Theorem pw1equn 4331

Description: A condition for a unit power class to equal a union. (Contributed by SF, 26-Jan-2015.)

Hypotheses

Ref	Expression
pw1equn.1	|- A e. _V
pw1equn.2	|- B e. _V

Assertion

Ref	Expression
pw1equn	|- (~P1 C = (A u. B) <-> E.xE.y(C = (x u. y) /\ A = ~P1 x /\ B = ~P1 y))

Distinct variable groups:   x,A,y   x,B,y   x,C,y

Proof of Theorem pw1equn

Step	Hyp	Ref	Expression
1		unipw1 4325	. . . 4 |- U.~P1 C = C
2		unieq 3900	. . . 4 |- (~P1 C = (A u. B) -> U.~P1 C = U.(A u. B))
3	1, 2	syl5eqr 2399	. . 3 |- (~P1 C = (A u. B) -> C = U.(A u. B))
4		ssun1 3426	. . . . . 6 |- A C_ (A u. B)
5		sseq2 3293	. . . . . 6 |- (~P1 C = (A u. B) -> (A C_ ~P1 C <-> A C_ (A u. B)))
6	4, 5	mpbiri 224	. . . . 5 |- (~P1 C = (A u. B) -> A C_ ~P1 C)
7		pw1ss1c 4158	. . . . 5 |- ~P1 C C_ 1c
8	6, 7	syl6ss 3284	. . . 4 |- (~P1 C = (A u. B) -> A C_ 1c)
9		eqpw1uni 4330	. . . 4 |- (A C_ 1c -> A = ~P1 U.A)
10	8, 9	syl 15	. . 3 |- (~P1 C = (A u. B) -> A = ~P1 U.A)
11		ssun2 3427	. . . . . 6 |- B C_ (A u. B)
12		sseq2 3293	. . . . . 6 |- (~P1 C = (A u. B) -> (B C_ ~P1 C <-> B C_ (A u. B)))
13	11, 12	mpbiri 224	. . . . 5 |- (~P1 C = (A u. B) -> B C_ ~P1 C)
14	13, 7	syl6ss 3284	. . . 4 |- (~P1 C = (A u. B) -> B C_ 1c)
15		eqpw1uni 4330	. . . 4 |- (B C_ 1c -> B = ~P1 U.B)
16	14, 15	syl 15	. . 3 |- (~P1 C = (A u. B) -> B = ~P1 U.B)
17		pw1equn.1	. . . . 5 |- A e. _V
18	17	uniex 4317	. . . 4 |- U.A e. _V
19		pw1equn.2	. . . . 5 |- B e. _V
20	19	uniex 4317	. . . 4 |- U.B e. _V
21		uneq12 3413	. . . . . . 7 |- ((x = U.A /\ y = U.B) -> (x u. y) = (U.A u. U.B))
22		uniun 3910	. . . . . . 7 |- U.(A u. B) = (U.A u. U.B)
23	21, 22	syl6eqr 2403	. . . . . 6 |- ((x = U.A /\ y = U.B) -> (x u. y) = U.(A u. B))
24	23	eqeq2d 2364	. . . . 5 |- ((x = U.A /\ y = U.B) -> (C = (x u. y) <-> C = U.(A u. B)))
25		pw1eq 4143	. . . . . . 7 |- (x = U.A -> ~P1 x = ~P1 U.A)
26	25	eqeq2d 2364	. . . . . 6 |- (x = U.A -> (A = ~P1 x <-> A = ~P1 U.A))
27	26	adantr 451	. . . . 5 |- ((x = U.A /\ y = U.B) -> (A = ~P1 x <-> A = ~P1 U.A))
28		pw1eq 4143	. . . . . . 7 |- (y = U.B -> ~P1 y = ~P1 U.B)
29	28	eqeq2d 2364	. . . . . 6 |- (y = U.B -> (B = ~P1 y <-> B = ~P1 U.B))
30	29	adantl 452	. . . . 5 |- ((x = U.A /\ y = U.B) -> (B = ~P1 y <-> B = ~P1 U.B))
31	24, 27, 30	3anbi123d 1252	. . . 4 |- ((x = U.A /\ y = U.B) -> ((C = (x u. y) /\ A = ~P1 x /\ B = ~P1 y) <-> (C = U.(A u. B) /\ A = ~P1 U.A /\ B = ~P1 U.B)))
32	18, 20, 31	spc2ev 2947	. . 3 |- ((C = U.(A u. B) /\ A = ~P1 U.A /\ B = ~P1 U.B) -> E.xE.y(C = (x u. y) /\ A = ~P1 x /\ B = ~P1 y))
33	3, 10, 16, 32	syl3anc 1182	. 2 |- (~P1 C = (A u. B) -> E.xE.y(C = (x u. y) /\ A = ~P1 x /\ B = ~P1 y))
34		pw1un 4163	. . . 4 |- ~P1 (x u. y) = (~P1 x u. ~P1 y)
35		pw1eq 4143	. . . . . 6 |- (C = (x u. y) -> ~P1 C = ~P1 (x u. y))
36		uneq12 3413	. . . . . 6 |- ((A = ~P1 x /\ B = ~P1 y) -> (A u. B) = (~P1 x u. ~P1 y))
37	35, 36	eqeqan12d 2368	. . . . 5 |- ((C = (x u. y) /\ (A = ~P1 x /\ B = ~P1 y)) -> (~P1 C = (A u. B) <-> ~P1 (x u. y) = (~P1 x u. ~P1 y)))
38	37	3impb 1147	. . . 4 |- ((C = (x u. y) /\ A = ~P1 x /\ B = ~P1 y) -> (~P1 C = (A u. B) <-> ~P1 (x u. y) = (~P1 x u. ~P1 y)))
39	34, 38	mpbiri 224	. . 3 |- ((C = (x u. y) /\ A = ~P1 x /\ B = ~P1 y) -> ~P1 C = (A u. B))
40	39	exlimivv 1635	. 2 |- (E.xE.y(C = (x u. y) /\ A = ~P1 x /\ B = ~P1 y) -> ~P1 C = (A u. B))
41	33, 40	impbii 180	1 |- (~P1 C = (A u. B) <-> E.xE.y(C = (x u. y) /\ A = ~P1 x /\ B = ~P1 y))

Syntax hints:   <-> wb 176   /\ wa 358   /\ w3a 934  E.wex 1541   = wceq 1642   e. wcel 1710  _Vcvv 2859   u. cun 3207   C_ wss 3257  U.cuni 3891  1_cc1c 4134  ~P₁ cpw1 4135

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 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-typlower 4086  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-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-imak 4189  df-p6 4191  df-sik 4192  df-ssetk 4193

This theorem is referenced by:  taddc  6229  letc  6231