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Theorem dfiota3 33863
Description: A definition 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 6297 . 2 (℩𝑥𝜑) = {𝑦 ∣ {𝑥𝜑} = {𝑦}}
2 abeq1 2865 . . . . 5 ({𝑦 ∣ {𝑥𝜑} = {𝑦}} = {𝑧 ∣ ∃𝑤(𝑧 = {𝑥𝜑} ∧ 𝑧 = {𝑤})} ↔ ∀𝑦({𝑥𝜑} = {𝑦} ↔ 𝑦 {𝑧 ∣ ∃𝑤(𝑧 = {𝑥𝜑} ∧ 𝑧 = {𝑤})}))
3 exdistr 1962 . . . . . 6 (∃𝑧𝑤(𝑦𝑧 ∧ (𝑧 = {𝑥𝜑} ∧ 𝑧 = {𝑤})) ↔ ∃𝑧(𝑦𝑧 ∧ ∃𝑤(𝑧 = {𝑥𝜑} ∧ 𝑧 = {𝑤})))
4 vex 3402 . . . . . . . . 9 𝑦 ∈ V
5 sneq 4526 . . . . . . . . . 10 (𝑤 = 𝑦 → {𝑤} = {𝑦})
65eqeq2d 2749 . . . . . . . . 9 (𝑤 = 𝑦 → ({𝑥𝜑} = {𝑤} ↔ {𝑥𝜑} = {𝑦}))
74, 6ceqsexv 3445 . . . . . . . 8 (∃𝑤(𝑤 = 𝑦 ∧ {𝑥𝜑} = {𝑤}) ↔ {𝑥𝜑} = {𝑦})
8 snex 5298 . . . . . . . . . . 11 {𝑤} ∈ V
9 eqeq1 2742 . . . . . . . . . . . . 13 (𝑧 = {𝑤} → (𝑧 = {𝑥𝜑} ↔ {𝑤} = {𝑥𝜑}))
10 eleq2 2821 . . . . . . . . . . . . 13 (𝑧 = {𝑤} → (𝑦𝑧𝑦 ∈ {𝑤}))
119, 10anbi12d 634 . . . . . . . . . . . 12 (𝑧 = {𝑤} → ((𝑧 = {𝑥𝜑} ∧ 𝑦𝑧) ↔ ({𝑤} = {𝑥𝜑} ∧ 𝑦 ∈ {𝑤})))
12 eqcom 2745 . . . . . . . . . . . . 13 ({𝑤} = {𝑥𝜑} ↔ {𝑥𝜑} = {𝑤})
13 velsn 4532 . . . . . . . . . . . . . 14 (𝑦 ∈ {𝑤} ↔ 𝑦 = 𝑤)
14 equcom 2030 . . . . . . . . . . . . . 14 (𝑦 = 𝑤𝑤 = 𝑦)
1513, 14bitri 278 . . . . . . . . . . . . 13 (𝑦 ∈ {𝑤} ↔ 𝑤 = 𝑦)
1612, 15anbi12ci 631 . . . . . . . . . . . 12 (({𝑤} = {𝑥𝜑} ∧ 𝑦 ∈ {𝑤}) ↔ (𝑤 = 𝑦 ∧ {𝑥𝜑} = {𝑤}))
1711, 16bitrdi 290 . . . . . . . . . . 11 (𝑧 = {𝑤} → ((𝑧 = {𝑥𝜑} ∧ 𝑦𝑧) ↔ (𝑤 = 𝑦 ∧ {𝑥𝜑} = {𝑤})))
188, 17ceqsexv 3445 . . . . . . . . . 10 (∃𝑧(𝑧 = {𝑤} ∧ (𝑧 = {𝑥𝜑} ∧ 𝑦𝑧)) ↔ (𝑤 = 𝑦 ∧ {𝑥𝜑} = {𝑤}))
19 an13 647 . . . . . . . . . . 11 ((𝑧 = {𝑤} ∧ (𝑧 = {𝑥𝜑} ∧ 𝑦𝑧)) ↔ (𝑦𝑧 ∧ (𝑧 = {𝑥𝜑} ∧ 𝑧 = {𝑤})))
2019exbii 1854 . . . . . . . . . 10 (∃𝑧(𝑧 = {𝑤} ∧ (𝑧 = {𝑥𝜑} ∧ 𝑦𝑧)) ↔ ∃𝑧(𝑦𝑧 ∧ (𝑧 = {𝑥𝜑} ∧ 𝑧 = {𝑤})))
2118, 20bitr3i 280 . . . . . . . . 9 ((𝑤 = 𝑦 ∧ {𝑥𝜑} = {𝑤}) ↔ ∃𝑧(𝑦𝑧 ∧ (𝑧 = {𝑥𝜑} ∧ 𝑧 = {𝑤})))
2221exbii 1854 . . . . . . . 8 (∃𝑤(𝑤 = 𝑦 ∧ {𝑥𝜑} = {𝑤}) ↔ ∃𝑤𝑧(𝑦𝑧 ∧ (𝑧 = {𝑥𝜑} ∧ 𝑧 = {𝑤})))
237, 22bitr3i 280 . . . . . . 7 ({𝑥𝜑} = {𝑦} ↔ ∃𝑤𝑧(𝑦𝑧 ∧ (𝑧 = {𝑥𝜑} ∧ 𝑧 = {𝑤})))
24 excom 2170 . . . . . . 7 (∃𝑤𝑧(𝑦𝑧 ∧ (𝑧 = {𝑥𝜑} ∧ 𝑧 = {𝑤})) ↔ ∃𝑧𝑤(𝑦𝑧 ∧ (𝑧 = {𝑥𝜑} ∧ 𝑧 = {𝑤})))
2523, 24bitri 278 . . . . . 6 ({𝑥𝜑} = {𝑦} ↔ ∃𝑧𝑤(𝑦𝑧 ∧ (𝑧 = {𝑥𝜑} ∧ 𝑧 = {𝑤})))
26 eluniab 4811 . . . . . 6 (𝑦 {𝑧 ∣ ∃𝑤(𝑧 = {𝑥𝜑} ∧ 𝑧 = {𝑤})} ↔ ∃𝑧(𝑦𝑧 ∧ ∃𝑤(𝑧 = {𝑥𝜑} ∧ 𝑧 = {𝑤})))
273, 25, 263bitr4i 306 . . . . 5 ({𝑥𝜑} = {𝑦} ↔ 𝑦 {𝑧 ∣ ∃𝑤(𝑧 = {𝑥𝜑} ∧ 𝑧 = {𝑤})})
282, 27mpgbir 1806 . . . 4 {𝑦 ∣ {𝑥𝜑} = {𝑦}} = {𝑧 ∣ ∃𝑤(𝑧 = {𝑥𝜑} ∧ 𝑧 = {𝑤})}
29 df-sn 4517 . . . . . . 7 {{𝑥𝜑}} = {𝑧𝑧 = {𝑥𝜑}}
30 dfsingles2 33861 . . . . . . 7 Singletons = {𝑧 ∣ ∃𝑤 𝑧 = {𝑤}}
3129, 30ineq12i 4101 . . . . . 6 ({{𝑥𝜑}} ∩ Singletons ) = ({𝑧𝑧 = {𝑥𝜑}} ∩ {𝑧 ∣ ∃𝑤 𝑧 = {𝑤}})
32 inab 4189 . . . . . . 7 ({𝑧𝑧 = {𝑥𝜑}} ∩ {𝑧 ∣ ∃𝑤 𝑧 = {𝑤}}) = {𝑧 ∣ (𝑧 = {𝑥𝜑} ∧ ∃𝑤 𝑧 = {𝑤})}
33 19.42v 1961 . . . . . . . . 9 (∃𝑤(𝑧 = {𝑥𝜑} ∧ 𝑧 = {𝑤}) ↔ (𝑧 = {𝑥𝜑} ∧ ∃𝑤 𝑧 = {𝑤}))
3433bicomi 227 . . . . . . . 8 ((𝑧 = {𝑥𝜑} ∧ ∃𝑤 𝑧 = {𝑤}) ↔ ∃𝑤(𝑧 = {𝑥𝜑} ∧ 𝑧 = {𝑤}))
3534abbii 2803 . . . . . . 7 {𝑧 ∣ (𝑧 = {𝑥𝜑} ∧ ∃𝑤 𝑧 = {𝑤})} = {𝑧 ∣ ∃𝑤(𝑧 = {𝑥𝜑} ∧ 𝑧 = {𝑤})}
3632, 35eqtri 2761 . . . . . 6 ({𝑧𝑧 = {𝑥𝜑}} ∩ {𝑧 ∣ ∃𝑤 𝑧 = {𝑤}}) = {𝑧 ∣ ∃𝑤(𝑧 = {𝑥𝜑} ∧ 𝑧 = {𝑤})}
3731, 36eqtri 2761 . . . . 5 ({{𝑥𝜑}} ∩ Singletons ) = {𝑧 ∣ ∃𝑤(𝑧 = {𝑥𝜑} ∧ 𝑧 = {𝑤})}
3837unieqi 4809 . . . 4 ({{𝑥𝜑}} ∩ Singletons ) = {𝑧 ∣ ∃𝑤(𝑧 = {𝑥𝜑} ∧ 𝑧 = {𝑤})}
3928, 38eqtr4i 2764 . . 3 {𝑦 ∣ {𝑥𝜑} = {𝑦}} = ({{𝑥𝜑}} ∩ Singletons )
4039unieqi 4809 . 2 {𝑦 ∣ {𝑥𝜑} = {𝑦}} = ({{𝑥𝜑}} ∩ Singletons )
411, 40eqtri 2761 1 (℩𝑥𝜑) = ({{𝑥𝜑}} ∩ Singletons )
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399   = wceq 1542  wex 1786  wcel 2114  {cab 2716  cin 3842  {csn 4516   cuni 4796  cio 6295   Singletons csingles 33779
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pr 5296  ax-un 7479
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3400  df-sbc 3681  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-symdif 4133  df-nul 4212  df-if 4415  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5429  df-eprel 5434  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-fo 6345  df-fv 6347  df-1st 7714  df-2nd 7715  df-txp 33794  df-singleton 33802  df-singles 33803
This theorem is referenced by:  dffv5  33864
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