴, 1, 0)}) ∪ 𝑥 ∈ (𝑂𝐴)({𝑥} × {if(𝑥𝐴, 1, 0)}))
97, 8syl6eq 2660 . 2 ((𝑂𝑉𝐴𝑂) → ((𝟭‘𝑂)‘𝐴) = ( 𝑥𝐴 ({𝑥} × {if(𝑥𝐴, 1, 0)}) ∪ 𝑥 ∈ (𝑂𝐴)({𝑥} × {if(𝑥𝐴, 1, 0)})))
10 iftrue 4042 . . . . . . 7 (𝑥𝐴 → if(𝑥𝐴, 1, 0) = 1)
1110sneqd 4137 . . . . . 6 (𝑥𝐴 → {if(𝑥𝐴, 1, 0)} = {1})
1211xpeq2d 5063 . . . . 5 (𝑥𝐴 → ({𝑥} × {if(𝑥𝐴, 1, 0)}) = ({𝑥} × {1}))
1312iuneq2i 4475 . . . 4 𝑥𝐴 ({𝑥} × {if(𝑥𝐴, 1, 0)}) = 𝑥𝐴 ({𝑥} × {1})
14 iunxpconst 5098 . . . 4 𝑥𝐴 ({𝑥} × {1}) = (𝐴 × {1})
1513, 14eqtri 2632 . . 3 𝑥𝐴 ({𝑥} × {if(𝑥𝐴, 1, 0)}) = (𝐴 × {1})
16 eldifn 3695 . . . . . . 7 (𝑥 ∈ (𝑂𝐴) → ¬ 𝑥𝐴)
17 iffalse 4045 . . . . . . . 8 𝑥𝐴 → if(𝑥𝐴, 1, 0) = 0)
1817sneqd 4137 . . . . . . 7 𝑥𝐴 → {if(𝑥𝐴, 1, 0)} = {0})
1916, 18syl 17 . . . . . 6 (𝑥 ∈ (𝑂𝐴) → {if(𝑥𝐴, 1, 0)} = {0})
2019xpeq2d 5063 . . . . 5 (𝑥 ∈ (𝑂𝐴) → ({𝑥} × {if(𝑥𝐴, 1, 0)}) = ({𝑥} × {0}))
2120iuneq2i 4475 . . . 4 𝑥 ∈ (𝑂𝐴)({𝑥} × {if(𝑥𝐴, 1, 0)}) = 𝑥 ∈ (𝑂𝐴)({𝑥} × {0})
22 iunxpconst 5098 . . . 4 𝑥 ∈ (𝑂𝐴)({𝑥} × {0}) = ((𝑂𝐴) × {0})
2321, 22eqtri 2632 . . 3 𝑥 ∈ (𝑂𝐴)({𝑥} × {if(𝑥𝐴, 1, 0)}) = ((𝑂𝐴) × {0})
2415, 23uneq12i 3727 . 2 ( 𝑥𝐴 ({𝑥} × {if(𝑥𝐴, 1, 0)}) ∪ 𝑥 ∈ (𝑂𝐴)({𝑥} × {if(𝑥𝐴, 1, 0)})) = ((𝐴 × {1}) ∪ ((𝑂𝐴) × {0}))
259, 24syl6eq 2660 1 ((𝑂𝑉𝐴𝑂) → ((𝟭‘𝑂)‘𝐴) = ((𝐴 × {1}) ∪ ((𝑂𝐴) × {0})))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ∖ cdif 3537   ∪ cun 3538   ⊆ wss 3540  ifcif 4036  {csn 4125  ∪ ciun 4455   ↦ cmpt 4643   × cxp 5036  ‘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: (None)
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