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Theorem dfiota3 33268
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 6312 . 2 (℩𝑥𝜑) = {𝑦 ∣ {𝑥𝜑} = {𝑦}}
2 abeq1 2951 . . . . 5 ({𝑦 ∣ {𝑥𝜑} = {𝑦}} = {𝑧 ∣ ∃𝑤(𝑧 = {𝑥𝜑} ∧ 𝑧 = {𝑤})} ↔ ∀𝑦({𝑥𝜑} = {𝑦} ↔ 𝑦 {𝑧 ∣ ∃𝑤(𝑧 = {𝑥𝜑} ∧ 𝑧 = {𝑤})}))
3 exdistr 1948 . . . . . 6 (∃𝑧𝑤(𝑦𝑧 ∧ (𝑧 = {𝑥𝜑} ∧ 𝑧 = {𝑤})) ↔ ∃𝑧(𝑦𝑧 ∧ ∃𝑤(𝑧 = {𝑥𝜑} ∧ 𝑧 = {𝑤})))
4 vex 3503 . . . . . . . . 9 𝑦 ∈ V
5 sneq 4574 . . . . . . . . . 10 (𝑤 = 𝑦 → {𝑤} = {𝑦})
65eqeq2d 2837 . . . . . . . . 9 (𝑤 = 𝑦 → ({𝑥𝜑} = {𝑤} ↔ {𝑥𝜑} = {𝑦}))
74, 6ceqsexv 3547 . . . . . . . 8 (∃𝑤(𝑤 = 𝑦 ∧ {𝑥𝜑} = {𝑤}) ↔ {𝑥𝜑} = {𝑦})
8 snex 5328 . . . . . . . . . . 11 {𝑤} ∈ V
9 eqeq1 2830 . . . . . . . . . . . . 13 (𝑧 = {𝑤} → (𝑧 = {𝑥𝜑} ↔ {𝑤} = {𝑥𝜑}))
10 eleq2 2906 . . . . . . . . . . . . 13 (𝑧 = {𝑤} → (𝑦𝑧𝑦 ∈ {𝑤}))
119, 10anbi12d 630 . . . . . . . . . . . 12 (𝑧 = {𝑤} → ((𝑧 = {𝑥𝜑} ∧ 𝑦𝑧) ↔ ({𝑤} = {𝑥𝜑} ∧ 𝑦 ∈ {𝑤})))
12 eqcom 2833 . . . . . . . . . . . . 13 ({𝑤} = {𝑥𝜑} ↔ {𝑥𝜑} = {𝑤})
13 velsn 4580 . . . . . . . . . . . . . 14 (𝑦 ∈ {𝑤} ↔ 𝑦 = 𝑤)
14 equcom 2018 . . . . . . . . . . . . . 14 (𝑦 = 𝑤𝑤 = 𝑦)
1513, 14bitri 276 . . . . . . . . . . . . 13 (𝑦 ∈ {𝑤} ↔ 𝑤 = 𝑦)
1612, 15anbi12ci 627 . . . . . . . . . . . 12 (({𝑤} = {𝑥𝜑} ∧ 𝑦 ∈ {𝑤}) ↔ (𝑤 = 𝑦 ∧ {𝑥𝜑} = {𝑤}))
1711, 16syl6bb 288 . . . . . . . . . . 11 (𝑧 = {𝑤} → ((𝑧 = {𝑥𝜑} ∧ 𝑦𝑧) ↔ (𝑤 = 𝑦 ∧ {𝑥𝜑} = {𝑤})))
188, 17ceqsexv 3547 . . . . . . . . . 10 (∃𝑧(𝑧 = {𝑤} ∧ (𝑧 = {𝑥𝜑} ∧ 𝑦𝑧)) ↔ (𝑤 = 𝑦 ∧ {𝑥𝜑} = {𝑤}))
19 an13 643 . . . . . . . . . . 11 ((𝑧 = {𝑤} ∧ (𝑧 = {𝑥𝜑} ∧ 𝑦𝑧)) ↔ (𝑦𝑧 ∧ (𝑧 = {𝑥𝜑} ∧ 𝑧 = {𝑤})))
2019exbii 1841 . . . . . . . . . 10 (∃𝑧(𝑧 = {𝑤} ∧ (𝑧 = {𝑥𝜑} ∧ 𝑦𝑧)) ↔ ∃𝑧(𝑦𝑧 ∧ (𝑧 = {𝑥𝜑} ∧ 𝑧 = {𝑤})))
2118, 20bitr3i 278 . . . . . . . . 9 ((𝑤 = 𝑦 ∧ {𝑥𝜑} = {𝑤}) ↔ ∃𝑧(𝑦𝑧 ∧ (𝑧 = {𝑥𝜑} ∧ 𝑧 = {𝑤})))
2221exbii 1841 . . . . . . . 8 (∃𝑤(𝑤 = 𝑦 ∧ {𝑥𝜑} = {𝑤}) ↔ ∃𝑤𝑧(𝑦𝑧 ∧ (𝑧 = {𝑥𝜑} ∧ 𝑧 = {𝑤})))
237, 22bitr3i 278 . . . . . . 7 ({𝑥𝜑} = {𝑦} ↔ ∃𝑤𝑧(𝑦𝑧 ∧ (𝑧 = {𝑥𝜑} ∧ 𝑧 = {𝑤})))
24 excom 2161 . . . . . . 7 (∃𝑤𝑧(𝑦𝑧 ∧ (𝑧 = {𝑥𝜑} ∧ 𝑧 = {𝑤})) ↔ ∃𝑧𝑤(𝑦𝑧 ∧ (𝑧 = {𝑥𝜑} ∧ 𝑧 = {𝑤})))
2523, 24bitri 276 . . . . . 6 ({𝑥𝜑} = {𝑦} ↔ ∃𝑧𝑤(𝑦𝑧 ∧ (𝑧 = {𝑥𝜑} ∧ 𝑧 = {𝑤})))
26 eluniab 4848 . . . . . 6 (𝑦 {𝑧 ∣ ∃𝑤(𝑧 = {𝑥𝜑} ∧ 𝑧 = {𝑤})} ↔ ∃𝑧(𝑦𝑧 ∧ ∃𝑤(𝑧 = {𝑥𝜑} ∧ 𝑧 = {𝑤})))
273, 25, 263bitr4i 304 . . . . 5 ({𝑥𝜑} = {𝑦} ↔ 𝑦 {𝑧 ∣ ∃𝑤(𝑧 = {𝑥𝜑} ∧ 𝑧 = {𝑤})})
282, 27mpgbir 1793 . . . 4 {𝑦 ∣ {𝑥𝜑} = {𝑦}} = {𝑧 ∣ ∃𝑤(𝑧 = {𝑥𝜑} ∧ 𝑧 = {𝑤})}
29 df-sn 4565 . . . . . . 7 {{𝑥𝜑}} = {𝑧𝑧 = {𝑥𝜑}}
30 dfsingles2 33266 . . . . . . 7 Singletons = {𝑧 ∣ ∃𝑤 𝑧 = {𝑤}}
3129, 30ineq12i 4191 . . . . . 6 ({{𝑥𝜑}} ∩ Singletons ) = ({𝑧𝑧 = {𝑥𝜑}} ∩ {𝑧 ∣ ∃𝑤 𝑧 = {𝑤}})
32 inab 4275 . . . . . . 7 ({𝑧𝑧 = {𝑥𝜑}} ∩ {𝑧 ∣ ∃𝑤 𝑧 = {𝑤}}) = {𝑧 ∣ (𝑧 = {𝑥𝜑} ∧ ∃𝑤 𝑧 = {𝑤})}
33 19.42v 1947 . . . . . . . . 9 (∃𝑤(𝑧 = {𝑥𝜑} ∧ 𝑧 = {𝑤}) ↔ (𝑧 = {𝑥𝜑} ∧ ∃𝑤 𝑧 = {𝑤}))
3433bicomi 225 . . . . . . . 8 ((𝑧 = {𝑥𝜑} ∧ ∃𝑤 𝑧 = {𝑤}) ↔ ∃𝑤(𝑧 = {𝑥𝜑} ∧ 𝑧 = {𝑤}))
3534abbii 2891 . . . . . . 7 {𝑧 ∣ (𝑧 = {𝑥𝜑} ∧ ∃𝑤 𝑧 = {𝑤})} = {𝑧 ∣ ∃𝑤(𝑧 = {𝑥𝜑} ∧ 𝑧 = {𝑤})}
3632, 35eqtri 2849 . . . . . 6 ({𝑧𝑧 = {𝑥𝜑}} ∩ {𝑧 ∣ ∃𝑤 𝑧 = {𝑤}}) = {𝑧 ∣ ∃𝑤(𝑧 = {𝑥𝜑} ∧ 𝑧 = {𝑤})}
3731, 36eqtri 2849 . . . . 5 ({{𝑥𝜑}} ∩ Singletons ) = {𝑧 ∣ ∃𝑤(𝑧 = {𝑥𝜑} ∧ 𝑧 = {𝑤})}
3837unieqi 4846 . . . 4 ({{𝑥𝜑}} ∩ Singletons ) = {𝑧 ∣ ∃𝑤(𝑧 = {𝑥𝜑} ∧ 𝑧 = {𝑤})}
3928, 38eqtr4i 2852 . . 3 {𝑦 ∣ {𝑥𝜑} = {𝑦}} = ({{𝑥𝜑}} ∩ Singletons )
4039unieqi 4846 . 2 {𝑦 ∣ {𝑥𝜑} = {𝑦}} = ({{𝑥𝜑}} ∩ Singletons )
411, 40eqtri 2849 1 (℩𝑥𝜑) = ({{𝑥𝜑}} ∩ Singletons )
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396   = wceq 1530  wex 1773  wcel 2107  {cab 2804  cin 3939  {csn 4564   cuni 4837  cio 6310   Singletons csingles 33184
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-sbc 3777  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-symdif 4223  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-eprel 5464  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-fo 6358  df-fv 6360  df-1st 7680  df-2nd 7681  df-txp 33199  df-singleton 33207  df-singles 33208
This theorem is referenced by:  dffv5  33269
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