Theorem indval2 32786

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 6683	. . . 4 ⊢ (𝑥 ∈ 𝑂 ↦ if(𝑥 ∈ 𝐴, 1, 0)) = ∪ 𝑥 ∈ 𝑂 ({𝑥} × {if(𝑥 ∈ 𝐴, 1, 0)})
2		indval 32785	. . . 4 ⊢ ((𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂) → ((𝟭‘𝑂)‘𝐴) = (𝑥 ∈ 𝑂 ↦ if(𝑥 ∈ 𝐴, 1, 0)))
3		undif 4464	. . . . . . 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 5001	. . . 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 2795	. . 3 ⊢ ((𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂) → ((𝟭‘𝑂)‘𝐴) = ∪ 𝑥 ∈ (𝐴 ∪ (𝑂 ∖ 𝐴))({𝑥} × {if(𝑥 ∈ 𝐴, 1, 0)}))
8		iunxun 5076	. . 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 2785	. 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 4513	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, 1, 0) = 1)
11	10	sneqd 4620	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 → {if(𝑥 ∈ 𝐴, 1, 0)} = {1})
12	11	xpeq2d 5697	. . . . 5 ⊢ (𝑥 ∈ 𝐴 → ({𝑥} × {if(𝑥 ∈ 𝐴, 1, 0)}) = ({𝑥} × {1}))
13	12	iuneq2i 4995	. . . 4 ⊢ ∪ 𝑥 ∈ 𝐴 ({𝑥} × {if(𝑥 ∈ 𝐴, 1, 0)}) = ∪ 𝑥 ∈ 𝐴 ({𝑥} × {1})
14		iunxpconst 5740	. . . 4 ⊢ ∪ 𝑥 ∈ 𝐴 ({𝑥} × {1}) = (𝐴 × {1})
15	13, 14	eqtri 2757	. . 3 ⊢ ∪ 𝑥 ∈ 𝐴 ({𝑥} × {if(𝑥 ∈ 𝐴, 1, 0)}) = (𝐴 × {1})
16		eldifn 4114	. . . . . . 7 ⊢ (𝑥 ∈ (𝑂 ∖ 𝐴) → ¬ 𝑥 ∈ 𝐴)
17		iffalse 4516	. . . . . . . 8 ⊢ (¬ 𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, 1, 0) = 0)
18	17	sneqd 4620	. . . . . . 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 5697	. . . . 5 ⊢ (𝑥 ∈ (𝑂 ∖ 𝐴) → ({𝑥} × {if(𝑥 ∈ 𝐴, 1, 0)}) = ({𝑥} × {0}))
21	20	iuneq2i 4995	. . . 4 ⊢ ∪ 𝑥 ∈ (𝑂 ∖ 𝐴)({𝑥} × {if(𝑥 ∈ 𝐴, 1, 0)}) = ∪ 𝑥 ∈ (𝑂 ∖ 𝐴)({𝑥} × {0})
22		iunxpconst 5740	. . . 4 ⊢ ∪ 𝑥 ∈ (𝑂 ∖ 𝐴)({𝑥} × {0}) = ((𝑂 ∖ 𝐴) × {0})
23	21, 22	eqtri 2757	. . 3 ⊢ ∪ 𝑥 ∈ (𝑂 ∖ 𝐴)({𝑥} × {if(𝑥 ∈ 𝐴, 1, 0)}) = ((𝑂 ∖ 𝐴) × {0})
24	15, 23	uneq12i 4148	. 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 2785	1 ⊢ ((𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂) → ((𝟭‘𝑂)‘𝐴) = ((𝐴 × {1}) ∪ ((𝑂 ∖ 𝐴) × {0})))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2107   ∖ cdif 3930   ∪ cun 3931   ⊆ wss 3933  ifcif 4507  {csn 4608  ∪ ciun 4973   ↦ cmpt 5207   × cxp 5665  ‘cfv 6542  0cc0 11138  1c1 11139  𝟭cind 32782

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-rep 5261  ax-sep 5278  ax-nul 5288  ax-pow 5347  ax-pr 5414

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3773  df-csb 3882  df-dif 3936  df-un 3938  df-in 3940  df-ss 3950  df-nul 4316  df-if 4508  df-pw 4584  df-sn 4609  df-pr 4611  df-op 4615  df-uni 4890  df-iun 4975  df-br 5126  df-opab 5188  df-mpt 5208  df-id 5560  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6495  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ind 32783

This theorem is referenced by: (None)