Theorem dfiota3 36685

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 6494	. 2 ⊢ (℩𝑥𝜑) = ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}}
2		eqabcb 2901	. . . . 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 1987	. . . . . 6 ⊢ (∃𝑧∃𝑤(𝑦 ∈ 𝑧 ∧ (𝑧 = {𝑥 ∣ 𝜑} ∧ 𝑧 = {𝑤})) ↔ ∃𝑧(𝑦 ∈ 𝑧 ∧ ∃𝑤(𝑧 = {𝑥 ∣ 𝜑} ∧ 𝑧 = {𝑤})))
4		vex 3455	. . . . . . . . 9 ⊢ 𝑦 ∈ V
5		sneq 4594	. . . . . . . . . 10 ⊢ (𝑤 = 𝑦 → {𝑤} = {𝑦})
6	5	eqeq2d 2772	. . . . . . . . 9 ⊢ (𝑤 = 𝑦 → ({𝑥 ∣ 𝜑} = {𝑤} ↔ {𝑥 ∣ 𝜑} = {𝑦}))
7	4, 6	ceqsexv 3499	. . . . . . . 8 ⊢ (∃𝑤(𝑤 = 𝑦 ∧ {𝑥 ∣ 𝜑} = {𝑤}) ↔ {𝑥 ∣ 𝜑} = {𝑦})
8		vsnex 5393	. . . . . . . . . . 11 ⊢ {𝑤} ∈ V
9		eqeq1 2765	. . . . . . . . . . . . 13 ⊢ (𝑧 = {𝑤} → (𝑧 = {𝑥 ∣ 𝜑} ↔ {𝑤} = {𝑥 ∣ 𝜑}))
10		eleq2 2850	. . . . . . . . . . . . 13 ⊢ (𝑧 = {𝑤} → (𝑦 ∈ 𝑧 ↔ 𝑦 ∈ {𝑤}))
11	9, 10	anbi12d 644	. . . . . . . . . . . 12 ⊢ (𝑧 = {𝑤} → ((𝑧 = {𝑥 ∣ 𝜑} ∧ 𝑦 ∈ 𝑧) ↔ ({𝑤} = {𝑥 ∣ 𝜑} ∧ 𝑦 ∈ {𝑤})))
12		eqcom 2768	. . . . . . . . . . . . 13 ⊢ ({𝑤} = {𝑥 ∣ 𝜑} ↔ {𝑥 ∣ 𝜑} = {𝑤})
13		velsn 4600	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ {𝑤} ↔ 𝑦 = 𝑤)
14		equcom 2051	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑤 ↔ 𝑤 = 𝑦)
15	13, 14	bitri 278	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ {𝑤} ↔ 𝑤 = 𝑦)
16	12, 15	anbi12ci 641	. . . . . . . . . . . 12 ⊢ (({𝑤} = {𝑥 ∣ 𝜑} ∧ 𝑦 ∈ {𝑤}) ↔ (𝑤 = 𝑦 ∧ {𝑥 ∣ 𝜑} = {𝑤}))
17	11, 16	bitrdi 290	. . . . . . . . . . 11 ⊢ (𝑧 = {𝑤} → ((𝑧 = {𝑥 ∣ 𝜑} ∧ 𝑦 ∈ 𝑧) ↔ (𝑤 = 𝑦 ∧ {𝑥 ∣ 𝜑} = {𝑤})))
18	8, 17	ceqsexv 3499	. . . . . . . . . 10 ⊢ (∃𝑧(𝑧 = {𝑤} ∧ (𝑧 = {𝑥 ∣ 𝜑} ∧ 𝑦 ∈ 𝑧)) ↔ (𝑤 = 𝑦 ∧ {𝑥 ∣ 𝜑} = {𝑤}))
19		an13 660	. . . . . . . . . . 11 ⊢ ((𝑧 = {𝑤} ∧ (𝑧 = {𝑥 ∣ 𝜑} ∧ 𝑦 ∈ 𝑧)) ↔ (𝑦 ∈ 𝑧 ∧ (𝑧 = {𝑥 ∣ 𝜑} ∧ 𝑧 = {𝑤})))
20	19	exbii 1881	. . . . . . . . . 10 ⊢ (∃𝑧(𝑧 = {𝑤} ∧ (𝑧 = {𝑥 ∣ 𝜑} ∧ 𝑦 ∈ 𝑧)) ↔ ∃𝑧(𝑦 ∈ 𝑧 ∧ (𝑧 = {𝑥 ∣ 𝜑} ∧ 𝑧 = {𝑤})))
21	18, 20	bitr3i 280	. . . . . . . . 9 ⊢ ((𝑤 = 𝑦 ∧ {𝑥 ∣ 𝜑} = {𝑤}) ↔ ∃𝑧(𝑦 ∈ 𝑧 ∧ (𝑧 = {𝑥 ∣ 𝜑} ∧ 𝑧 = {𝑤})))
22	21	exbii 1881	. . . . . . . 8 ⊢ (∃𝑤(𝑤 = 𝑦 ∧ {𝑥 ∣ 𝜑} = {𝑤}) ↔ ∃𝑤∃𝑧(𝑦 ∈ 𝑧 ∧ (𝑧 = {𝑥 ∣ 𝜑} ∧ 𝑧 = {𝑤})))
23	7, 22	bitr3i 280	. . . . . . 7 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} ↔ ∃𝑤∃𝑧(𝑦 ∈ 𝑧 ∧ (𝑧 = {𝑥 ∣ 𝜑} ∧ 𝑧 = {𝑤})))
24		excom 2199	. . . . . . 7 ⊢ (∃𝑤∃𝑧(𝑦 ∈ 𝑧 ∧ (𝑧 = {𝑥 ∣ 𝜑} ∧ 𝑧 = {𝑤})) ↔ ∃𝑧∃𝑤(𝑦 ∈ 𝑧 ∧ (𝑧 = {𝑥 ∣ 𝜑} ∧ 𝑧 = {𝑤})))
25	23, 24	bitri 278	. . . . . 6 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} ↔ ∃𝑧∃𝑤(𝑦 ∈ 𝑧 ∧ (𝑧 = {𝑥 ∣ 𝜑} ∧ 𝑧 = {𝑤})))
26		eluniab 4881	. . . . . 6 ⊢ (𝑦 ∈ ∪ {𝑧 ∣ ∃𝑤(𝑧 = {𝑥 ∣ 𝜑} ∧ 𝑧 = {𝑤})} ↔ ∃𝑧(𝑦 ∈ 𝑧 ∧ ∃𝑤(𝑧 = {𝑥 ∣ 𝜑} ∧ 𝑧 = {𝑤})))
27	3, 25, 26	3bitr4i 306	. . . . 5 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} ↔ 𝑦 ∈ ∪ {𝑧 ∣ ∃𝑤(𝑧 = {𝑥 ∣ 𝜑} ∧ 𝑧 = {𝑤})})
28	2, 27	mpgbir 1832	. . . 4 ⊢ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = ∪ {𝑧 ∣ ∃𝑤(𝑧 = {𝑥 ∣ 𝜑} ∧ 𝑧 = {𝑤})}
29		df-sn 4585	. . . . . . 7 ⊢ {{𝑥 ∣ 𝜑}} = {𝑧 ∣ 𝑧 = {𝑥 ∣ 𝜑}}
30		dfsingles2 36683	. . . . . . 7 ⊢ Singletons = {𝑧 ∣ ∃𝑤 𝑧 = {𝑤}}
31	29, 30	ineq12i 4164	. . . . . 6 ⊢ ({{𝑥 ∣ 𝜑}} ∩ Singletons ) = ({𝑧 ∣ 𝑧 = {𝑥 ∣ 𝜑}} ∩ {𝑧 ∣ ∃𝑤 𝑧 = {𝑤}})
32		inab 4255	. . . . . . 7 ⊢ ({𝑧 ∣ 𝑧 = {𝑥 ∣ 𝜑}} ∩ {𝑧 ∣ ∃𝑤 𝑧 = {𝑤}}) = {𝑧 ∣ (𝑧 = {𝑥 ∣ 𝜑} ∧ ∃𝑤 𝑧 = {𝑤})}
33		19.42v 1986	. . . . . . . . 9 ⊢ (∃𝑤(𝑧 = {𝑥 ∣ 𝜑} ∧ 𝑧 = {𝑤}) ↔ (𝑧 = {𝑥 ∣ 𝜑} ∧ ∃𝑤 𝑧 = {𝑤}))
34	33	bicomi 227	. . . . . . . 8 ⊢ ((𝑧 = {𝑥 ∣ 𝜑} ∧ ∃𝑤 𝑧 = {𝑤}) ↔ ∃𝑤(𝑧 = {𝑥 ∣ 𝜑} ∧ 𝑧 = {𝑤}))
35	34	abbii 2828	. . . . . . 7 ⊢ {𝑧 ∣ (𝑧 = {𝑥 ∣ 𝜑} ∧ ∃𝑤 𝑧 = {𝑤})} = {𝑧 ∣ ∃𝑤(𝑧 = {𝑥 ∣ 𝜑} ∧ 𝑧 = {𝑤})}
36	32, 35	eqtri 2784	. . . . . 6 ⊢ ({𝑧 ∣ 𝑧 = {𝑥 ∣ 𝜑}} ∩ {𝑧 ∣ ∃𝑤 𝑧 = {𝑤}}) = {𝑧 ∣ ∃𝑤(𝑧 = {𝑥 ∣ 𝜑} ∧ 𝑧 = {𝑤})}
37	31, 36	eqtri 2784	. . . . 5 ⊢ ({{𝑥 ∣ 𝜑}} ∩ Singletons ) = {𝑧 ∣ ∃𝑤(𝑧 = {𝑥 ∣ 𝜑} ∧ 𝑧 = {𝑤})}
38	37	unieqi 4879	. . . 4 ⊢ ∪ ({{𝑥 ∣ 𝜑}} ∩ Singletons ) = ∪ {𝑧 ∣ ∃𝑤(𝑧 = {𝑥 ∣ 𝜑} ∧ 𝑧 = {𝑤})}
39	28, 38	eqtr4i 2787	. . 3 ⊢ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = ∪ ({{𝑥 ∣ 𝜑}} ∩ Singletons )
40	39	unieqi 4879	. 2 ⊢ ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = ∪ ∪ ({{𝑥 ∣ 𝜑}} ∩ Singletons )
41	1, 40	eqtri 2784	1 ⊢ (℩𝑥𝜑) = ∪ ∪ ({{𝑥 ∣ 𝜑}} ∩ Singletons )

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2145  {cab 2739   ∩ cin 3898  {csn 4584  ∪ cuni 4867  ℩cio 6492   Singletons csingles 36601

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pr 5391  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-symdif 4199  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-eprel 5551  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-fo 6544  df-fv 6546  df-1st 8001  df-2nd 8002  df-txp 36616  df-singleton 36624  df-singles 36625

This theorem is used by:  dffv5  36686