Theorem indval2 12211

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 6659	. . . 4 ⊢ (𝑥 ∈ 𝑂 ↦ if(𝑥 ∈ 𝐴, 1, 0)) = ∪ 𝑥 ∈ 𝑂 ({𝑥} × {if(𝑥 ∈ 𝐴, 1, 0)})
2		indval 12209	. . . 4 ⊢ ((𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂) → ((𝟭‘𝑂)‘𝐴) = (𝑥 ∈ 𝑂 ↦ if(𝑥 ∈ 𝐴, 1, 0)))
3		undif 4439	. . . . . 6 ⊢ (𝐴 ⊆ 𝑂 ↔ (𝐴 ∪ (𝑂 ∖ 𝐴)) = 𝑂)
4	3	bilani 509	. . . . 5 ⊢ ((𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂) → (𝐴 ∪ (𝑂 ∖ 𝐴)) = 𝑂)
5	4	iuneq1d 4979	. . . 4 ⊢ ((𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂) → ∪ 𝑥 ∈ (𝐴 ∪ (𝑂 ∖ 𝐴))({𝑥} × {if(𝑥 ∈ 𝐴, 1, 0)}) = ∪ 𝑥 ∈ 𝑂 ({𝑥} × {if(𝑥 ∈ 𝐴, 1, 0)}))
6	1, 2, 5	3eqtr4a 2826	. . 3 ⊢ ((𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂) → ((𝟭‘𝑂)‘𝐴) = ∪ 𝑥 ∈ (𝐴 ∪ (𝑂 ∖ 𝐴))({𝑥} × {if(𝑥 ∈ 𝐴, 1, 0)}))
7		iunxun 5055	. . 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)}))
8	6, 7	eqtrdi 2816	. 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)})))
9		iftrue 4489	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, 1, 0) = 1)
10	9	sneqd 4597	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 → {if(𝑥 ∈ 𝐴, 1, 0)} = {1})
11	10	xpeq2d 5681	. . . . 5 ⊢ (𝑥 ∈ 𝐴 → ({𝑥} × {if(𝑥 ∈ 𝐴, 1, 0)}) = ({𝑥} × {1}))
12	11	iuneq2i 4973	. . . 4 ⊢ ∪ 𝑥 ∈ 𝐴 ({𝑥} × {if(𝑥 ∈ 𝐴, 1, 0)}) = ∪ 𝑥 ∈ 𝐴 ({𝑥} × {1})
13		iunxpconst 5724	. . . 4 ⊢ ∪ 𝑥 ∈ 𝐴 ({𝑥} × {1}) = (𝐴 × {1})
14	12, 13	eqtri 2788	. . 3 ⊢ ∪ 𝑥 ∈ 𝐴 ({𝑥} × {if(𝑥 ∈ 𝐴, 1, 0)}) = (𝐴 × {1})
15		eldifn 4088	. . . . . . 7 ⊢ (𝑥 ∈ (𝑂 ∖ 𝐴) → ¬ 𝑥 ∈ 𝐴)
16		iffalse 4492	. . . . . . . 8 ⊢ (¬ 𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, 1, 0) = 0)
17	16	sneqd 4597	. . . . . . 7 ⊢ (¬ 𝑥 ∈ 𝐴 → {if(𝑥 ∈ 𝐴, 1, 0)} = {0})
18	15, 17	syl 18	. . . . . 6 ⊢ (𝑥 ∈ (𝑂 ∖ 𝐴) → {if(𝑥 ∈ 𝐴, 1, 0)} = {0})
19	18	xpeq2d 5681	. . . . 5 ⊢ (𝑥 ∈ (𝑂 ∖ 𝐴) → ({𝑥} × {if(𝑥 ∈ 𝐴, 1, 0)}) = ({𝑥} × {0}))
20	19	iuneq2i 4973	. . . 4 ⊢ ∪ 𝑥 ∈ (𝑂 ∖ 𝐴)({𝑥} × {if(𝑥 ∈ 𝐴, 1, 0)}) = ∪ 𝑥 ∈ (𝑂 ∖ 𝐴)({𝑥} × {0})
21		iunxpconst 5724	. . . 4 ⊢ ∪ 𝑥 ∈ (𝑂 ∖ 𝐴)({𝑥} × {0}) = ((𝑂 ∖ 𝐴) × {0})
22	20, 21	eqtri 2788	. . 3 ⊢ ∪ 𝑥 ∈ (𝑂 ∖ 𝐴)({𝑥} × {if(𝑥 ∈ 𝐴, 1, 0)}) = ((𝑂 ∖ 𝐴) × {0})
23	14, 22	uneq12i 4122	. 2 ⊢ (∪ 𝑥 ∈ 𝐴 ({𝑥} × {if(𝑥 ∈ 𝐴, 1, 0)}) ∪ ∪ 𝑥 ∈ (𝑂 ∖ 𝐴)({𝑥} × {if(𝑥 ∈ 𝐴, 1, 0)})) = ((𝐴 × {1}) ∪ ((𝑂 ∖ 𝐴) × {0}))
24	8, 23	eqtrdi 2816	1 ⊢ ((𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂) → ((𝟭‘𝑂)‘𝐴) = ((𝐴 × {1}) ∪ ((𝑂 ∖ 𝐴) × {0})))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   = wceq 1563   ∈ wcel 2145   ∖ cdif 3904   ∪ cun 3905   ⊆ wss 3907  ifcif 4483  {csn 4585  ∪ ciun 4951   ↦ cmpt 5185   × cxp 5649  ‘cfv 6525  0cc0 11088  1c1 11089  𝟭cind 12206

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5231  ax-sep 5250  ax-nul 5260  ax-pow 5326  ax-pr 5394

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5105  df-opab 5167  df-mpt 5186  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-ind 12207

This theorem is referenced by:  indconst0  12218  indconst1  12219  gsumind  33575