Theorem indval2 32797

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 6616	. . . 4 ⊢ (𝑥 ∈ 𝑂 ↦ if(𝑥 ∈ 𝐴, 1, 0)) = ∪ 𝑥 ∈ 𝑂 ({𝑥} × {if(𝑥 ∈ 𝐴, 1, 0)})
2		indval 32796	. . . 4 ⊢ ((𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂) → ((𝟭‘𝑂)‘𝐴) = (𝑥 ∈ 𝑂 ↦ if(𝑥 ∈ 𝐴, 1, 0)))
3		undif 4433	. . . . . . 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 4969	. . . 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 2790	. . 3 ⊢ ((𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂) → ((𝟭‘𝑂)‘𝐴) = ∪ 𝑥 ∈ (𝐴 ∪ (𝑂 ∖ 𝐴))({𝑥} × {if(𝑥 ∈ 𝐴, 1, 0)}))
8		iunxun 5043	. . 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 2780	. 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 4482	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, 1, 0) = 1)
11	10	sneqd 4589	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 → {if(𝑥 ∈ 𝐴, 1, 0)} = {1})
12	11	xpeq2d 5649	. . . . 5 ⊢ (𝑥 ∈ 𝐴 → ({𝑥} × {if(𝑥 ∈ 𝐴, 1, 0)}) = ({𝑥} × {1}))
13	12	iuneq2i 4963	. . . 4 ⊢ ∪ 𝑥 ∈ 𝐴 ({𝑥} × {if(𝑥 ∈ 𝐴, 1, 0)}) = ∪ 𝑥 ∈ 𝐴 ({𝑥} × {1})
14		iunxpconst 5692	. . . 4 ⊢ ∪ 𝑥 ∈ 𝐴 ({𝑥} × {1}) = (𝐴 × {1})
15	13, 14	eqtri 2752	. . 3 ⊢ ∪ 𝑥 ∈ 𝐴 ({𝑥} × {if(𝑥 ∈ 𝐴, 1, 0)}) = (𝐴 × {1})
16		eldifn 4083	. . . . . . 7 ⊢ (𝑥 ∈ (𝑂 ∖ 𝐴) → ¬ 𝑥 ∈ 𝐴)
17		iffalse 4485	. . . . . . . 8 ⊢ (¬ 𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, 1, 0) = 0)
18	17	sneqd 4589	. . . . . . 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 5649	. . . . 5 ⊢ (𝑥 ∈ (𝑂 ∖ 𝐴) → ({𝑥} × {if(𝑥 ∈ 𝐴, 1, 0)}) = ({𝑥} × {0}))
21	20	iuneq2i 4963	. . . 4 ⊢ ∪ 𝑥 ∈ (𝑂 ∖ 𝐴)({𝑥} × {if(𝑥 ∈ 𝐴, 1, 0)}) = ∪ 𝑥 ∈ (𝑂 ∖ 𝐴)({𝑥} × {0})
22		iunxpconst 5692	. . . 4 ⊢ ∪ 𝑥 ∈ (𝑂 ∖ 𝐴)({𝑥} × {0}) = ((𝑂 ∖ 𝐴) × {0})
23	21, 22	eqtri 2752	. . 3 ⊢ ∪ 𝑥 ∈ (𝑂 ∖ 𝐴)({𝑥} × {if(𝑥 ∈ 𝐴, 1, 0)}) = ((𝑂 ∖ 𝐴) × {0})
24	15, 23	uneq12i 4117	. 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 2780	1 ⊢ ((𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂) → ((𝟭‘𝑂)‘𝐴) = ((𝐴 × {1}) ∪ ((𝑂 ∖ 𝐴) × {0})))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ∖ cdif 3900   ∪ cun 3901   ⊆ wss 3903  ifcif 4476  {csn 4577  ∪ ciun 4941   ↦ cmpt 5173   × cxp 5617  ‘cfv 6482  0cc0 11009  1c1 11010  𝟭cind 32793

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-ind 32794

This theorem is referenced by: (None)