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Theorem dfimak2 4298
 Description: Alternate definition of Kuratowski image. This is the first of a series of definitions throughout the file designed to prove existence of various operations. (Contributed by SF, 14-Jan-2015.)
Assertion
Ref Expression
dfimak2 (Ak B) = ∼ P6 ( ∼ (1c ×k V) ∪ SIk ∼ (A ∩ (B ×k V)))

Proof of Theorem dfimak2
Dummy variables x y z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-rex 2620 . . . 4 (y By, x Ay(y B y, x A))
2 exancom 1586 . . . 4 (y(y B y, x A) ↔ y(⟪y, x A y B))
3 vex 2862 . . . . . . 7 x V
4 elp6 4263 . . . . . . 7 (x V → (x P6 ( ∼ (1c ×k V) ∪ SIk ∼ (A ∩ (B ×k V))) ↔ zz, {x}⟫ ( ∼ (1c ×k V) ∪ SIk ∼ (A ∩ (B ×k V)))))
53, 4ax-mp 8 . . . . . 6 (x P6 ( ∼ (1c ×k V) ∪ SIk ∼ (A ∩ (B ×k V))) ↔ zz, {x}⟫ ( ∼ (1c ×k V) ∪ SIk ∼ (A ∩ (B ×k V))))
6 elun 3220 . . . . . . . 8 (⟪z, {x}⟫ ( ∼ (1c ×k V) ∪ SIk ∼ (A ∩ (B ×k V))) ↔ (⟪z, {x}⟫ ∼ (1c ×k V) z, {x}⟫ SIk ∼ (A ∩ (B ×k V))))
7 opkex 4113 . . . . . . . . . . 11 z, {x}⟫ V
87elcompl 3225 . . . . . . . . . 10 (⟪z, {x}⟫ ∼ (1c ×k V) ↔ ¬ ⟪z, {x}⟫ (1c ×k V))
9 snex 4111 . . . . . . . . . . 11 {x} V
10 vex 2862 . . . . . . . . . . . 12 z V
1110, 9opkelxpk 4248 . . . . . . . . . . 11 (⟪z, {x}⟫ (1c ×k V) ↔ (z 1c {x} V))
129, 11mpbiran2 885 . . . . . . . . . 10 (⟪z, {x}⟫ (1c ×k V) ↔ z 1c)
138, 12xchbinx 301 . . . . . . . . 9 (⟪z, {x}⟫ ∼ (1c ×k V) ↔ ¬ z 1c)
1413orbi1i 506 . . . . . . . 8 ((⟪z, {x}⟫ ∼ (1c ×k V) z, {x}⟫ SIk ∼ (A ∩ (B ×k V))) ↔ (¬ z 1c z, {x}⟫ SIk ∼ (A ∩ (B ×k V))))
15 iman 413 . . . . . . . . 9 ((z 1c → ⟪z, {x}⟫ SIk ∼ (A ∩ (B ×k V))) ↔ ¬ (z 1c ¬ ⟪z, {x}⟫ SIk ∼ (A ∩ (B ×k V))))
16 imor 401 . . . . . . . . 9 ((z 1c → ⟪z, {x}⟫ SIk ∼ (A ∩ (B ×k V))) ↔ (¬ z 1c z, {x}⟫ SIk ∼ (A ∩ (B ×k V))))
17 el1c 4139 . . . . . . . . . . . 12 (z 1cy z = {y})
1817anbi1i 676 . . . . . . . . . . 11 ((z 1c ¬ ⟪z, {x}⟫ SIk ∼ (A ∩ (B ×k V))) ↔ (y z = {y} ¬ ⟪z, {x}⟫ SIk ∼ (A ∩ (B ×k V))))
19 19.41v 1901 . . . . . . . . . . 11 (y(z = {y} ¬ ⟪z, {x}⟫ SIk ∼ (A ∩ (B ×k V))) ↔ (y z = {y} ¬ ⟪z, {x}⟫ SIk ∼ (A ∩ (B ×k V))))
2018, 19bitr4i 243 . . . . . . . . . 10 ((z 1c ¬ ⟪z, {x}⟫ SIk ∼ (A ∩ (B ×k V))) ↔ y(z = {y} ¬ ⟪z, {x}⟫ SIk ∼ (A ∩ (B ×k V))))
2120notbii 287 . . . . . . . . 9 (¬ (z 1c ¬ ⟪z, {x}⟫ SIk ∼ (A ∩ (B ×k V))) ↔ ¬ y(z = {y} ¬ ⟪z, {x}⟫ SIk ∼ (A ∩ (B ×k V))))
2215, 16, 213bitr3i 266 . . . . . . . 8 ((¬ z 1c z, {x}⟫ SIk ∼ (A ∩ (B ×k V))) ↔ ¬ y(z = {y} ¬ ⟪z, {x}⟫ SIk ∼ (A ∩ (B ×k V))))
236, 14, 223bitri 262 . . . . . . 7 (⟪z, {x}⟫ ( ∼ (1c ×k V) ∪ SIk ∼ (A ∩ (B ×k V))) ↔ ¬ y(z = {y} ¬ ⟪z, {x}⟫ SIk ∼ (A ∩ (B ×k V))))
2423albii 1566 . . . . . 6 (zz, {x}⟫ ( ∼ (1c ×k V) ∪ SIk ∼ (A ∩ (B ×k V))) ↔ z ¬ y(z = {y} ¬ ⟪z, {x}⟫ SIk ∼ (A ∩ (B ×k V))))
25 alnex 1543 . . . . . . 7 (z ¬ y(z = {y} ¬ ⟪z, {x}⟫ SIk ∼ (A ∩ (B ×k V))) ↔ ¬ zy(z = {y} ¬ ⟪z, {x}⟫ SIk ∼ (A ∩ (B ×k V))))
26 excom 1741 . . . . . . . 8 (zy(z = {y} ¬ ⟪z, {x}⟫ SIk ∼ (A ∩ (B ×k V))) ↔ yz(z = {y} ¬ ⟪z, {x}⟫ SIk ∼ (A ∩ (B ×k V))))
27 snex 4111 . . . . . . . . . . 11 {y} V
28 opkeq1 4059 . . . . . . . . . . . . . 14 (z = {y} → ⟪z, {x}⟫ = ⟪{y}, {x}⟫)
2928eleq1d 2419 . . . . . . . . . . . . 13 (z = {y} → (⟪z, {x}⟫ SIk ∼ (A ∩ (B ×k V)) ↔ ⟪{y}, {x}⟫ SIk ∼ (A ∩ (B ×k V))))
30 vex 2862 . . . . . . . . . . . . . . 15 y V
3130, 3opksnelsik 4265 . . . . . . . . . . . . . 14 (⟪{y}, {x}⟫ SIk ∼ (A ∩ (B ×k V)) ↔ ⟪y, x ∼ (A ∩ (B ×k V)))
32 opkex 4113 . . . . . . . . . . . . . . 15 y, x V
3332elcompl 3225 . . . . . . . . . . . . . 14 (⟪y, x ∼ (A ∩ (B ×k V)) ↔ ¬ ⟪y, x (A ∩ (B ×k V)))
3431, 33bitri 240 . . . . . . . . . . . . 13 (⟪{y}, {x}⟫ SIk ∼ (A ∩ (B ×k V)) ↔ ¬ ⟪y, x (A ∩ (B ×k V)))
3529, 34syl6bb 252 . . . . . . . . . . . 12 (z = {y} → (⟪z, {x}⟫ SIk ∼ (A ∩ (B ×k V)) ↔ ¬ ⟪y, x (A ∩ (B ×k V))))
3635notbid 285 . . . . . . . . . . 11 (z = {y} → (¬ ⟪z, {x}⟫ SIk ∼ (A ∩ (B ×k V)) ↔ ¬ ¬ ⟪y, x (A ∩ (B ×k V))))
3727, 36ceqsexv 2894 . . . . . . . . . 10 (z(z = {y} ¬ ⟪z, {x}⟫ SIk ∼ (A ∩ (B ×k V))) ↔ ¬ ¬ ⟪y, x (A ∩ (B ×k V)))
38 elin 3219 . . . . . . . . . . 11 (⟪y, x (A ∩ (B ×k V)) ↔ (⟪y, x A y, x (B ×k V)))
39 notnot 282 . . . . . . . . . . 11 (⟪y, x (A ∩ (B ×k V)) ↔ ¬ ¬ ⟪y, x (A ∩ (B ×k V)))
4030, 3opkelxpk 4248 . . . . . . . . . . . . 13 (⟪y, x (B ×k V) ↔ (y B x V))
413, 40mpbiran2 885 . . . . . . . . . . . 12 (⟪y, x (B ×k V) ↔ y B)
4241anbi2i 675 . . . . . . . . . . 11 ((⟪y, x A y, x (B ×k V)) ↔ (⟪y, x A y B))
4338, 39, 423bitr3i 266 . . . . . . . . . 10 (¬ ¬ ⟪y, x (A ∩ (B ×k V)) ↔ (⟪y, x A y B))
4437, 43bitri 240 . . . . . . . . 9 (z(z = {y} ¬ ⟪z, {x}⟫ SIk ∼ (A ∩ (B ×k V))) ↔ (⟪y, x A y B))
4544exbii 1582 . . . . . . . 8 (yz(z = {y} ¬ ⟪z, {x}⟫ SIk ∼ (A ∩ (B ×k V))) ↔ y(⟪y, x A y B))
4626, 45bitri 240 . . . . . . 7 (zy(z = {y} ¬ ⟪z, {x}⟫ SIk ∼ (A ∩ (B ×k V))) ↔ y(⟪y, x A y B))
4725, 46xchbinx 301 . . . . . 6 (z ¬ y(z = {y} ¬ ⟪z, {x}⟫ SIk ∼ (A ∩ (B ×k V))) ↔ ¬ y(⟪y, x A y B))
485, 24, 473bitri 262 . . . . 5 (x P6 ( ∼ (1c ×k V) ∪ SIk ∼ (A ∩ (B ×k V))) ↔ ¬ y(⟪y, x A y B))
4948con2bii 322 . . . 4 (y(⟪y, x A y B) ↔ ¬ x P6 ( ∼ (1c ×k V) ∪ SIk ∼ (A ∩ (B ×k V))))
501, 2, 493bitri 262 . . 3 (y By, x A ↔ ¬ x P6 ( ∼ (1c ×k V) ∪ SIk ∼ (A ∩ (B ×k V))))
513elimak 4259 . . 3 (x (Ak B) ↔ y By, x A)
523elcompl 3225 . . 3 (x P6 ( ∼ (1c ×k V) ∪ SIk ∼ (A ∩ (B ×k V))) ↔ ¬ x P6 ( ∼ (1c ×k V) ∪ SIk ∼ (A ∩ (B ×k V))))
5350, 51, 523bitr4i 268 . 2 (x (Ak B) ↔ x P6 ( ∼ (1c ×k V) ∪ SIk ∼ (A ∩ (B ×k V))))
5453eqriv 2350 1 (Ak B) = ∼ P6 ( ∼ (1c ×k V) ∪ SIk ∼ (A ∩ (B ×k V)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∨ wo 357   ∧ wa 358  ∀wal 1540  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃wrex 2615  Vcvv 2859   ∼ ccompl 3205   ∪ cun 3207   ∩ cin 3208  {csn 3737  ⟪copk 4057  1cc1c 4134   ×k cxpk 4174   P6 cp6 4178   “k cimak 4179   SIk csik 4181 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-sn 4087 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-ss 3259  df-nul 3551  df-sn 3741  df-pr 3742  df-opk 4058  df-1c 4136  df-xpk 4185  df-imak 4189  df-p6 4191  df-sik 4192 This theorem is referenced by:  imakexg  4299
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