Theorem indval2 33995

Description: Alternate value of the indicator function generator. (Contributed by Thierry Arnoux, 2-Feb-2017.)

Assertion

Ref	Expression
indval2	⊢ ((𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂) → ((𝟭‘𝑂)‘𝐴) = ((𝐴 × {1}) ∪ ((𝑂 ∖ 𝐴) × {0})))

Proof of Theorem indval2

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfmpt3 6703	. . . 4 ⊢ (𝑥 ∈ 𝑂 ↦ if(𝑥 ∈ 𝐴, 1, 0)) = ∪ 𝑥 ∈ 𝑂 ({𝑥} × {if(𝑥 ∈ 𝐴, 1, 0)})
2		indval 33994	. . . 4 ⊢ ((𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂) → ((𝟭‘𝑂)‘𝐴) = (𝑥 ∈ 𝑂 ↦ if(𝑥 ∈ 𝐴, 1, 0)))
3		undif 4488	. . . . . . 7 ⊢ (𝐴 ⊆ 𝑂 ↔ (𝐴 ∪ (𝑂 ∖ 𝐴)) = 𝑂)
4	3	biimpi 216	. . . . . 6 ⊢ (𝐴 ⊆ 𝑂 → (𝐴 ∪ (𝑂 ∖ 𝐴)) = 𝑂)
5	4	adantl 481	. . . . 5 ⊢ ((𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂) → (𝐴 ∪ (𝑂 ∖ 𝐴)) = 𝑂)
6	5	iuneq1d 5024	. . . 4 ⊢ ((𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂) → ∪ 𝑥 ∈ (𝐴 ∪ (𝑂 ∖ 𝐴))({𝑥} × {if(𝑥 ∈ 𝐴, 1, 0)}) = ∪ 𝑥 ∈ 𝑂 ({𝑥} × {if(𝑥 ∈ 𝐴, 1, 0)}))
7	1, 2, 6	3eqtr4a 2801	. . 3 ⊢ ((𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂) → ((𝟭‘𝑂)‘𝐴) = ∪ 𝑥 ∈ (𝐴 ∪ (𝑂 ∖ 𝐴))({𝑥} × {if(𝑥 ∈ 𝐴, 1, 0)}))
8		iunxun 5099	. . 3 ⊢ ∪ 𝑥 ∈ (𝐴 ∪ (𝑂 ∖ 𝐴))({𝑥} × {if(𝑥 ∈ 𝐴, 1, 0)}) = (∪ 𝑥 ∈ 𝐴 ({𝑥} × {if(𝑥 ∈ 𝐴, 1, 0)}) ∪ ∪ 𝑥 ∈ (𝑂 ∖ 𝐴)({𝑥} × {if(𝑥 ∈ 𝐴, 1, 0)}))
9	7, 8	eqtrdi 2791	. 2 ⊢ ((𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂) → ((𝟭‘𝑂)‘𝐴) = (∪ 𝑥 ∈ 𝐴 ({𝑥} × {if(𝑥 ∈ 𝐴, 1, 0)}) ∪ ∪ 𝑥 ∈ (𝑂 ∖ 𝐴)({𝑥} × {if(𝑥 ∈ 𝐴, 1, 0)})))
10		iftrue 4537	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, 1, 0) = 1)
11	10	sneqd 4643	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 → {if(𝑥 ∈ 𝐴, 1, 0)} = {1})
12	11	xpeq2d 5719	. . . . 5 ⊢ (𝑥 ∈ 𝐴 → ({𝑥} × {if(𝑥 ∈ 𝐴, 1, 0)}) = ({𝑥} × {1}))
13	12	iuneq2i 5018	. . . 4 ⊢ ∪ 𝑥 ∈ 𝐴 ({𝑥} × {if(𝑥 ∈ 𝐴, 1, 0)}) = ∪ 𝑥 ∈ 𝐴 ({𝑥} × {1})
14		iunxpconst 5761	. . . 4 ⊢ ∪ 𝑥 ∈ 𝐴 ({𝑥} × {1}) = (𝐴 × {1})
15	13, 14	eqtri 2763	. . 3 ⊢ ∪ 𝑥 ∈ 𝐴 ({𝑥} × {if(𝑥 ∈ 𝐴, 1, 0)}) = (𝐴 × {1})
16		eldifn 4142	. . . . . . 7 ⊢ (𝑥 ∈ (𝑂 ∖ 𝐴) → ¬ 𝑥 ∈ 𝐴)
17		iffalse 4540	. . . . . . . 8 ⊢ (¬ 𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, 1, 0) = 0)
18	17	sneqd 4643	. . . . . . 7 ⊢ (¬ 𝑥 ∈ 𝐴 → {if(𝑥 ∈ 𝐴, 1, 0)} = {0})
19	16, 18	syl 17	. . . . . 6 ⊢ (𝑥 ∈ (𝑂 ∖ 𝐴) → {if(𝑥 ∈ 𝐴, 1, 0)} = {0})
20	19	xpeq2d 5719	. . . . 5 ⊢ (𝑥 ∈ (𝑂 ∖ 𝐴) → ({𝑥} × {if(𝑥 ∈ 𝐴, 1, 0)}) = ({𝑥} × {0}))
21	20	iuneq2i 5018	. . . 4 ⊢ ∪ 𝑥 ∈ (𝑂 ∖ 𝐴)({𝑥} × {if(𝑥 ∈ 𝐴, 1, 0)}) = ∪ 𝑥 ∈ (𝑂 ∖ 𝐴)({𝑥} × {0})
22		iunxpconst 5761	. . . 4 ⊢ ∪ 𝑥 ∈ (𝑂 ∖ 𝐴)({𝑥} × {0}) = ((𝑂 ∖ 𝐴) × {0})
23	21, 22	eqtri 2763	. . 3 ⊢ ∪ 𝑥 ∈ (𝑂 ∖ 𝐴)({𝑥} × {if(𝑥 ∈ 𝐴, 1, 0)}) = ((𝑂 ∖ 𝐴) × {0})
24	15, 23	uneq12i 4176	. 2 ⊢ (∪ 𝑥 ∈ 𝐴 ({𝑥} × {if(𝑥 ∈ 𝐴, 1, 0)}) ∪ ∪ 𝑥 ∈ (𝑂 ∖ 𝐴)({𝑥} × {if(𝑥 ∈ 𝐴, 1, 0)})) = ((𝐴 × {1}) ∪ ((𝑂 ∖ 𝐴) × {0}))
25	9, 24	eqtrdi 2791	1 ⊢ ((𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂) → ((𝟭‘𝑂)‘𝐴) = ((𝐴 × {1}) ∪ ((𝑂 ∖ 𝐴) × {0})))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2106   ∖ cdif 3960   ∪ cun 3961   ⊆ wss 3963  ifcif 4531  {csn 4631  ∪ ciun 4996   ↦ cmpt 5231   × cxp 5687  ‘cfv 6563  0cc0 11153  1c1 11154  𝟭cind 33991

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ind 33992

This theorem is referenced by: (None)