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Theorem sikexg 4296
 Description: The Kuratowski singleton image of a set is a set. (Contributed by SF, 14-Jan-2015.)
Assertion
Ref Expression
sikexg (A VSIk A V)

Proof of Theorem sikexg
Dummy variables x y z w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sikeq 4241 . . 3 (x = ASIk x = SIk A)
21eleq1d 2419 . 2 (x = A → ( SIk x V ↔ SIk A V))
3 ax-si 4083 . . 3 yzw(⟪{z}, {w}⟫ y ↔ ⟪z, w x)
4 inss1 3475 . . . . . . . 8 ((1c ×k 1c) ∩ y) (1c ×k 1c)
5 sikss1c1c 4267 . . . . . . . 8 SIk x (1c ×k 1c)
64, 5sikexlem 4295 . . . . . . 7 (((1c ×k 1c) ∩ y) = SIk xzw(⟪{z}, {w}⟫ ((1c ×k 1c) ∩ y) ↔ ⟪{z}, {w}⟫ SIk x))
7 vex 2862 . . . . . . . . . . . 12 z V
87snel1c 4140 . . . . . . . . . . 11 {z} 1c
9 vex 2862 . . . . . . . . . . . 12 w V
109snel1c 4140 . . . . . . . . . . 11 {w} 1c
11 snex 4111 . . . . . . . . . . . 12 {z} V
12 snex 4111 . . . . . . . . . . . 12 {w} V
1311, 12opkelxpk 4248 . . . . . . . . . . 11 (⟪{z}, {w}⟫ (1c ×k 1c) ↔ ({z} 1c {w} 1c))
148, 10, 13mpbir2an 886 . . . . . . . . . 10 ⟪{z}, {w}⟫ (1c ×k 1c)
15 elin 3219 . . . . . . . . . 10 (⟪{z}, {w}⟫ ((1c ×k 1c) ∩ y) ↔ (⟪{z}, {w}⟫ (1c ×k 1c) ⟪{z}, {w}⟫ y))
1614, 15mpbiran 884 . . . . . . . . 9 (⟪{z}, {w}⟫ ((1c ×k 1c) ∩ y) ↔ ⟪{z}, {w}⟫ y)
177, 9opksnelsik 4265 . . . . . . . . 9 (⟪{z}, {w}⟫ SIk x ↔ ⟪z, w x)
1816, 17bibi12i 306 . . . . . . . 8 ((⟪{z}, {w}⟫ ((1c ×k 1c) ∩ y) ↔ ⟪{z}, {w}⟫ SIk x) ↔ (⟪{z}, {w}⟫ y ↔ ⟪z, w x))
19182albii 1567 . . . . . . 7 (zw(⟪{z}, {w}⟫ ((1c ×k 1c) ∩ y) ↔ ⟪{z}, {w}⟫ SIk x) ↔ zw(⟪{z}, {w}⟫ y ↔ ⟪z, w x))
206, 19bitri 240 . . . . . 6 (((1c ×k 1c) ∩ y) = SIk xzw(⟪{z}, {w}⟫ y ↔ ⟪z, w x))
2120biimpri 197 . . . . 5 (zw(⟪{z}, {w}⟫ y ↔ ⟪z, w x) → ((1c ×k 1c) ∩ y) = SIk x)
22 1cex 4142 . . . . . . 7 1c V
2322, 22xpkex 4289 . . . . . 6 (1c ×k 1c) V
24 vex 2862 . . . . . 6 y V
2523, 24inex 4105 . . . . 5 ((1c ×k 1c) ∩ y) V
2621, 25syl6eqelr 2442 . . . 4 (zw(⟪{z}, {w}⟫ y ↔ ⟪z, w x) → SIk x V)
2726exlimiv 1634 . . 3 (yzw(⟪{z}, {w}⟫ y ↔ ⟪z, w x) → SIk x V)
283, 27ax-mp 8 . 2 SIk x V
292, 28vtoclg 2914 1 (A VSIk A V)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540  ∃wex 1541   = wceq 1642   ∈ wcel 1710  Vcvv 2859   ∩ cin 3208  {csn 3737  ⟪copk 4057  1cc1c 4134   ×k cxpk 4174   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-xp 4079  ax-cnv 4080  ax-1c 4081  ax-si 4083  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-cnvk 4186  df-sik 4192 This theorem is referenced by:  sikex  4297  imakexg  4299  pw1exg  4302  imagekexg  4311  siexg  4752
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