Theorem dfiota3 32030

Description: A definiton 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 5851	. 2 ⊢ (℩𝑥𝜑) = ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}}
2		abeq1 2733	. . . . 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 1919	. . . . . 6 ⊢ (∃𝑧∃𝑤(𝑦 ∈ 𝑧 ∧ (𝑧 = {𝑥 ∣ 𝜑} ∧ 𝑧 = {𝑤})) ↔ ∃𝑧(𝑦 ∈ 𝑧 ∧ ∃𝑤(𝑧 = {𝑥 ∣ 𝜑} ∧ 𝑧 = {𝑤})))
4		vex 3203	. . . . . . . . 9 ⊢ 𝑦 ∈ V
5		sneq 4187	. . . . . . . . . 10 ⊢ (𝑤 = 𝑦 → {𝑤} = {𝑦})
6	5	eqeq2d 2632	. . . . . . . . 9 ⊢ (𝑤 = 𝑦 → ({𝑥 ∣ 𝜑} = {𝑤} ↔ {𝑥 ∣ 𝜑} = {𝑦}))
7	4, 6	ceqsexv 3242	. . . . . . . 8 ⊢ (∃𝑤(𝑤 = 𝑦 ∧ {𝑥 ∣ 𝜑} = {𝑤}) ↔ {𝑥 ∣ 𝜑} = {𝑦})
8		snex 4908	. . . . . . . . . . 11 ⊢ {𝑤} ∈ V
9		eqeq1 2626	. . . . . . . . . . . . 13 ⊢ (𝑧 = {𝑤} → (𝑧 = {𝑥 ∣ 𝜑} ↔ {𝑤} = {𝑥 ∣ 𝜑}))
10		eleq2 2690	. . . . . . . . . . . . 13 ⊢ (𝑧 = {𝑤} → (𝑦 ∈ 𝑧 ↔ 𝑦 ∈ {𝑤}))
11	9, 10	anbi12d 747	. . . . . . . . . . . 12 ⊢ (𝑧 = {𝑤} → ((𝑧 = {𝑥 ∣ 𝜑} ∧ 𝑦 ∈ 𝑧) ↔ ({𝑤} = {𝑥 ∣ 𝜑} ∧ 𝑦 ∈ {𝑤})))
12		eqcom 2629	. . . . . . . . . . . . 13 ⊢ ({𝑤} = {𝑥 ∣ 𝜑} ↔ {𝑥 ∣ 𝜑} = {𝑤})
13		velsn 4193	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ {𝑤} ↔ 𝑦 = 𝑤)
14		equcom 1945	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑤 ↔ 𝑤 = 𝑦)
15	13, 14	bitri 264	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ {𝑤} ↔ 𝑤 = 𝑦)
16	12, 15	anbi12ci 734	. . . . . . . . . . . 12 ⊢ (({𝑤} = {𝑥 ∣ 𝜑} ∧ 𝑦 ∈ {𝑤}) ↔ (𝑤 = 𝑦 ∧ {𝑥 ∣ 𝜑} = {𝑤}))
17	11, 16	syl6bb 276	. . . . . . . . . . 11 ⊢ (𝑧 = {𝑤} → ((𝑧 = {𝑥 ∣ 𝜑} ∧ 𝑦 ∈ 𝑧) ↔ (𝑤 = 𝑦 ∧ {𝑥 ∣ 𝜑} = {𝑤})))
18	8, 17	ceqsexv 3242	. . . . . . . . . 10 ⊢ (∃𝑧(𝑧 = {𝑤} ∧ (𝑧 = {𝑥 ∣ 𝜑} ∧ 𝑦 ∈ 𝑧)) ↔ (𝑤 = 𝑦 ∧ {𝑥 ∣ 𝜑} = {𝑤}))
19		an13 840	. . . . . . . . . . 11 ⊢ ((𝑧 = {𝑤} ∧ (𝑧 = {𝑥 ∣ 𝜑} ∧ 𝑦 ∈ 𝑧)) ↔ (𝑦 ∈ 𝑧 ∧ (𝑧 = {𝑥 ∣ 𝜑} ∧ 𝑧 = {𝑤})))
20	19	exbii 1774	. . . . . . . . . 10 ⊢ (∃𝑧(𝑧 = {𝑤} ∧ (𝑧 = {𝑥 ∣ 𝜑} ∧ 𝑦 ∈ 𝑧)) ↔ ∃𝑧(𝑦 ∈ 𝑧 ∧ (𝑧 = {𝑥 ∣ 𝜑} ∧ 𝑧 = {𝑤})))
21	18, 20	bitr3i 266	. . . . . . . . 9 ⊢ ((𝑤 = 𝑦 ∧ {𝑥 ∣ 𝜑} = {𝑤}) ↔ ∃𝑧(𝑦 ∈ 𝑧 ∧ (𝑧 = {𝑥 ∣ 𝜑} ∧ 𝑧 = {𝑤})))
22	21	exbii 1774	. . . . . . . 8 ⊢ (∃𝑤(𝑤 = 𝑦 ∧ {𝑥 ∣ 𝜑} = {𝑤}) ↔ ∃𝑤∃𝑧(𝑦 ∈ 𝑧 ∧ (𝑧 = {𝑥 ∣ 𝜑} ∧ 𝑧 = {𝑤})))
23	7, 22	bitr3i 266	. . . . . . 7 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} ↔ ∃𝑤∃𝑧(𝑦 ∈ 𝑧 ∧ (𝑧 = {𝑥 ∣ 𝜑} ∧ 𝑧 = {𝑤})))
24		excom 2042	. . . . . . 7 ⊢ (∃𝑤∃𝑧(𝑦 ∈ 𝑧 ∧ (𝑧 = {𝑥 ∣ 𝜑} ∧ 𝑧 = {𝑤})) ↔ ∃𝑧∃𝑤(𝑦 ∈ 𝑧 ∧ (𝑧 = {𝑥 ∣ 𝜑} ∧ 𝑧 = {𝑤})))
25	23, 24	bitri 264	. . . . . 6 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} ↔ ∃𝑧∃𝑤(𝑦 ∈ 𝑧 ∧ (𝑧 = {𝑥 ∣ 𝜑} ∧ 𝑧 = {𝑤})))
26		eluniab 4447	. . . . . 6 ⊢ (𝑦 ∈ ∪ {𝑧 ∣ ∃𝑤(𝑧 = {𝑥 ∣ 𝜑} ∧ 𝑧 = {𝑤})} ↔ ∃𝑧(𝑦 ∈ 𝑧 ∧ ∃𝑤(𝑧 = {𝑥 ∣ 𝜑} ∧ 𝑧 = {𝑤})))
27	3, 25, 26	3bitr4i 292	. . . . 5 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} ↔ 𝑦 ∈ ∪ {𝑧 ∣ ∃𝑤(𝑧 = {𝑥 ∣ 𝜑} ∧ 𝑧 = {𝑤})})
28	2, 27	mpgbir 1726	. . . 4 ⊢ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = ∪ {𝑧 ∣ ∃𝑤(𝑧 = {𝑥 ∣ 𝜑} ∧ 𝑧 = {𝑤})}
29		df-sn 4178	. . . . . . 7 ⊢ {{𝑥 ∣ 𝜑}} = {𝑧 ∣ 𝑧 = {𝑥 ∣ 𝜑}}
30		dfsingles2 32028	. . . . . . 7 ⊢ Singletons = {𝑧 ∣ ∃𝑤 𝑧 = {𝑤}}
31	29, 30	ineq12i 3812	. . . . . 6 ⊢ ({{𝑥 ∣ 𝜑}} ∩ Singletons ) = ({𝑧 ∣ 𝑧 = {𝑥 ∣ 𝜑}} ∩ {𝑧 ∣ ∃𝑤 𝑧 = {𝑤}})
32		inab 3895	. . . . . . 7 ⊢ ({𝑧 ∣ 𝑧 = {𝑥 ∣ 𝜑}} ∩ {𝑧 ∣ ∃𝑤 𝑧 = {𝑤}}) = {𝑧 ∣ (𝑧 = {𝑥 ∣ 𝜑} ∧ ∃𝑤 𝑧 = {𝑤})}
33		19.42v 1918	. . . . . . . . 9 ⊢ (∃𝑤(𝑧 = {𝑥 ∣ 𝜑} ∧ 𝑧 = {𝑤}) ↔ (𝑧 = {𝑥 ∣ 𝜑} ∧ ∃𝑤 𝑧 = {𝑤}))
34	33	bicomi 214	. . . . . . . 8 ⊢ ((𝑧 = {𝑥 ∣ 𝜑} ∧ ∃𝑤 𝑧 = {𝑤}) ↔ ∃𝑤(𝑧 = {𝑥 ∣ 𝜑} ∧ 𝑧 = {𝑤}))
35	34	abbii 2739	. . . . . . 7 ⊢ {𝑧 ∣ (𝑧 = {𝑥 ∣ 𝜑} ∧ ∃𝑤 𝑧 = {𝑤})} = {𝑧 ∣ ∃𝑤(𝑧 = {𝑥 ∣ 𝜑} ∧ 𝑧 = {𝑤})}
36	32, 35	eqtri 2644	. . . . . 6 ⊢ ({𝑧 ∣ 𝑧 = {𝑥 ∣ 𝜑}} ∩ {𝑧 ∣ ∃𝑤 𝑧 = {𝑤}}) = {𝑧 ∣ ∃𝑤(𝑧 = {𝑥 ∣ 𝜑} ∧ 𝑧 = {𝑤})}
37	31, 36	eqtri 2644	. . . . 5 ⊢ ({{𝑥 ∣ 𝜑}} ∩ Singletons ) = {𝑧 ∣ ∃𝑤(𝑧 = {𝑥 ∣ 𝜑} ∧ 𝑧 = {𝑤})}
38	37	unieqi 4445	. . . 4 ⊢ ∪ ({{𝑥 ∣ 𝜑}} ∩ Singletons ) = ∪ {𝑧 ∣ ∃𝑤(𝑧 = {𝑥 ∣ 𝜑} ∧ 𝑧 = {𝑤})}
39	28, 38	eqtr4i 2647	. . 3 ⊢ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = ∪ ({{𝑥 ∣ 𝜑}} ∩ Singletons )
40	39	unieqi 4445	. 2 ⊢ ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = ∪ ∪ ({{𝑥 ∣ 𝜑}} ∩ Singletons )
41	1, 40	eqtri 2644	1 ⊢ (℩𝑥𝜑) = ∪ ∪ ({{𝑥 ∣ 𝜑}} ∩ Singletons )

Syntax hints:   ↔ wb 196   ∧ wa 384   = wceq 1483  ∃wex 1704   ∈ wcel 1990  {cab 2608   ∩ cin 3573  {csn 4177  ∪ cuni 4436  ℩cio 5849   Singletons csingles 31946

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-symdif 3844  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-eprel 5029  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fo 5894  df-fv 5896  df-1st 7168  df-2nd 7169  df-txp 31961  df-singleton 31969  df-singles 31970

This theorem is referenced by:  dffv5  32031