Theorem eqpw1relk 4480

Description: Represent equality to unit power class via a Kuratowski relationship. (Contributed by SF, 21-Jan-2015.)

Hypotheses

Ref	Expression
eqpw1relk.1	|- A e. _V
eqpw1relk.2	|- B e. _V

Assertion

Ref	Expression
eqpw1relk	|- (<<A, {B}>> e. ((~P1c X.k _V) \ (( Ins3k Sk (+) Ins2k SIk Sk )"k~P1 ~P1 ~P1 1c)) <-> A = ~P1 B)

Proof of Theorem eqpw1relk

Dummy variables x t are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		snex 4112	. . . . 5 |- {B} e. _V
2		eqpw1relk.1	. . . . . 6 |- A e. _V
3	2, 1	opkelxpk 4249	. . . . 5 |- (<<A, {B}>> e. (~P1c X.k _V) <-> (A e. ~P1c /\ {B} e. _V))
4	1, 3	mpbiran2 885	. . . 4 |- (<<A, {B}>> e. (~P1c X.k _V) <-> A e. ~P1c)
5	2	elpw 3729	. . . 4 |- (A e. ~P1c <-> A C_ 1c)
6	4, 5	bitri 240	. . 3 |- (<<A, {B}>> e. (~P1c X.k _V) <-> A C_ 1c)
7		opkex 4114	. . . . . . 7 |- <<A, {B}>> e. _V
8	7	elimak 4260	. . . . . 6 |- (<<A, {B}>> e. (( Ins3k Sk (+) Ins2k SIk Sk )"k~P1 ~P1 ~P1 1c) <-> E.t e. ~P1 ~P1 ~P1 1c<<t, <<A, {B}>>>> e. ( Ins3k Sk (+) Ins2k SIk Sk ))
9		elpw131c 4150	. . . . . . . . . 10 |- (t e. ~P1 ~P1 ~P1 1c <-> E.x t = {{{{x}}}})
10	9	anbi1i 676	. . . . . . . . 9 |- ((t e. ~P1 ~P1 ~P1 1c /\ <<t, <<A, {B}>>>> e. ( Ins3k Sk (+) Ins2k SIk Sk )) <-> (E.x t = {{{{x}}}} /\ <<t, <<A, {B}>>>> e. ( Ins3k Sk (+) Ins2k SIk Sk )))
11		19.41v 1901	. . . . . . . . 9 |- (E.x(t = {{{{x}}}} /\ <<t, <<A, {B}>>>> e. ( Ins3k Sk (+) Ins2k SIk Sk )) <-> (E.x t = {{{{x}}}} /\ <<t, <<A, {B}>>>> e. ( Ins3k Sk (+) Ins2k SIk Sk )))
12	10, 11	bitr4i 243	. . . . . . . 8 |- ((t e. ~P1 ~P1 ~P1 1c /\ <<t, <<A, {B}>>>> e. ( Ins3k Sk (+) Ins2k SIk Sk )) <-> E.x(t = {{{{x}}}} /\ <<t, <<A, {B}>>>> e. ( Ins3k Sk (+) Ins2k SIk Sk )))
13	12	exbii 1582	. . . . . . 7 |- (E.t(t e. ~P1 ~P1 ~P1 1c /\ <<t, <<A, {B}>>>> e. ( Ins3k Sk (+) Ins2k SIk Sk )) <-> E.tE.x(t = {{{{x}}}} /\ <<t, <<A, {B}>>>> e. ( Ins3k Sk (+) Ins2k SIk Sk )))
14		df-rex 2621	. . . . . . 7 |- (E.t e. ~P1 ~P1 ~P1 1c<<t, <<A, {B}>>>> e. ( Ins3k Sk (+) Ins2k SIk Sk ) <-> E.t(t e. ~P1 ~P1 ~P1 1c /\ <<t, <<A, {B}>>>> e. ( Ins3k Sk (+) Ins2k SIk Sk )))
15		excom 1741	. . . . . . 7 |- (E.xE.t(t = {{{{x}}}} /\ <<t, <<A, {B}>>>> e. ( Ins3k Sk (+) Ins2k SIk Sk )) <-> E.tE.x(t = {{{{x}}}} /\ <<t, <<A, {B}>>>> e. ( Ins3k Sk (+) Ins2k SIk Sk )))
16	13, 14, 15	3bitr4i 268	. . . . . 6 |- (E.t e. ~P1 ~P1 ~P1 1c<<t, <<A, {B}>>>> e. ( Ins3k Sk (+) Ins2k SIk Sk ) <-> E.xE.t(t = {{{{x}}}} /\ <<t, <<A, {B}>>>> e. ( Ins3k Sk (+) Ins2k SIk Sk )))
17	8, 16	bitri 240	. . . . 5 |- (<<A, {B}>> e. (( Ins3k Sk (+) Ins2k SIk Sk )"k~P1 ~P1 ~P1 1c) <-> E.xE.t(t = {{{{x}}}} /\ <<t, <<A, {B}>>>> e. ( Ins3k Sk (+) Ins2k SIk Sk )))
18		snex 4112	. . . . . . . 8 |- {{{{x}}}} e. _V
19		opkeq1 4060	. . . . . . . . 9 |- (t = {{{{x}}}} -> <<t, <<A, {B}>>>> = <<{{{{x}}}}, <<A, {B}>>>>)
20	19	eleq1d 2419	. . . . . . . 8 |- (t = {{{{x}}}} -> (<<t, <<A, {B}>>>> e. ( Ins3k Sk (+) Ins2k SIk Sk ) <-> <<{{{{x}}}}, <<A, {B}>>>> e. ( Ins3k Sk (+) Ins2k SIk Sk )))
21	18, 20	ceqsexv 2895	. . . . . . 7 |- (E.t(t = {{{{x}}}} /\ <<t, <<A, {B}>>>> e. ( Ins3k Sk (+) Ins2k SIk Sk )) <-> <<{{{{x}}}}, <<A, {B}>>>> e. ( Ins3k Sk (+) Ins2k SIk Sk ))
22		elsymdif 3224	. . . . . . . 8 |- (<<{{{{x}}}}, <<A, {B}>>>> e. ( Ins3k Sk (+) Ins2k SIk Sk ) <-> -. (<<{{{{x}}}}, <<A, {B}>>>> e. Ins3k Sk <-> <<{{{{x}}}}, <<A, {B}>>>> e. Ins2k SIk Sk ))
23		snex 4112	. . . . . . . . . . 11 |- {{x}} e. _V
24	23, 2, 1	otkelins3k 4257	. . . . . . . . . 10 |- (<<{{{{x}}}}, <<A, {B}>>>> e. Ins3k Sk <-> <<{{x}}, A>> e. Sk )
25		snex 4112	. . . . . . . . . . 11 |- {x} e. _V
26	25, 2	elssetk 4271	. . . . . . . . . 10 |- (<<{{x}}, A>> e. Sk <-> {x} e. A)
27	24, 26	bitri 240	. . . . . . . . 9 |- (<<{{{{x}}}}, <<A, {B}>>>> e. Ins3k Sk <-> {x} e. A)
28	23, 2, 1	otkelins2k 4256	. . . . . . . . . 10 |- (<<{{{{x}}}}, <<A, {B}>>>> e. Ins2k SIk Sk <-> <<{{x}}, {B}>> e. SIk Sk )
29		eqpw1relk.2	. . . . . . . . . . . 12 |- B e. _V
30	25, 29	opksnelsik 4266	. . . . . . . . . . 11 |- (<<{{x}}, {B}>> e. SIk Sk <-> <<{x}, B>> e. Sk )
31		vex 2863	. . . . . . . . . . . 12 |- x e. _V
32	31, 29	elssetk 4271	. . . . . . . . . . 11 |- (<<{x}, B>> e. Sk <-> x e. B)
33	30, 32	bitri 240	. . . . . . . . . 10 |- (<<{{x}}, {B}>> e. SIk Sk <-> x e. B)
34	28, 33	bitri 240	. . . . . . . . 9 |- (<<{{{{x}}}}, <<A, {B}>>>> e. Ins2k SIk Sk <-> x e. B)
35	27, 34	bibi12i 306	. . . . . . . 8 |- ((<<{{{{x}}}}, <<A, {B}>>>> e. Ins3k Sk <-> <<{{{{x}}}}, <<A, {B}>>>> e. Ins2k SIk Sk ) <-> ({x} e. A <-> x e. B))
36	22, 35	xchbinx 301	. . . . . . 7 |- (<<{{{{x}}}}, <<A, {B}>>>> e. ( Ins3k Sk (+) Ins2k SIk Sk ) <-> -. ({x} e. A <-> x e. B))
37	21, 36	bitri 240	. . . . . 6 |- (E.t(t = {{{{x}}}} /\ <<t, <<A, {B}>>>> e. ( Ins3k Sk (+) Ins2k SIk Sk )) <-> -. ({x} e. A <-> x e. B))
38	37	exbii 1582	. . . . 5 |- (E.xE.t(t = {{{{x}}}} /\ <<t, <<A, {B}>>>> e. ( Ins3k Sk (+) Ins2k SIk Sk )) <-> E.x -. ({x} e. A <-> x e. B))
39		exnal 1574	. . . . 5 |- (E.x -. ({x} e. A <-> x e. B) <-> -. A.x({x} e. A <-> x e. B))
40	17, 38, 39	3bitrri 263	. . . 4 |- (-. A.x({x} e. A <-> x e. B) <-> <<A, {B}>> e. (( Ins3k Sk (+) Ins2k SIk Sk )"k~P1 ~P1 ~P1 1c))
41	40	con1bii 321	. . 3 |- (-. <<A, {B}>> e. (( Ins3k Sk (+) Ins2k SIk Sk )"k~P1 ~P1 ~P1 1c) <-> A.x({x} e. A <-> x e. B))
42	6, 41	anbi12i 678	. 2 |- ((<<A, {B}>> e. (~P1c X.k _V) /\ -. <<A, {B}>> e. (( Ins3k Sk (+) Ins2k SIk Sk )"k~P1 ~P1 ~P1 1c)) <-> (A C_ 1c /\ A.x({x} e. A <-> x e. B)))
43		eldif 3222	. 2 |- (<<A, {B}>> e. ((~P1c X.k _V) \ (( Ins3k Sk (+) Ins2k SIk Sk )"k~P1 ~P1 ~P1 1c)) <-> (<<A, {B}>> e. (~P1c X.k _V) /\ -. <<A, {B}>> e. (( Ins3k Sk (+) Ins2k SIk Sk )"k~P1 ~P1 ~P1 1c)))
44		eqpw1 4163	. 2 |- (A = ~P1 B <-> (A C_ 1c /\ A.x({x} e. A <-> x e. B)))
45	42, 43, 44	3bitr4i 268	1 |- (<<A, {B}>> e. ((~P1c X.k _V) \ (( Ins3k Sk (+) Ins2k SIk Sk )"k~P1 ~P1 ~P1 1c)) <-> A = ~P1 B)

Syntax hints:  -. wn 3   <-> wb 176   /\ wa 358  A.wal 1540  E.wex 1541   = wceq 1642   e. wcel 1710  E.wrex 2616  _Vcvv 2860   \ cdif 3207   (+) csymdif 3210   C_ wss 3258  ~Pcpw 3723  {csn 3738  <<copk 4058  1_cc1c 4135  ~P₁ cpw1 4136   X._k cxpk 4175   Ins2_k cins2k 4177   Ins3_k cins3k 4178  "_kcimak 4180   SI_k csik 4182   S_k cssetk 4184

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-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-symdif 3217  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-ins2k 4188  df-ins3k 4189  df-imak 4190  df-sik 4193  df-ssetk 4194

This theorem is referenced by:  ncfinraiselem2  4481  ncfinlowerlem1  4483  eqtfinrelk  4487  srelk  4525