Theorem dfiota3 34225

Description: A definition of iota using minimal quantifiers. (Contributed by Scott Fenton, 19-Feb-2013.)

Assertion

Ref	Expression
dfiota3	⊢ (℩𝑥𝜑) = ∪ ∪ ({{𝑥 ∣ 𝜑}} ∩ Singletons )

Proof of Theorem dfiota3

Dummy variables 𝑦 𝑧 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-iota 6391	. 2 ⊢ (℩𝑥𝜑) = ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}}
2		abeq1 2873	. . . . 5 ⊢ ({𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = ∪ {𝑧 ∣ ∃𝑤(𝑧 = {𝑥 ∣ 𝜑} ∧ 𝑧 = {𝑤})} ↔ ∀𝑦({𝑥 ∣ 𝜑} = {𝑦} ↔ 𝑦 ∈ ∪ {𝑧 ∣ ∃𝑤(𝑧 = {𝑥 ∣ 𝜑} ∧ 𝑧 = {𝑤})}))
3		exdistr 1958	. . . . . 6 ⊢ (∃𝑧∃𝑤(𝑦 ∈ 𝑧 ∧ (𝑧 = {𝑥 ∣ 𝜑} ∧ 𝑧 = {𝑤})) ↔ ∃𝑧(𝑦 ∈ 𝑧 ∧ ∃𝑤(𝑧 = {𝑥 ∣ 𝜑} ∧ 𝑧 = {𝑤})))
4		vex 3436	. . . . . . . . 9 ⊢ 𝑦 ∈ V
5		sneq 4571	. . . . . . . . . 10 ⊢ (𝑤 = 𝑦 → {𝑤} = {𝑦})
6	5	eqeq2d 2749	. . . . . . . . 9 ⊢ (𝑤 = 𝑦 → ({𝑥 ∣ 𝜑} = {𝑤} ↔ {𝑥 ∣ 𝜑} = {𝑦}))
7	4, 6	ceqsexv 3479	. . . . . . . 8 ⊢ (∃𝑤(𝑤 = 𝑦 ∧ {𝑥 ∣ 𝜑} = {𝑤}) ↔ {𝑥 ∣ 𝜑} = {𝑦})
8		snex 5354	. . . . . . . . . . 11 ⊢ {𝑤} ∈ V
9		eqeq1 2742	. . . . . . . . . . . . 13 ⊢ (𝑧 = {𝑤} → (𝑧 = {𝑥 ∣ 𝜑} ↔ {𝑤} = {𝑥 ∣ 𝜑}))
10		eleq2 2827	. . . . . . . . . . . . 13 ⊢ (𝑧 = {𝑤} → (𝑦 ∈ 𝑧 ↔ 𝑦 ∈ {𝑤}))
11	9, 10	anbi12d 631	. . . . . . . . . . . 12 ⊢ (𝑧 = {𝑤} → ((𝑧 = {𝑥 ∣ 𝜑} ∧ 𝑦 ∈ 𝑧) ↔ ({𝑤} = {𝑥 ∣ 𝜑} ∧ 𝑦 ∈ {𝑤})))
12		eqcom 2745	. . . . . . . . . . . . 13 ⊢ ({𝑤} = {𝑥 ∣ 𝜑} ↔ {𝑥 ∣ 𝜑} = {𝑤})
13		velsn 4577	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ {𝑤} ↔ 𝑦 = 𝑤)
14		equcom 2021	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑤 ↔ 𝑤 = 𝑦)
15	13, 14	bitri 274	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ {𝑤} ↔ 𝑤 = 𝑦)
16	12, 15	anbi12ci 628	. . . . . . . . . . . 12 ⊢ (({𝑤} = {𝑥 ∣ 𝜑} ∧ 𝑦 ∈ {𝑤}) ↔ (𝑤 = 𝑦 ∧ {𝑥 ∣ 𝜑} = {𝑤}))
17	11, 16	bitrdi 287	. . . . . . . . . . 11 ⊢ (𝑧 = {𝑤} → ((𝑧 = {𝑥 ∣ 𝜑} ∧ 𝑦 ∈ 𝑧) ↔ (𝑤 = 𝑦 ∧ {𝑥 ∣ 𝜑} = {𝑤})))
18	8, 17	ceqsexv 3479	. . . . . . . . . 10 ⊢ (∃𝑧(𝑧 = {𝑤} ∧ (𝑧 = {𝑥 ∣ 𝜑} ∧ 𝑦 ∈ 𝑧)) ↔ (𝑤 = 𝑦 ∧ {𝑥 ∣ 𝜑} = {𝑤}))
19		an13 644	. . . . . . . . . . 11 ⊢ ((𝑧 = {𝑤} ∧ (𝑧 = {𝑥 ∣ 𝜑} ∧ 𝑦 ∈ 𝑧)) ↔ (𝑦 ∈ 𝑧 ∧ (𝑧 = {𝑥 ∣ 𝜑} ∧ 𝑧 = {𝑤})))
20	19	exbii 1850	. . . . . . . . . 10 ⊢ (∃𝑧(𝑧 = {𝑤} ∧ (𝑧 = {𝑥 ∣ 𝜑} ∧ 𝑦 ∈ 𝑧)) ↔ ∃𝑧(𝑦 ∈ 𝑧 ∧ (𝑧 = {𝑥 ∣ 𝜑} ∧ 𝑧 = {𝑤})))
21	18, 20	bitr3i 276	. . . . . . . . 9 ⊢ ((𝑤 = 𝑦 ∧ {𝑥 ∣ 𝜑} = {𝑤}) ↔ ∃𝑧(𝑦 ∈ 𝑧 ∧ (𝑧 = {𝑥 ∣ 𝜑} ∧ 𝑧 = {𝑤})))
22	21	exbii 1850	. . . . . . . 8 ⊢ (∃𝑤(𝑤 = 𝑦 ∧ {𝑥 ∣ 𝜑} = {𝑤}) ↔ ∃𝑤∃𝑧(𝑦 ∈ 𝑧 ∧ (𝑧 = {𝑥 ∣ 𝜑} ∧ 𝑧 = {𝑤})))
23	7, 22	bitr3i 276	. . . . . . 7 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} ↔ ∃𝑤∃𝑧(𝑦 ∈ 𝑧 ∧ (𝑧 = {𝑥 ∣ 𝜑} ∧ 𝑧 = {𝑤})))
24		excom 2162	. . . . . . 7 ⊢ (∃𝑤∃𝑧(𝑦 ∈ 𝑧 ∧ (𝑧 = {𝑥 ∣ 𝜑} ∧ 𝑧 = {𝑤})) ↔ ∃𝑧∃𝑤(𝑦 ∈ 𝑧 ∧ (𝑧 = {𝑥 ∣ 𝜑} ∧ 𝑧 = {𝑤})))
25	23, 24	bitri 274	. . . . . 6 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} ↔ ∃𝑧∃𝑤(𝑦 ∈ 𝑧 ∧ (𝑧 = {𝑥 ∣ 𝜑} ∧ 𝑧 = {𝑤})))
26		eluniab 4854	. . . . . 6 ⊢ (𝑦 ∈ ∪ {𝑧 ∣ ∃𝑤(𝑧 = {𝑥 ∣ 𝜑} ∧ 𝑧 = {𝑤})} ↔ ∃𝑧(𝑦 ∈ 𝑧 ∧ ∃𝑤(𝑧 = {𝑥 ∣ 𝜑} ∧ 𝑧 = {𝑤})))
27	3, 25, 26	3bitr4i 303	. . . . 5 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} ↔ 𝑦 ∈ ∪ {𝑧 ∣ ∃𝑤(𝑧 = {𝑥 ∣ 𝜑} ∧ 𝑧 = {𝑤})})
28	2, 27	mpgbir 1802	. . . 4 ⊢ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = ∪ {𝑧 ∣ ∃𝑤(𝑧 = {𝑥 ∣ 𝜑} ∧ 𝑧 = {𝑤})}
29		df-sn 4562	. . . . . . 7 ⊢ {{𝑥 ∣ 𝜑}} = {𝑧 ∣ 𝑧 = {𝑥 ∣ 𝜑}}
30		dfsingles2 34223	. . . . . . 7 ⊢ Singletons = {𝑧 ∣ ∃𝑤 𝑧 = {𝑤}}
31	29, 30	ineq12i 4144	. . . . . 6 ⊢ ({{𝑥 ∣ 𝜑}} ∩ Singletons ) = ({𝑧 ∣ 𝑧 = {𝑥 ∣ 𝜑}} ∩ {𝑧 ∣ ∃𝑤 𝑧 = {𝑤}})
32		inab 4233	. . . . . . 7 ⊢ ({𝑧 ∣ 𝑧 = {𝑥 ∣ 𝜑}} ∩ {𝑧 ∣ ∃𝑤 𝑧 = {𝑤}}) = {𝑧 ∣ (𝑧 = {𝑥 ∣ 𝜑} ∧ ∃𝑤 𝑧 = {𝑤})}
33		19.42v 1957	. . . . . . . . 9 ⊢ (∃𝑤(𝑧 = {𝑥 ∣ 𝜑} ∧ 𝑧 = {𝑤}) ↔ (𝑧 = {𝑥 ∣ 𝜑} ∧ ∃𝑤 𝑧 = {𝑤}))
34	33	bicomi 223	. . . . . . . 8 ⊢ ((𝑧 = {𝑥 ∣ 𝜑} ∧ ∃𝑤 𝑧 = {𝑤}) ↔ ∃𝑤(𝑧 = {𝑥 ∣ 𝜑} ∧ 𝑧 = {𝑤}))
35	34	abbii 2808	. . . . . . 7 ⊢ {𝑧 ∣ (𝑧 = {𝑥 ∣ 𝜑} ∧ ∃𝑤 𝑧 = {𝑤})} = {𝑧 ∣ ∃𝑤(𝑧 = {𝑥 ∣ 𝜑} ∧ 𝑧 = {𝑤})}
36	32, 35	eqtri 2766	. . . . . 6 ⊢ ({𝑧 ∣ 𝑧 = {𝑥 ∣ 𝜑}} ∩ {𝑧 ∣ ∃𝑤 𝑧 = {𝑤}}) = {𝑧 ∣ ∃𝑤(𝑧 = {𝑥 ∣ 𝜑} ∧ 𝑧 = {𝑤})}
37	31, 36	eqtri 2766	. . . . 5 ⊢ ({{𝑥 ∣ 𝜑}} ∩ Singletons ) = {𝑧 ∣ ∃𝑤(𝑧 = {𝑥 ∣ 𝜑} ∧ 𝑧 = {𝑤})}
38	37	unieqi 4852	. . . 4 ⊢ ∪ ({{𝑥 ∣ 𝜑}} ∩ Singletons ) = ∪ {𝑧 ∣ ∃𝑤(𝑧 = {𝑥 ∣ 𝜑} ∧ 𝑧 = {𝑤})}
39	28, 38	eqtr4i 2769	. . 3 ⊢ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = ∪ ({{𝑥 ∣ 𝜑}} ∩ Singletons )
40	39	unieqi 4852	. 2 ⊢ ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = ∪ ∪ ({{𝑥 ∣ 𝜑}} ∩ Singletons )
41	1, 40	eqtri 2766	1 ⊢ (℩𝑥𝜑) = ∪ ∪ ({{𝑥 ∣ 𝜑}} ∩ Singletons )

Syntax hints:   ↔ wb 205   ∧ wa 396   = wceq 1539  ∃wex 1782   ∈ wcel 2106  {cab 2715   ∩ cin 3886  {csn 4561  ∪ cuni 4839  ℩cio 6389   Singletons csingles 34141

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-symdif 4176  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-eprel 5495  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-fo 6439  df-fv 6441  df-1st 7831  df-2nd 7832  df-txp 34156  df-singleton 34164  df-singles 34165

This theorem is referenced by:  dffv5  34226