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Theorem dfiota3 31707
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 5815 . 2 (℩𝑥𝜑) = {𝑦 ∣ {𝑥𝜑} = {𝑦}}
2 abeq1 2730 . . . . 5 ({𝑦 ∣ {𝑥𝜑} = {𝑦}} = {𝑧 ∣ ∃𝑤(𝑧 = {𝑥𝜑} ∧ 𝑧 = {𝑤})} ↔ ∀𝑦({𝑥𝜑} = {𝑦} ↔ 𝑦 {𝑧 ∣ ∃𝑤(𝑧 = {𝑥𝜑} ∧ 𝑧 = {𝑤})}))
3 exdistr 1916 . . . . . 6 (∃𝑧𝑤(𝑦𝑧 ∧ (𝑧 = {𝑥𝜑} ∧ 𝑧 = {𝑤})) ↔ ∃𝑧(𝑦𝑧 ∧ ∃𝑤(𝑧 = {𝑥𝜑} ∧ 𝑧 = {𝑤})))
4 vex 3192 . . . . . . . . 9 𝑦 ∈ V
5 sneq 4163 . . . . . . . . . 10 (𝑤 = 𝑦 → {𝑤} = {𝑦})
65eqeq2d 2631 . . . . . . . . 9 (𝑤 = 𝑦 → ({𝑥𝜑} = {𝑤} ↔ {𝑥𝜑} = {𝑦}))
74, 6ceqsexv 3231 . . . . . . . 8 (∃𝑤(𝑤 = 𝑦 ∧ {𝑥𝜑} = {𝑤}) ↔ {𝑥𝜑} = {𝑦})
8 snex 4874 . . . . . . . . . . 11 {𝑤} ∈ V
9 eqeq1 2625 . . . . . . . . . . . . 13 (𝑧 = {𝑤} → (𝑧 = {𝑥𝜑} ↔ {𝑤} = {𝑥𝜑}))
10 eleq2 2687 . . . . . . . . . . . . 13 (𝑧 = {𝑤} → (𝑦𝑧𝑦 ∈ {𝑤}))
119, 10anbi12d 746 . . . . . . . . . . . 12 (𝑧 = {𝑤} → ((𝑧 = {𝑥𝜑} ∧ 𝑦𝑧) ↔ ({𝑤} = {𝑥𝜑} ∧ 𝑦 ∈ {𝑤})))
12 eqcom 2628 . . . . . . . . . . . . 13 ({𝑤} = {𝑥𝜑} ↔ {𝑥𝜑} = {𝑤})
13 velsn 4169 . . . . . . . . . . . . . 14 (𝑦 ∈ {𝑤} ↔ 𝑦 = 𝑤)
14 equcom 1942 . . . . . . . . . . . . . 14 (𝑦 = 𝑤𝑤 = 𝑦)
1513, 14bitri 264 . . . . . . . . . . . . 13 (𝑦 ∈ {𝑤} ↔ 𝑤 = 𝑦)
1612, 15anbi12ci 733 . . . . . . . . . . . 12 (({𝑤} = {𝑥𝜑} ∧ 𝑦 ∈ {𝑤}) ↔ (𝑤 = 𝑦 ∧ {𝑥𝜑} = {𝑤}))
1711, 16syl6bb 276 . . . . . . . . . . 11 (𝑧 = {𝑤} → ((𝑧 = {𝑥𝜑} ∧ 𝑦𝑧) ↔ (𝑤 = 𝑦 ∧ {𝑥𝜑} = {𝑤})))
188, 17ceqsexv 3231 . . . . . . . . . 10 (∃𝑧(𝑧 = {𝑤} ∧ (𝑧 = {𝑥𝜑} ∧ 𝑦𝑧)) ↔ (𝑤 = 𝑦 ∧ {𝑥𝜑} = {𝑤}))
19 an13 839 . . . . . . . . . . 11 ((𝑧 = {𝑤} ∧ (𝑧 = {𝑥𝜑} ∧ 𝑦𝑧)) ↔ (𝑦𝑧 ∧ (𝑧 = {𝑥𝜑} ∧ 𝑧 = {𝑤})))
2019exbii 1771 . . . . . . . . . 10 (∃𝑧(𝑧 = {𝑤} ∧ (𝑧 = {𝑥𝜑} ∧ 𝑦𝑧)) ↔ ∃𝑧(𝑦𝑧 ∧ (𝑧 = {𝑥𝜑} ∧ 𝑧 = {𝑤})))
2118, 20bitr3i 266 . . . . . . . . 9 ((𝑤 = 𝑦 ∧ {𝑥𝜑} = {𝑤}) ↔ ∃𝑧(𝑦𝑧 ∧ (𝑧 = {𝑥𝜑} ∧ 𝑧 = {𝑤})))
2221exbii 1771 . . . . . . . 8 (∃𝑤(𝑤 = 𝑦 ∧ {𝑥𝜑} = {𝑤}) ↔ ∃𝑤𝑧(𝑦𝑧 ∧ (𝑧 = {𝑥𝜑} ∧ 𝑧 = {𝑤})))
237, 22bitr3i 266 . . . . . . 7 ({𝑥𝜑} = {𝑦} ↔ ∃𝑤𝑧(𝑦𝑧 ∧ (𝑧 = {𝑥𝜑} ∧ 𝑧 = {𝑤})))
24 excom 2039 . . . . . . 7 (∃𝑤𝑧(𝑦𝑧 ∧ (𝑧 = {𝑥𝜑} ∧ 𝑧 = {𝑤})) ↔ ∃𝑧𝑤(𝑦𝑧 ∧ (𝑧 = {𝑥𝜑} ∧ 𝑧 = {𝑤})))
2523, 24bitri 264 . . . . . 6 ({𝑥𝜑} = {𝑦} ↔ ∃𝑧𝑤(𝑦𝑧 ∧ (𝑧 = {𝑥𝜑} ∧ 𝑧 = {𝑤})))
26 eluniab 4418 . . . . . 6 (𝑦 {𝑧 ∣ ∃𝑤(𝑧 = {𝑥𝜑} ∧ 𝑧 = {𝑤})} ↔ ∃𝑧(𝑦𝑧 ∧ ∃𝑤(𝑧 = {𝑥𝜑} ∧ 𝑧 = {𝑤})))
273, 25, 263bitr4i 292 . . . . 5 ({𝑥𝜑} = {𝑦} ↔ 𝑦 {𝑧 ∣ ∃𝑤(𝑧 = {𝑥𝜑} ∧ 𝑧 = {𝑤})})
282, 27mpgbir 1723 . . . 4 {𝑦 ∣ {𝑥𝜑} = {𝑦}} = {𝑧 ∣ ∃𝑤(𝑧 = {𝑥𝜑} ∧ 𝑧 = {𝑤})}
29 df-sn 4154 . . . . . . 7 {{𝑥𝜑}} = {𝑧𝑧 = {𝑥𝜑}}
30 dfsingles2 31705 . . . . . . 7 Singletons = {𝑧 ∣ ∃𝑤 𝑧 = {𝑤}}
3129, 30ineq12i 3795 . . . . . 6 ({{𝑥𝜑}} ∩ Singletons ) = ({𝑧𝑧 = {𝑥𝜑}} ∩ {𝑧 ∣ ∃𝑤 𝑧 = {𝑤}})
32 inab 3876 . . . . . . 7 ({𝑧𝑧 = {𝑥𝜑}} ∩ {𝑧 ∣ ∃𝑤 𝑧 = {𝑤}}) = {𝑧 ∣ (𝑧 = {𝑥𝜑} ∧ ∃𝑤 𝑧 = {𝑤})}
33 19.42v 1915 . . . . . . . . 9 (∃𝑤(𝑧 = {𝑥𝜑} ∧ 𝑧 = {𝑤}) ↔ (𝑧 = {𝑥𝜑} ∧ ∃𝑤 𝑧 = {𝑤}))
3433bicomi 214 . . . . . . . 8 ((𝑧 = {𝑥𝜑} ∧ ∃𝑤 𝑧 = {𝑤}) ↔ ∃𝑤(𝑧 = {𝑥𝜑} ∧ 𝑧 = {𝑤}))
3534abbii 2736 . . . . . . 7 {𝑧 ∣ (𝑧 = {𝑥𝜑} ∧ ∃𝑤 𝑧 = {𝑤})} = {𝑧 ∣ ∃𝑤(𝑧 = {𝑥𝜑} ∧ 𝑧 = {𝑤})}
3632, 35eqtri 2643 . . . . . 6 ({𝑧𝑧 = {𝑥𝜑}} ∩ {𝑧 ∣ ∃𝑤 𝑧 = {𝑤}}) = {𝑧 ∣ ∃𝑤(𝑧 = {𝑥𝜑} ∧ 𝑧 = {𝑤})}
3731, 36eqtri 2643 . . . . 5 ({{𝑥𝜑}} ∩ Singletons ) = {𝑧 ∣ ∃𝑤(𝑧 = {𝑥𝜑} ∧ 𝑧 = {𝑤})}
3837unieqi 4416 . . . 4 ({{𝑥𝜑}} ∩ Singletons ) = {𝑧 ∣ ∃𝑤(𝑧 = {𝑥𝜑} ∧ 𝑧 = {𝑤})}
3928, 38eqtr4i 2646 . . 3 {𝑦 ∣ {𝑥𝜑} = {𝑦}} = ({{𝑥𝜑}} ∩ Singletons )
4039unieqi 4416 . 2 {𝑦 ∣ {𝑥𝜑} = {𝑦}} = ({{𝑥𝜑}} ∩ Singletons )
411, 40eqtri 2643 1 (℩𝑥𝜑) = ({{𝑥𝜑}} ∩ Singletons )
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384   = wceq 1480  wex 1701  wcel 1987  {cab 2607  cin 3558  {csn 4153   cuni 4407  cio 5813   Singletons csingles 31622
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-symdif 3827  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-eprel 4990  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-fo 5858  df-fv 5860  df-1st 7120  df-2nd 7121  df-txp 31637  df-singleton 31645  df-singles 31646
This theorem is referenced by:  dffv5  31708
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