Theorem dfiota3 36137

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 6456	. 2 ⊢ (℩𝑥𝜑) = ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}}
2		eqabcb 2877	. . . . 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 1956	. . . . . 6 ⊢ (∃𝑧∃𝑤(𝑦 ∈ 𝑧 ∧ (𝑧 = {𝑥 ∣ 𝜑} ∧ 𝑧 = {𝑤})) ↔ ∃𝑧(𝑦 ∈ 𝑧 ∧ ∃𝑤(𝑧 = {𝑥 ∣ 𝜑} ∧ 𝑧 = {𝑤})))
4		vex 3446	. . . . . . . . 9 ⊢ 𝑦 ∈ V
5		sneq 4592	. . . . . . . . . 10 ⊢ (𝑤 = 𝑦 → {𝑤} = {𝑦})
6	5	eqeq2d 2748	. . . . . . . . 9 ⊢ (𝑤 = 𝑦 → ({𝑥 ∣ 𝜑} = {𝑤} ↔ {𝑥 ∣ 𝜑} = {𝑦}))
7	4, 6	ceqsexv 3492	. . . . . . . 8 ⊢ (∃𝑤(𝑤 = 𝑦 ∧ {𝑥 ∣ 𝜑} = {𝑤}) ↔ {𝑥 ∣ 𝜑} = {𝑦})
8		vsnex 5381	. . . . . . . . . . 11 ⊢ {𝑤} ∈ V
9		eqeq1 2741	. . . . . . . . . . . . 13 ⊢ (𝑧 = {𝑤} → (𝑧 = {𝑥 ∣ 𝜑} ↔ {𝑤} = {𝑥 ∣ 𝜑}))
10		eleq2 2826	. . . . . . . . . . . . 13 ⊢ (𝑧 = {𝑤} → (𝑦 ∈ 𝑧 ↔ 𝑦 ∈ {𝑤}))
11	9, 10	anbi12d 633	. . . . . . . . . . . 12 ⊢ (𝑧 = {𝑤} → ((𝑧 = {𝑥 ∣ 𝜑} ∧ 𝑦 ∈ 𝑧) ↔ ({𝑤} = {𝑥 ∣ 𝜑} ∧ 𝑦 ∈ {𝑤})))
12		eqcom 2744	. . . . . . . . . . . . 13 ⊢ ({𝑤} = {𝑥 ∣ 𝜑} ↔ {𝑥 ∣ 𝜑} = {𝑤})
13		velsn 4598	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ {𝑤} ↔ 𝑦 = 𝑤)
14		equcom 2020	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑤 ↔ 𝑤 = 𝑦)
15	13, 14	bitri 275	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ {𝑤} ↔ 𝑤 = 𝑦)
16	12, 15	anbi12ci 630	. . . . . . . . . . . 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 3492	. . . . . . . . . 10 ⊢ (∃𝑧(𝑧 = {𝑤} ∧ (𝑧 = {𝑥 ∣ 𝜑} ∧ 𝑦 ∈ 𝑧)) ↔ (𝑤 = 𝑦 ∧ {𝑥 ∣ 𝜑} = {𝑤}))
19		an13 648	. . . . . . . . . . 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 277	. . . . . . . . 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 277	. . . . . . 7 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} ↔ ∃𝑤∃𝑧(𝑦 ∈ 𝑧 ∧ (𝑧 = {𝑥 ∣ 𝜑} ∧ 𝑧 = {𝑤})))
24		excom 2168	. . . . . . 7 ⊢ (∃𝑤∃𝑧(𝑦 ∈ 𝑧 ∧ (𝑧 = {𝑥 ∣ 𝜑} ∧ 𝑧 = {𝑤})) ↔ ∃𝑧∃𝑤(𝑦 ∈ 𝑧 ∧ (𝑧 = {𝑥 ∣ 𝜑} ∧ 𝑧 = {𝑤})))
25	23, 24	bitri 275	. . . . . 6 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} ↔ ∃𝑧∃𝑤(𝑦 ∈ 𝑧 ∧ (𝑧 = {𝑥 ∣ 𝜑} ∧ 𝑧 = {𝑤})))
26		eluniab 4879	. . . . . 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 1801	. . . 4 ⊢ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = ∪ {𝑧 ∣ ∃𝑤(𝑧 = {𝑥 ∣ 𝜑} ∧ 𝑧 = {𝑤})}
29		df-sn 4583	. . . . . . 7 ⊢ {{𝑥 ∣ 𝜑}} = {𝑧 ∣ 𝑧 = {𝑥 ∣ 𝜑}}
30		dfsingles2 36135	. . . . . . 7 ⊢ Singletons = {𝑧 ∣ ∃𝑤 𝑧 = {𝑤}}
31	29, 30	ineq12i 4172	. . . . . 6 ⊢ ({{𝑥 ∣ 𝜑}} ∩ Singletons ) = ({𝑧 ∣ 𝑧 = {𝑥 ∣ 𝜑}} ∩ {𝑧 ∣ ∃𝑤 𝑧 = {𝑤}})
32		inab 4263	. . . . . . 7 ⊢ ({𝑧 ∣ 𝑧 = {𝑥 ∣ 𝜑}} ∩ {𝑧 ∣ ∃𝑤 𝑧 = {𝑤}}) = {𝑧 ∣ (𝑧 = {𝑥 ∣ 𝜑} ∧ ∃𝑤 𝑧 = {𝑤})}
33		19.42v 1955	. . . . . . . . 9 ⊢ (∃𝑤(𝑧 = {𝑥 ∣ 𝜑} ∧ 𝑧 = {𝑤}) ↔ (𝑧 = {𝑥 ∣ 𝜑} ∧ ∃𝑤 𝑧 = {𝑤}))
34	33	bicomi 224	. . . . . . . 8 ⊢ ((𝑧 = {𝑥 ∣ 𝜑} ∧ ∃𝑤 𝑧 = {𝑤}) ↔ ∃𝑤(𝑧 = {𝑥 ∣ 𝜑} ∧ 𝑧 = {𝑤}))
35	34	abbii 2804	. . . . . . 7 ⊢ {𝑧 ∣ (𝑧 = {𝑥 ∣ 𝜑} ∧ ∃𝑤 𝑧 = {𝑤})} = {𝑧 ∣ ∃𝑤(𝑧 = {𝑥 ∣ 𝜑} ∧ 𝑧 = {𝑤})}
36	32, 35	eqtri 2760	. . . . . 6 ⊢ ({𝑧 ∣ 𝑧 = {𝑥 ∣ 𝜑}} ∩ {𝑧 ∣ ∃𝑤 𝑧 = {𝑤}}) = {𝑧 ∣ ∃𝑤(𝑧 = {𝑥 ∣ 𝜑} ∧ 𝑧 = {𝑤})}
37	31, 36	eqtri 2760	. . . . 5 ⊢ ({{𝑥 ∣ 𝜑}} ∩ Singletons ) = {𝑧 ∣ ∃𝑤(𝑧 = {𝑥 ∣ 𝜑} ∧ 𝑧 = {𝑤})}
38	37	unieqi 4877	. . . 4 ⊢ ∪ ({{𝑥 ∣ 𝜑}} ∩ Singletons ) = ∪ {𝑧 ∣ ∃𝑤(𝑧 = {𝑥 ∣ 𝜑} ∧ 𝑧 = {𝑤})}
39	28, 38	eqtr4i 2763	. . 3 ⊢ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = ∪ ({{𝑥 ∣ 𝜑}} ∩ Singletons )
40	39	unieqi 4877	. 2 ⊢ ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = ∪ ∪ ({{𝑥 ∣ 𝜑}} ∩ Singletons )
41	1, 40	eqtri 2760	1 ⊢ (℩𝑥𝜑) = ∪ ∪ ({{𝑥 ∣ 𝜑}} ∩ Singletons )

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1542  ∃wex 1781   ∈ wcel 2114  {cab 2715   ∩ cin 3902  {csn 4582  ∪ cuni 4865  ℩cio 6454   Singletons csingles 36053

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-symdif 4207  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-eprel 5532  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-fo 6506  df-fv 6508  df-1st 7943  df-2nd 7944  df-txp 36068  df-singleton 36076  df-singles 36077

This theorem is referenced by:  dffv5  36138