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 ∈ V → SIk A ∈ 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 ∈ V ↔ SIk A ∈ V))
3		ax-si 4084	. . 3 ⊢ ∃y∀z∀w(⟪{z}, {w}⟫ ∈ y ↔ ⟪z, w⟫ ∈ x)
4		inss1 3476	. . . . . . . 8 ⊢ ((1c ×k 1c) ∩ y) ⊆ (1c ×k 1c)
5		sikss1c1c 4268	. . . . . . . 8 ⊢ SIk x ⊆ (1c ×k 1c)
6	4, 5	sikexlem 4296	. . . . . . 7 ⊢ (((1c ×k 1c) ∩ y) = SIk x ↔ ∀z∀w(⟪{z}, {w}⟫ ∈ ((1c ×k 1c) ∩ y) ↔ ⟪{z}, {w}⟫ ∈ SIk x))
7		vex 2863	. . . . . . . . . . . 12 ⊢ z ∈ V
8	7	snel1c 4141	. . . . . . . . . . 11 ⊢ {z} ∈ 1c
9		vex 2863	. . . . . . . . . . . 12 ⊢ w ∈ V
10	9	snel1c 4141	. . . . . . . . . . 11 ⊢ {w} ∈ 1c
11		snex 4112	. . . . . . . . . . . 12 ⊢ {z} ∈ V
12		snex 4112	. . . . . . . . . . . 12 ⊢ {w} ∈ V
13	11, 12	opkelxpk 4249	. . . . . . . . . . 11 ⊢ (⟪{z}, {w}⟫ ∈ (1c ×k 1c) ↔ ({z} ∈ 1c ∧ {w} ∈ 1c))
14	8, 10, 13	mpbir2an 886	. . . . . . . . . 10 ⊢ ⟪{z}, {w}⟫ ∈ (1c ×k 1c)
15		elin 3220	. . . . . . . . . 10 ⊢ (⟪{z}, {w}⟫ ∈ ((1c ×k 1c) ∩ y) ↔ (⟪{z}, {w}⟫ ∈ (1c ×k 1c) ∧ ⟪{z}, {w}⟫ ∈ y))
16	14, 15	mpbiran 884	. . . . . . . . 9 ⊢ (⟪{z}, {w}⟫ ∈ ((1c ×k 1c) ∩ y) ↔ ⟪{z}, {w}⟫ ∈ y)
17	7, 9	opksnelsik 4266	. . . . . . . . 9 ⊢ (⟪{z}, {w}⟫ ∈ SIk x ↔ ⟪z, w⟫ ∈ x)
18	16, 17	bibi12i 306	. . . . . . . 8 ⊢ ((⟪{z}, {w}⟫ ∈ ((1c ×k 1c) ∩ y) ↔ ⟪{z}, {w}⟫ ∈ SIk x) ↔ (⟪{z}, {w}⟫ ∈ y ↔ ⟪z, w⟫ ∈ x))
19	18	2albii 1567	. . . . . . 7 ⊢ (∀z∀w(⟪{z}, {w}⟫ ∈ ((1c ×k 1c) ∩ y) ↔ ⟪{z}, {w}⟫ ∈ SIk x) ↔ ∀z∀w(⟪{z}, {w}⟫ ∈ y ↔ ⟪z, w⟫ ∈ x))
20	6, 19	bitri 240	. . . . . 6 ⊢ (((1c ×k 1c) ∩ y) = SIk x ↔ ∀z∀w(⟪{z}, {w}⟫ ∈ y ↔ ⟪z, w⟫ ∈ x))
21	20	biimpri 197	. . . . 5 ⊢ (∀z∀w(⟪{z}, {w}⟫ ∈ y ↔ ⟪z, w⟫ ∈ x) → ((1c ×k 1c) ∩ y) = SIk x)
22		1cex 4143	. . . . . . 7 ⊢ 1c ∈ V
23	22, 22	xpkex 4290	. . . . . 6 ⊢ (1c ×k 1c) ∈ V
24		vex 2863	. . . . . 6 ⊢ y ∈ V
25	23, 24	inex 4106	. . . . 5 ⊢ ((1c ×k 1c) ∩ y) ∈ V
26	21, 25	syl6eqelr 2442	. . . 4 ⊢ (∀z∀w(⟪{z}, {w}⟫ ∈ y ↔ ⟪z, w⟫ ∈ x) → SIk x ∈ V)
27	26	exlimiv 1634	. . 3 ⊢ (∃y∀z∀w(⟪{z}, {w}⟫ ∈ y ↔ ⟪z, w⟫ ∈ x) → SIk x ∈ V)
28	3, 27	ax-mp 5	. 2 ⊢ SIk x ∈ V
29	2, 28	vtoclg 2915	1 ⊢ (A ∈ V → SIk A ∈ V)

Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540  ∃wex 1541   = wceq 1642   ∈ wcel 1710  Vcvv 2860   ∩ cin 3209  {csn 3738  ⟪copk 4058  1_cc1c 4135   ×_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