Theorem sikexg 4297

Description: The Kuratowski singleton image of a set is a set. (Contributed by SF, 14-Jan-2015.)

Assertion

Ref	Expression
sikexg	|- (A e. V -> SIk A e. _V)

Proof of Theorem sikexg

Dummy variables x y z w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		sikeq 4242	. . 3 |- (x = A -> SIk x = SIk A)
2	1	eleq1d 2419	. 2 |- (x = A -> ( SIk x e. _V <-> SIk A e. _V))
3		ax-si 4084	. . 3 |- E.yA.zA.w(<<{z}, {w}>> e. y <-> <<z, w>> e. x)
4		inss1 3476	. . . . . . . 8 |- ((1c X.k 1c) i^i y) C_ (1c X.k 1c)
5		sikss1c1c 4268	. . . . . . . 8 |- SIk x C_ (1c X.k 1c)
6	4, 5	sikexlem 4296	. . . . . . 7 |- (((1c X.k 1c) i^i y) = SIk x <-> A.zA.w(<<{z}, {w}>> e. ((1c X.k 1c) i^i y) <-> <<{z}, {w}>> e. SIk x))
7		vex 2863	. . . . . . . . . . . 12 |- z e. _V
8	7	snel1c 4141	. . . . . . . . . . 11 |- {z} e. 1c
9		vex 2863	. . . . . . . . . . . 12 |- w e. _V
10	9	snel1c 4141	. . . . . . . . . . 11 |- {w} e. 1c
11		snex 4112	. . . . . . . . . . . 12 |- {z} e. _V
12		snex 4112	. . . . . . . . . . . 12 |- {w} e. _V
13	11, 12	opkelxpk 4249	. . . . . . . . . . 11 |- (<<{z}, {w}>> e. (1c X.k 1c) <-> ({z} e. 1c /\ {w} e. 1c))
14	8, 10, 13	mpbir2an 886	. . . . . . . . . 10 |- <<{z}, {w}>> e. (1c X.k 1c)
15		elin 3220	. . . . . . . . . 10 |- (<<{z}, {w}>> e. ((1c X.k 1c) i^i y) <-> (<<{z}, {w}>> e. (1c X.k 1c) /\ <<{z}, {w}>> e. y))
16	14, 15	mpbiran 884	. . . . . . . . 9 |- (<<{z}, {w}>> e. ((1c X.k 1c) i^i y) <-> <<{z}, {w}>> e. y)
17	7, 9	opksnelsik 4266	. . . . . . . . 9 |- (<<{z}, {w}>> e. SIk x <-> <<z, w>> e. x)
18	16, 17	bibi12i 306	. . . . . . . 8 |- ((<<{z}, {w}>> e. ((1c X.k 1c) i^i y) <-> <<{z}, {w}>> e. SIk x) <-> (<<{z}, {w}>> e. y <-> <<z, w>> e. x))
19	18	2albii 1567	. . . . . . 7 |- (A.zA.w(<<{z}, {w}>> e. ((1c X.k 1c) i^i y) <-> <<{z}, {w}>> e. SIk x) <-> A.zA.w(<<{z}, {w}>> e. y <-> <<z, w>> e. x))
20	6, 19	bitri 240	. . . . . 6 |- (((1c X.k 1c) i^i y) = SIk x <-> A.zA.w(<<{z}, {w}>> e. y <-> <<z, w>> e. x))
21	20	biimpri 197	. . . . 5 |- (A.zA.w(<<{z}, {w}>> e. y <-> <<z, w>> e. x) -> ((1c X.k 1c) i^i y) = SIk x)
22		1cex 4143	. . . . . . 7 |- 1c e. _V
23	22, 22	xpkex 4290	. . . . . 6 |- (1c X.k 1c) e. _V
24		vex 2863	. . . . . 6 |- y e. _V
25	23, 24	inex 4106	. . . . 5 |- ((1c X.k 1c) i^i y) e. _V
26	21, 25	syl6eqelr 2442	. . . 4 |- (A.zA.w(<<{z}, {w}>> e. y <-> <<z, w>> e. x) -> SIk x e. _V)
27	26	exlimiv 1634	. . 3 |- (E.yA.zA.w(<<{z}, {w}>> e. y <-> <<z, w>> e. x) -> SIk x e. _V)
28	3, 27	ax-mp 5	. 2 |- SIk x e. _V
29	2, 28	vtoclg 2915	1 |- (A e. V -> SIk A e. _V)

Syntax hints:   -> wi 4   <-> wb 176  A.wal 1540  E.wex 1541   = wceq 1642   e. wcel 1710  _Vcvv 2860   i^i cin 3209  {csn 3738  <<copk 4058  1_cc1c 4135   X._k cxpk 4175   SI_k csik 4182

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-si 4084  ax-sn 4088

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 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-ss 3260  df-nul 3552  df-sn 3742  df-pr 3743  df-opk 4059  df-1c 4137  df-xpk 4186  df-cnvk 4187  df-sik 4193

This theorem is referenced by:  sikex  4298  imakexg  4300  pw1exg  4303  imagekexg  4312  siexg  4753