Theorem dfpw12 4302

Description: Alternate expression for unit power classes. (Contributed by SF, 26-Jan-2015.)

Assertion

Ref	Expression
dfpw12	⊢ ℘1A = ( SIk (A ×k A) “k V)

Proof of Theorem dfpw12

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

Step	Hyp	Ref	Expression
1		elpw1 4145	. . 3 ⊢ (x ∈ ℘1A ↔ ∃y ∈ A x = {y})
2		vex 2863	. . . . 5 ⊢ x ∈ V
3	2	elimakv 4261	. . . 4 ⊢ (x ∈ ( SIk (A ×k A) “k V) ↔ ∃z⟪z, x⟫ ∈ SIk (A ×k A))
4		vex 2863	. . . . . . 7 ⊢ z ∈ V
5		opkelsikg 4265	. . . . . . 7 ⊢ ((z ∈ V ∧ x ∈ V) → (⟪z, x⟫ ∈ SIk (A ×k A) ↔ ∃w∃y(z = {w} ∧ x = {y} ∧ ⟪w, y⟫ ∈ (A ×k A))))
6	4, 2, 5	mp2an 653	. . . . . 6 ⊢ (⟪z, x⟫ ∈ SIk (A ×k A) ↔ ∃w∃y(z = {w} ∧ x = {y} ∧ ⟪w, y⟫ ∈ (A ×k A)))
7	6	exbii 1582	. . . . 5 ⊢ (∃z⟪z, x⟫ ∈ SIk (A ×k A) ↔ ∃z∃w∃y(z = {w} ∧ x = {y} ∧ ⟪w, y⟫ ∈ (A ×k A)))
8		exrot3 1744	. . . . 5 ⊢ (∃y∃z∃w(z = {w} ∧ x = {y} ∧ ⟪w, y⟫ ∈ (A ×k A)) ↔ ∃z∃w∃y(z = {w} ∧ x = {y} ∧ ⟪w, y⟫ ∈ (A ×k A)))
9	7, 8	bitr4i 243	. . . 4 ⊢ (∃z⟪z, x⟫ ∈ SIk (A ×k A) ↔ ∃y∃z∃w(z = {w} ∧ x = {y} ∧ ⟪w, y⟫ ∈ (A ×k A)))
10		df-3an 936	. . . . . . . . 9 ⊢ ((z = {w} ∧ x = {y} ∧ ⟪w, y⟫ ∈ (A ×k A)) ↔ ((z = {w} ∧ x = {y}) ∧ ⟪w, y⟫ ∈ (A ×k A)))
11		vex 2863	. . . . . . . . . . 11 ⊢ w ∈ V
12		vex 2863	. . . . . . . . . . 11 ⊢ y ∈ V
13	11, 12	opkelxpk 4249	. . . . . . . . . 10 ⊢ (⟪w, y⟫ ∈ (A ×k A) ↔ (w ∈ A ∧ y ∈ A))
14	13	anbi2i 675	. . . . . . . . 9 ⊢ (((z = {w} ∧ x = {y}) ∧ ⟪w, y⟫ ∈ (A ×k A)) ↔ ((z = {w} ∧ x = {y}) ∧ (w ∈ A ∧ y ∈ A)))
15		an4 797	. . . . . . . . 9 ⊢ (((z = {w} ∧ x = {y}) ∧ (w ∈ A ∧ y ∈ A)) ↔ ((z = {w} ∧ w ∈ A) ∧ (x = {y} ∧ y ∈ A)))
16	10, 14, 15	3bitri 262	. . . . . . . 8 ⊢ ((z = {w} ∧ x = {y} ∧ ⟪w, y⟫ ∈ (A ×k A)) ↔ ((z = {w} ∧ w ∈ A) ∧ (x = {y} ∧ y ∈ A)))
17	16	2exbii 1583	. . . . . . 7 ⊢ (∃z∃w(z = {w} ∧ x = {y} ∧ ⟪w, y⟫ ∈ (A ×k A)) ↔ ∃z∃w((z = {w} ∧ w ∈ A) ∧ (x = {y} ∧ y ∈ A)))
18		19.41vv 1902	. . . . . . 7 ⊢ (∃z∃w((z = {w} ∧ w ∈ A) ∧ (x = {y} ∧ y ∈ A)) ↔ (∃z∃w(z = {w} ∧ w ∈ A) ∧ (x = {y} ∧ y ∈ A)))
19		sneq 3745	. . . . . . . . . . . 12 ⊢ (w = y → {w} = {y})
20		eqeq12 2365	. . . . . . . . . . . 12 ⊢ ((z = x ∧ {w} = {y}) → (z = {w} ↔ x = {y}))
21	19, 20	sylan2 460	. . . . . . . . . . 11 ⊢ ((z = x ∧ w = y) → (z = {w} ↔ x = {y}))
22		eleq1 2413	. . . . . . . . . . . 12 ⊢ (w = y → (w ∈ A ↔ y ∈ A))
23	22	adantl 452	. . . . . . . . . . 11 ⊢ ((z = x ∧ w = y) → (w ∈ A ↔ y ∈ A))
24	21, 23	anbi12d 691	. . . . . . . . . 10 ⊢ ((z = x ∧ w = y) → ((z = {w} ∧ w ∈ A) ↔ (x = {y} ∧ y ∈ A)))
25	2, 12, 24	spc2ev 2948	. . . . . . . . 9 ⊢ ((x = {y} ∧ y ∈ A) → ∃z∃w(z = {w} ∧ w ∈ A))
26	25	pm4.71ri 614	. . . . . . . 8 ⊢ ((x = {y} ∧ y ∈ A) ↔ (∃z∃w(z = {w} ∧ w ∈ A) ∧ (x = {y} ∧ y ∈ A)))
27		ancom 437	. . . . . . . 8 ⊢ ((x = {y} ∧ y ∈ A) ↔ (y ∈ A ∧ x = {y}))
28	26, 27	bitr3i 242	. . . . . . 7 ⊢ ((∃z∃w(z = {w} ∧ w ∈ A) ∧ (x = {y} ∧ y ∈ A)) ↔ (y ∈ A ∧ x = {y}))
29	17, 18, 28	3bitri 262	. . . . . 6 ⊢ (∃z∃w(z = {w} ∧ x = {y} ∧ ⟪w, y⟫ ∈ (A ×k A)) ↔ (y ∈ A ∧ x = {y}))
30	29	exbii 1582	. . . . 5 ⊢ (∃y∃z∃w(z = {w} ∧ x = {y} ∧ ⟪w, y⟫ ∈ (A ×k A)) ↔ ∃y(y ∈ A ∧ x = {y}))
31		df-rex 2621	. . . . 5 ⊢ (∃y ∈ A x = {y} ↔ ∃y(y ∈ A ∧ x = {y}))
32	30, 31	bitr4i 243	. . . 4 ⊢ (∃y∃z∃w(z = {w} ∧ x = {y} ∧ ⟪w, y⟫ ∈ (A ×k A)) ↔ ∃y ∈ A x = {y})
33	3, 9, 32	3bitri 262	. . 3 ⊢ (x ∈ ( SIk (A ×k A) “k V) ↔ ∃y ∈ A x = {y})
34	1, 33	bitr4i 243	. 2 ⊢ (x ∈ ℘1A ↔ x ∈ ( SIk (A ×k A) “k V))
35	34	eqriv 2350	1 ⊢ ℘1A = ( SIk (A ×k A) “k V)

This proof depends on syntax axioms:   ↔ wb 176   ∧ wa 358   ∧ w3a 934  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃wrex 2616  Vcvv 2860  {csn 3738  ⟪copk 4058  ℘₁cpw1 4136   ×_k cxpk 4175   “_k cimak 4180   SI_k csik 4182

This proof depends on 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-sn 4088

This proof 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-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-pw 3725  df-sn 3742  df-pr 3743  df-opk 4059  df-1c 4137  df-pw1 4138  df-xpk 4186  df-imak 4190  df-sik 4193

This theorem is used by:  pw1exg  4303