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Theorem dfiota3 32406
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 6031 . 2 (℩𝑥𝜑) = {𝑦 ∣ {𝑥𝜑} = {𝑦}}
2 abeq1 2876 . . . . 5 ({𝑦 ∣ {𝑥𝜑} = {𝑦}} = {𝑧 ∣ ∃𝑤(𝑧 = {𝑥𝜑} ∧ 𝑧 = {𝑤})} ↔ ∀𝑦({𝑥𝜑} = {𝑦} ↔ 𝑦 {𝑧 ∣ ∃𝑤(𝑧 = {𝑥𝜑} ∧ 𝑧 = {𝑤})}))
3 exdistr 2049 . . . . . 6 (∃𝑧𝑤(𝑦𝑧 ∧ (𝑧 = {𝑥𝜑} ∧ 𝑧 = {𝑤})) ↔ ∃𝑧(𝑦𝑧 ∧ ∃𝑤(𝑧 = {𝑥𝜑} ∧ 𝑧 = {𝑤})))
4 vex 3353 . . . . . . . . 9 𝑦 ∈ V
5 sneq 4344 . . . . . . . . . 10 (𝑤 = 𝑦 → {𝑤} = {𝑦})
65eqeq2d 2775 . . . . . . . . 9 (𝑤 = 𝑦 → ({𝑥𝜑} = {𝑤} ↔ {𝑥𝜑} = {𝑦}))
74, 6ceqsexv 3395 . . . . . . . 8 (∃𝑤(𝑤 = 𝑦 ∧ {𝑥𝜑} = {𝑤}) ↔ {𝑥𝜑} = {𝑦})
8 snex 5064 . . . . . . . . . . 11 {𝑤} ∈ V
9 eqeq1 2769 . . . . . . . . . . . . 13 (𝑧 = {𝑤} → (𝑧 = {𝑥𝜑} ↔ {𝑤} = {𝑥𝜑}))
10 eleq2 2833 . . . . . . . . . . . . 13 (𝑧 = {𝑤} → (𝑦𝑧𝑦 ∈ {𝑤}))
119, 10anbi12d 624 . . . . . . . . . . . 12 (𝑧 = {𝑤} → ((𝑧 = {𝑥𝜑} ∧ 𝑦𝑧) ↔ ({𝑤} = {𝑥𝜑} ∧ 𝑦 ∈ {𝑤})))
12 eqcom 2772 . . . . . . . . . . . . 13 ({𝑤} = {𝑥𝜑} ↔ {𝑥𝜑} = {𝑤})
13 velsn 4350 . . . . . . . . . . . . . 14 (𝑦 ∈ {𝑤} ↔ 𝑦 = 𝑤)
14 equcom 2115 . . . . . . . . . . . . . 14 (𝑦 = 𝑤𝑤 = 𝑦)
1513, 14bitri 266 . . . . . . . . . . . . 13 (𝑦 ∈ {𝑤} ↔ 𝑤 = 𝑦)
1612, 15anbi12ci 621 . . . . . . . . . . . 12 (({𝑤} = {𝑥𝜑} ∧ 𝑦 ∈ {𝑤}) ↔ (𝑤 = 𝑦 ∧ {𝑥𝜑} = {𝑤}))
1711, 16syl6bb 278 . . . . . . . . . . 11 (𝑧 = {𝑤} → ((𝑧 = {𝑥𝜑} ∧ 𝑦𝑧) ↔ (𝑤 = 𝑦 ∧ {𝑥𝜑} = {𝑤})))
188, 17ceqsexv 3395 . . . . . . . . . 10 (∃𝑧(𝑧 = {𝑤} ∧ (𝑧 = {𝑥𝜑} ∧ 𝑦𝑧)) ↔ (𝑤 = 𝑦 ∧ {𝑥𝜑} = {𝑤}))
19 an13 637 . . . . . . . . . . 11 ((𝑧 = {𝑤} ∧ (𝑧 = {𝑥𝜑} ∧ 𝑦𝑧)) ↔ (𝑦𝑧 ∧ (𝑧 = {𝑥𝜑} ∧ 𝑧 = {𝑤})))
2019exbii 1943 . . . . . . . . . 10 (∃𝑧(𝑧 = {𝑤} ∧ (𝑧 = {𝑥𝜑} ∧ 𝑦𝑧)) ↔ ∃𝑧(𝑦𝑧 ∧ (𝑧 = {𝑥𝜑} ∧ 𝑧 = {𝑤})))
2118, 20bitr3i 268 . . . . . . . . 9 ((𝑤 = 𝑦 ∧ {𝑥𝜑} = {𝑤}) ↔ ∃𝑧(𝑦𝑧 ∧ (𝑧 = {𝑥𝜑} ∧ 𝑧 = {𝑤})))
2221exbii 1943 . . . . . . . 8 (∃𝑤(𝑤 = 𝑦 ∧ {𝑥𝜑} = {𝑤}) ↔ ∃𝑤𝑧(𝑦𝑧 ∧ (𝑧 = {𝑥𝜑} ∧ 𝑧 = {𝑤})))
237, 22bitr3i 268 . . . . . . 7 ({𝑥𝜑} = {𝑦} ↔ ∃𝑤𝑧(𝑦𝑧 ∧ (𝑧 = {𝑥𝜑} ∧ 𝑧 = {𝑤})))
24 excom 2206 . . . . . . 7 (∃𝑤𝑧(𝑦𝑧 ∧ (𝑧 = {𝑥𝜑} ∧ 𝑧 = {𝑤})) ↔ ∃𝑧𝑤(𝑦𝑧 ∧ (𝑧 = {𝑥𝜑} ∧ 𝑧 = {𝑤})))
2523, 24bitri 266 . . . . . 6 ({𝑥𝜑} = {𝑦} ↔ ∃𝑧𝑤(𝑦𝑧 ∧ (𝑧 = {𝑥𝜑} ∧ 𝑧 = {𝑤})))
26 eluniab 4605 . . . . . 6 (𝑦 {𝑧 ∣ ∃𝑤(𝑧 = {𝑥𝜑} ∧ 𝑧 = {𝑤})} ↔ ∃𝑧(𝑦𝑧 ∧ ∃𝑤(𝑧 = {𝑥𝜑} ∧ 𝑧 = {𝑤})))
273, 25, 263bitr4i 294 . . . . 5 ({𝑥𝜑} = {𝑦} ↔ 𝑦 {𝑧 ∣ ∃𝑤(𝑧 = {𝑥𝜑} ∧ 𝑧 = {𝑤})})
282, 27mpgbir 1894 . . . 4 {𝑦 ∣ {𝑥𝜑} = {𝑦}} = {𝑧 ∣ ∃𝑤(𝑧 = {𝑥𝜑} ∧ 𝑧 = {𝑤})}
29 df-sn 4335 . . . . . . 7 {{𝑥𝜑}} = {𝑧𝑧 = {𝑥𝜑}}
30 dfsingles2 32404 . . . . . . 7 Singletons = {𝑧 ∣ ∃𝑤 𝑧 = {𝑤}}
3129, 30ineq12i 3974 . . . . . 6 ({{𝑥𝜑}} ∩ Singletons ) = ({𝑧𝑧 = {𝑥𝜑}} ∩ {𝑧 ∣ ∃𝑤 𝑧 = {𝑤}})
32 inab 4059 . . . . . . 7 ({𝑧𝑧 = {𝑥𝜑}} ∩ {𝑧 ∣ ∃𝑤 𝑧 = {𝑤}}) = {𝑧 ∣ (𝑧 = {𝑥𝜑} ∧ ∃𝑤 𝑧 = {𝑤})}
33 19.42v 2048 . . . . . . . . 9 (∃𝑤(𝑧 = {𝑥𝜑} ∧ 𝑧 = {𝑤}) ↔ (𝑧 = {𝑥𝜑} ∧ ∃𝑤 𝑧 = {𝑤}))
3433bicomi 215 . . . . . . . 8 ((𝑧 = {𝑥𝜑} ∧ ∃𝑤 𝑧 = {𝑤}) ↔ ∃𝑤(𝑧 = {𝑥𝜑} ∧ 𝑧 = {𝑤}))
3534abbii 2882 . . . . . . 7 {𝑧 ∣ (𝑧 = {𝑥𝜑} ∧ ∃𝑤 𝑧 = {𝑤})} = {𝑧 ∣ ∃𝑤(𝑧 = {𝑥𝜑} ∧ 𝑧 = {𝑤})}
3632, 35eqtri 2787 . . . . . 6 ({𝑧𝑧 = {𝑥𝜑}} ∩ {𝑧 ∣ ∃𝑤 𝑧 = {𝑤}}) = {𝑧 ∣ ∃𝑤(𝑧 = {𝑥𝜑} ∧ 𝑧 = {𝑤})}
3731, 36eqtri 2787 . . . . 5 ({{𝑥𝜑}} ∩ Singletons ) = {𝑧 ∣ ∃𝑤(𝑧 = {𝑥𝜑} ∧ 𝑧 = {𝑤})}
3837unieqi 4603 . . . 4 ({{𝑥𝜑}} ∩ Singletons ) = {𝑧 ∣ ∃𝑤(𝑧 = {𝑥𝜑} ∧ 𝑧 = {𝑤})}
3928, 38eqtr4i 2790 . . 3 {𝑦 ∣ {𝑥𝜑} = {𝑦}} = ({{𝑥𝜑}} ∩ Singletons )
4039unieqi 4603 . 2 {𝑦 ∣ {𝑥𝜑} = {𝑦}} = ({{𝑥𝜑}} ∩ Singletons )
411, 40eqtri 2787 1 (℩𝑥𝜑) = ({{𝑥𝜑}} ∩ Singletons )
Colors of variables: wff setvar class
Syntax hints:  wb 197  wa 384   = wceq 1652  wex 1874  wcel 2155  {cab 2751  cin 3731  {csn 4334   cuni 4594  cio 6029   Singletons csingles 32322
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3597  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-symdif 4005  df-nul 4080  df-if 4244  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-eprel 5190  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-fo 6074  df-fv 6076  df-1st 7366  df-2nd 7367  df-txp 32337  df-singleton 32345  df-singles 32346
This theorem is referenced by:  dffv5  32407
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