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Theorem dfiota3 31200
 Description: A definiton of iota using minimal quantifiers. (Contributed by Scott Fenton, 19-Feb-2013.)
Assertion
Ref Expression
dfiota3 (℩𝑥𝜑) = ({{𝑥𝜑}} ∩ Singletons )

Proof of Theorem dfiota3
Dummy variables 𝑦 𝑧 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-iota 5768 . 2 (℩𝑥𝜑) = {𝑦 ∣ {𝑥𝜑} = {𝑦}}
2 abeq1 2720 . . . . 5 ({𝑦 ∣ {𝑥𝜑} = {𝑦}} = {𝑧 ∣ ∃𝑤(𝑧 = {𝑥𝜑} ∧ 𝑧 = {𝑤})} ↔ ∀𝑦({𝑥𝜑} = {𝑦} ↔ 𝑦 {𝑧 ∣ ∃𝑤(𝑧 = {𝑥𝜑} ∧ 𝑧 = {𝑤})}))
3 exdistr 1906 . . . . . 6 (∃𝑧𝑤(𝑦𝑧 ∧ (𝑧 = {𝑥𝜑} ∧ 𝑧 = {𝑤})) ↔ ∃𝑧(𝑦𝑧 ∧ ∃𝑤(𝑧 = {𝑥𝜑} ∧ 𝑧 = {𝑤})))
4 vex 3176 . . . . . . . . 9 𝑦 ∈ V
5 sneq 4135 . . . . . . . . . 10 (𝑤 = 𝑦 → {𝑤} = {𝑦})
65eqeq2d 2620 . . . . . . . . 9 (𝑤 = 𝑦 → ({𝑥𝜑} = {𝑤} ↔ {𝑥𝜑} = {𝑦}))
74, 6ceqsexv 3215 . . . . . . . 8 (∃𝑤(𝑤 = 𝑦 ∧ {𝑥𝜑} = {𝑤}) ↔ {𝑥𝜑} = {𝑦})
8 snex 4835 . . . . . . . . . . 11 {𝑤} ∈ V
9 eqeq1 2614 . . . . . . . . . . . . 13 (𝑧 = {𝑤} → (𝑧 = {𝑥𝜑} ↔ {𝑤} = {𝑥𝜑}))
10 eleq2 2677 . . . . . . . . . . . . 13 (𝑧 = {𝑤} → (𝑦𝑧𝑦 ∈ {𝑤}))
119, 10anbi12d 743 . . . . . . . . . . . 12 (𝑧 = {𝑤} → ((𝑧 = {𝑥𝜑} ∧ 𝑦𝑧) ↔ ({𝑤} = {𝑥𝜑} ∧ 𝑦 ∈ {𝑤})))
12 eqcom 2617 . . . . . . . . . . . . 13 ({𝑤} = {𝑥𝜑} ↔ {𝑥𝜑} = {𝑤})
13 velsn 4141 . . . . . . . . . . . . . 14 (𝑦 ∈ {𝑤} ↔ 𝑦 = 𝑤)
14 equcom 1932 . . . . . . . . . . . . . 14 (𝑦 = 𝑤𝑤 = 𝑦)
1513, 14bitri 263 . . . . . . . . . . . . 13 (𝑦 ∈ {𝑤} ↔ 𝑤 = 𝑦)
1612, 15anbi12ci 730 . . . . . . . . . . . 12 (({𝑤} = {𝑥𝜑} ∧ 𝑦 ∈ {𝑤}) ↔ (𝑤 = 𝑦 ∧ {𝑥𝜑} = {𝑤}))
1711, 16syl6bb 275 . . . . . . . . . . 11 (𝑧 = {𝑤} → ((𝑧 = {𝑥𝜑} ∧ 𝑦𝑧) ↔ (𝑤 = 𝑦 ∧ {𝑥𝜑} = {𝑤})))
188, 17ceqsexv 3215 . . . . . . . . . 10 (∃𝑧(𝑧 = {𝑤} ∧ (𝑧 = {𝑥𝜑} ∧ 𝑦𝑧)) ↔ (𝑤 = 𝑦 ∧ {𝑥𝜑} = {𝑤}))
19 an13 836 . . . . . . . . . . 11 ((𝑧 = {𝑤} ∧ (𝑧 = {𝑥𝜑} ∧ 𝑦𝑧)) ↔ (𝑦𝑧 ∧ (𝑧 = {𝑥𝜑} ∧ 𝑧 = {𝑤})))
2019exbii 1764 . . . . . . . . . 10 (∃𝑧(𝑧 = {𝑤} ∧ (𝑧 = {𝑥𝜑} ∧ 𝑦𝑧)) ↔ ∃𝑧(𝑦𝑧 ∧ (𝑧 = {𝑥𝜑} ∧ 𝑧 = {𝑤})))
2118, 20bitr3i 265 . . . . . . . . 9 ((𝑤 = 𝑦 ∧ {𝑥𝜑} = {𝑤}) ↔ ∃𝑧(𝑦𝑧 ∧ (𝑧 = {𝑥𝜑} ∧ 𝑧 = {𝑤})))
2221exbii 1764 . . . . . . . 8 (∃𝑤(𝑤 = 𝑦 ∧ {𝑥𝜑} = {𝑤}) ↔ ∃𝑤𝑧(𝑦𝑧 ∧ (𝑧 = {𝑥𝜑} ∧ 𝑧 = {𝑤})))
237, 22bitr3i 265 . . . . . . 7 ({𝑥𝜑} = {𝑦} ↔ ∃𝑤𝑧(𝑦𝑧 ∧ (𝑧 = {𝑥𝜑} ∧ 𝑧 = {𝑤})))
24 excom 2029 . . . . . . 7 (∃𝑤𝑧(𝑦𝑧 ∧ (𝑧 = {𝑥𝜑} ∧ 𝑧 = {𝑤})) ↔ ∃𝑧𝑤(𝑦𝑧 ∧ (𝑧 = {𝑥𝜑} ∧ 𝑧 = {𝑤})))
2523, 24bitri 263 . . . . . 6 ({𝑥𝜑} = {𝑦} ↔ ∃𝑧𝑤(𝑦𝑧 ∧ (𝑧 = {𝑥𝜑} ∧ 𝑧 = {𝑤})))
26 eluniab 4383 . . . . . 6 (𝑦 {𝑧 ∣ ∃𝑤(𝑧 = {𝑥𝜑} ∧ 𝑧 = {𝑤})} ↔ ∃𝑧(𝑦𝑧 ∧ ∃𝑤(𝑧 = {𝑥𝜑} ∧ 𝑧 = {𝑤})))
273, 25, 263bitr4i 291 . . . . 5 ({𝑥𝜑} = {𝑦} ↔ 𝑦 {𝑧 ∣ ∃𝑤(𝑧 = {𝑥𝜑} ∧ 𝑧 = {𝑤})})
282, 27mpgbir 1717 . . . 4 {𝑦 ∣ {𝑥𝜑} = {𝑦}} = {𝑧 ∣ ∃𝑤(𝑧 = {𝑥𝜑} ∧ 𝑧 = {𝑤})}
29 df-sn 4126 . . . . . . 7 {{𝑥𝜑}} = {𝑧𝑧 = {𝑥𝜑}}
30 dfsingles2 31198 . . . . . . 7 Singletons = {𝑧 ∣ ∃𝑤 𝑧 = {𝑤}}
3129, 30ineq12i 3774 . . . . . 6 ({{𝑥𝜑}} ∩ Singletons ) = ({𝑧𝑧 = {𝑥𝜑}} ∩ {𝑧 ∣ ∃𝑤 𝑧 = {𝑤}})
32 inab 3854 . . . . . . 7 ({𝑧𝑧 = {𝑥𝜑}} ∩ {𝑧 ∣ ∃𝑤 𝑧 = {𝑤}}) = {𝑧 ∣ (𝑧 = {𝑥𝜑} ∧ ∃𝑤 𝑧 = {𝑤})}
33 19.42v 1905 . . . . . . . . 9 (∃𝑤(𝑧 = {𝑥𝜑} ∧ 𝑧 = {𝑤}) ↔ (𝑧 = {𝑥𝜑} ∧ ∃𝑤 𝑧 = {𝑤}))
3433bicomi 213 . . . . . . . 8 ((𝑧 = {𝑥𝜑} ∧ ∃𝑤 𝑧 = {𝑤}) ↔ ∃𝑤(𝑧 = {𝑥𝜑} ∧ 𝑧 = {𝑤}))
3534abbii 2726 . . . . . . 7 {𝑧 ∣ (𝑧 = {𝑥𝜑} ∧ ∃𝑤 𝑧 = {𝑤})} = {𝑧 ∣ ∃𝑤(𝑧 = {𝑥𝜑} ∧ 𝑧 = {𝑤})}
3632, 35eqtri 2632 . . . . . 6 ({𝑧𝑧 = {𝑥𝜑}} ∩ {𝑧 ∣ ∃𝑤 𝑧 = {𝑤}}) = {𝑧 ∣ ∃𝑤(𝑧 = {𝑥𝜑} ∧ 𝑧 = {𝑤})}
3731, 36eqtri 2632 . . . . 5 ({{𝑥𝜑}} ∩ Singletons ) = {𝑧 ∣ ∃𝑤(𝑧 = {𝑥𝜑} ∧ 𝑧 = {𝑤})}
3837unieqi 4381 . . . 4 ({{𝑥𝜑}} ∩ Singletons ) = {𝑧 ∣ ∃𝑤(𝑧 = {𝑥𝜑} ∧ 𝑧 = {𝑤})}
3928, 38eqtr4i 2635 . . 3 {𝑦 ∣ {𝑥𝜑} = {𝑦}} = ({{𝑥𝜑}} ∩ Singletons )
4039unieqi 4381 . 2 {𝑦 ∣ {𝑥𝜑} = {𝑦}} = ({{𝑥𝜑}} ∩ Singletons )
411, 40eqtri 2632 1 (℩𝑥𝜑) = ({{𝑥𝜑}} ∩ Singletons )
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977  {cab 2596   ∩ cin 3539  {csn 4125  ∪ cuni 4372  ℩cio 5766   Singletons csingles 31115 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-symdif 3806  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-eprel 4949  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fo 5810  df-fv 5812  df-1st 7059  df-2nd 7060  df-txp 31130  df-singleton 31138  df-singles 31139 This theorem is referenced by:  dffv5  31201
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