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Theorem indval2 30204
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 6052 . . . 4 (𝑥𝑂 ↦ if(𝑥𝐴, 1, 0)) = 𝑥𝑂 ({𝑥} × {if(𝑥𝐴, 1, 0)})
2 indval 30203 . . . 4 ((𝑂𝑉𝐴𝑂) → ((𝟭‘𝑂)‘𝐴) = (𝑥𝑂 ↦ if(𝑥𝐴, 1, 0)))
3 undif 4082 . . . . . . 7 (𝐴𝑂 ↔ (𝐴 ∪ (𝑂𝐴)) = 𝑂)
43biimpi 206 . . . . . 6 (𝐴𝑂 → (𝐴 ∪ (𝑂𝐴)) = 𝑂)
54adantl 481 . . . . 5 ((𝑂𝑉𝐴𝑂) → (𝐴 ∪ (𝑂𝐴)) = 𝑂)
65iuneq1d 4577 . . . 4 ((𝑂𝑉𝐴𝑂) → 𝑥 ∈ (𝐴 ∪ (𝑂𝐴))({𝑥} × {if(𝑥𝐴, 1, 0)}) = 𝑥𝑂 ({𝑥} × {if(𝑥𝐴, 1, 0)}))
71, 2, 63eqtr4a 2711 . . 3 ((𝑂𝑉𝐴𝑂) → ((𝟭‘𝑂)‘𝐴) = 𝑥 ∈ (𝐴 ∪ (𝑂𝐴))({𝑥} × {if(𝑥𝐴, 1, 0)}))
8 iunxun 4637 . . 3 𝑥 ∈ (𝐴 ∪ (𝑂𝐴))({𝑥} × {if(𝑥𝐴, 1, 0)}) = ( 𝑥𝐴 ({𝑥} × {if(𝑥𝐴, 1, 0)}) ∪ 𝑥 ∈ (𝑂𝐴)({𝑥} × {if(𝑥𝐴, 1, 0)}))
97, 8syl6eq 2701 . 2 ((𝑂𝑉𝐴𝑂) → ((𝟭‘𝑂)‘𝐴) = ( 𝑥𝐴 ({𝑥} × {if(𝑥𝐴, 1, 0)}) ∪ 𝑥 ∈ (𝑂𝐴)({𝑥} × {if(𝑥𝐴, 1, 0)})))
10 iftrue 4125 . . . . . . 7 (𝑥𝐴 → if(𝑥𝐴, 1, 0) = 1)
1110sneqd 4222 . . . . . 6 (𝑥𝐴 → {if(𝑥𝐴, 1, 0)} = {1})
1211xpeq2d 5173 . . . . 5 (𝑥𝐴 → ({𝑥} × {if(𝑥𝐴, 1, 0)}) = ({𝑥} × {1}))
1312iuneq2i 4571 . . . 4 𝑥𝐴 ({𝑥} × {if(𝑥𝐴, 1, 0)}) = 𝑥𝐴 ({𝑥} × {1})
14 iunxpconst 5209 . . . 4 𝑥𝐴 ({𝑥} × {1}) = (𝐴 × {1})
1513, 14eqtri 2673 . . 3 𝑥𝐴 ({𝑥} × {if(𝑥𝐴, 1, 0)}) = (𝐴 × {1})
16 eldifn 3766 . . . . . . 7 (𝑥 ∈ (𝑂𝐴) → ¬ 𝑥𝐴)
17 iffalse 4128 . . . . . . . 8 𝑥𝐴 → if(𝑥𝐴, 1, 0) = 0)
1817sneqd 4222 . . . . . . 7 𝑥𝐴 → {if(𝑥𝐴, 1, 0)} = {0})
1916, 18syl 17 . . . . . 6 (𝑥 ∈ (𝑂𝐴) → {if(𝑥𝐴, 1, 0)} = {0})
2019xpeq2d 5173 . . . . 5 (𝑥 ∈ (𝑂𝐴) → ({𝑥} × {if(𝑥𝐴, 1, 0)}) = ({𝑥} × {0}))
2120iuneq2i 4571 . . . 4 𝑥 ∈ (𝑂𝐴)({𝑥} × {if(𝑥𝐴, 1, 0)}) = 𝑥 ∈ (𝑂𝐴)({𝑥} × {0})
22 iunxpconst 5209 . . . 4 𝑥 ∈ (𝑂𝐴)({𝑥} × {0}) = ((𝑂𝐴) × {0})
2321, 22eqtri 2673 . . 3 𝑥 ∈ (𝑂𝐴)({𝑥} × {if(𝑥𝐴, 1, 0)}) = ((𝑂𝐴) × {0})
2415, 23uneq12i 3798 . 2 ( 𝑥𝐴 ({𝑥} × {if(𝑥𝐴, 1, 0)}) ∪ 𝑥 ∈ (𝑂𝐴)({𝑥} × {if(𝑥𝐴, 1, 0)})) = ((𝐴 × {1}) ∪ ((𝑂𝐴) × {0}))
259, 24syl6eq 2701 1 ((𝑂𝑉𝐴𝑂) → ((𝟭‘𝑂)‘𝐴) = ((𝐴 × {1}) ∪ ((𝑂𝐴) × {0})))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383   = wceq 1523  wcel 2030  cdif 3604  cun 3605  wss 3607  ifcif 4119  {csn 4210   ciun 4552  cmpt 4762   × cxp 5141  cfv 5926  0cc0 9974  1c1 9975  𝟭cind 30200
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ind 30201
This theorem is referenced by: (None)
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