Theorem nenpw1pwlem2 6086

Description: Lemma for nenpw1pw 6087. Establish the main theorem with an extra hypothesis. (Contributed by SF, 10-Mar-2015.)

Hypothesis

Ref	Expression
nenpw1pwlem2.1	|- S = {x e. A | -. x e. (r` {x})}

Assertion

Ref	Expression
nenpw1pwlem2	|- -. ~P1 A ~~ ~PA

Distinct variable group:   A,r,x

Allowed substitution hints:   S(x,r)

Proof of Theorem nenpw1pwlem2

Dummy variables u y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		pm5.19 349	. . . . 5 |- -. (y e. S <-> -. y e. S)
2	1	a1i 10	. . . 4 |- (y e. A -> -. (y e. S <-> -. y e. S))
3	2	nrex 2717	. . 3 |- -. E.y e. A (y e. S <-> -. y e. S)
4	3	nex 1555	. 2 |- -. E.rE.y e. A (y e. S <-> -. y e. S)
5		bren 6031	. . 3 |- (~P1 A ~~ ~PA <-> E.r r:~P1 A-1-1-onto->~PA)
6		f1odm 5291	. . . . . . . . 9 |- (r:~P1 A-1-1-onto->~PA -> dom r = ~P1 A)
7		vex 2863	. . . . . . . . . 10 |- r e. _V
8	7	dmex 5107	. . . . . . . . 9 |- dom r e. _V
9	6, 8	syl6eqelr 2442	. . . . . . . 8 |- (r:~P1 A-1-1-onto->~PA -> ~P1 A e. _V)
10		pw1exb 4327	. . . . . . . 8 |- (~P1 A e. _V <-> A e. _V)
11	9, 10	sylib 188	. . . . . . 7 |- (r:~P1 A-1-1-onto->~PA -> A e. _V)
12		nenpw1pwlem2.1	. . . . . . . 8 |- S = {x e. A | -. x e. (r` {x})}
13	12	nenpw1pwlem1 6085	. . . . . . 7 |- (A e. _V -> S e. _V)
14	11, 13	syl 15	. . . . . 6 |- (r:~P1 A-1-1-onto->~PA -> S e. _V)
15		ssrab2 3352	. . . . . . . 8 |- {x e. A | -. x e. (r` {x})} C_ A
16	12, 15	eqsstri 3302	. . . . . . 7 |- S C_ A
17		elpwg 3730	. . . . . . 7 |- (S e. _V -> (S e. ~PA <-> S C_ A))
18	16, 17	mpbiri 224	. . . . . 6 |- (S e. _V -> S e. ~PA)
19	14, 18	syl 15	. . . . 5 |- (r:~P1 A-1-1-onto->~PA -> S e. ~PA)
20		f1ofo 5294	. . . . . . . 8 |- (r:~P1 A-1-1-onto->~PA -> r:~P1 A-onto->~PA)
21		forn 5273	. . . . . . . 8 |- (r:~P1 A-onto->~PA -> ran r = ~PA)
22	20, 21	syl 15	. . . . . . 7 |- (r:~P1 A-1-1-onto->~PA -> ran r = ~PA)
23	22	eleq2d 2420	. . . . . 6 |- (r:~P1 A-1-1-onto->~PA -> (S e. ran r <-> S e. ~PA))
24		elrn 4897	. . . . . . 7 |- (S e. ran r <-> E.u urS)
25		breldm 4912	. . . . . . . . . . . 12 |- (urS -> u e. dom r)
26	25	adantl 452	. . . . . . . . . . 11 |- ((r:~P1 A-1-1-onto->~PA /\ urS) -> u e. dom r)
27	6	adantr 451	. . . . . . . . . . 11 |- ((r:~P1 A-1-1-onto->~PA /\ urS) -> dom r = ~P1 A)
28	26, 27	eleqtrd 2429	. . . . . . . . . 10 |- ((r:~P1 A-1-1-onto->~PA /\ urS) -> u e. ~P1 A)
29		elpw1 4145	. . . . . . . . . . 11 |- (u e. ~P1 A <-> E.y e. A u = {y})
30		breq1 4643	. . . . . . . . . . . . . . . 16 |- (u = {y} -> (urS <-> {y}rS))
31	30	3anbi2d 1257	. . . . . . . . . . . . . . 15 |- (u = {y} -> ((r:~P1 A-1-1-onto->~PA /\ urS /\ y e. A) <-> (r:~P1 A-1-1-onto->~PA /\ {y}rS /\ y e. A)))
32		id 19	. . . . . . . . . . . . . . . . . . 19 |- (x = y -> x = y)
33		sneq 3745	. . . . . . . . . . . . . . . . . . . 20 |- (x = y -> {x} = {y})
34	33	fveq2d 5333	. . . . . . . . . . . . . . . . . . 19 |- (x = y -> (r` {x}) = (r` {y}))
35	32, 34	eleq12d 2421	. . . . . . . . . . . . . . . . . 18 |- (x = y -> (x e. (r` {x}) <-> y e. (r` {y})))
36	35	notbid 285	. . . . . . . . . . . . . . . . 17 |- (x = y -> (-. x e. (r` {x}) <-> -. y e. (r` {y})))
37	36, 12	elrab2 2997	. . . . . . . . . . . . . . . 16 |- (y e. S <-> (y e. A /\ -. y e. (r` {y})))
38		simp3 957	. . . . . . . . . . . . . . . . . 18 |- ((r:~P1 A-1-1-onto->~PA /\ {y}rS /\ y e. A) -> y e. A)
39	38	biantrurd 494	. . . . . . . . . . . . . . . . 17 |- ((r:~P1 A-1-1-onto->~PA /\ {y}rS /\ y e. A) -> (-. y e. (r` {y}) <-> (y e. A /\ -. y e. (r` {y}))))
40		simp2 956	. . . . . . . . . . . . . . . . . . . 20 |- ((r:~P1 A-1-1-onto->~PA /\ {y}rS /\ y e. A) -> {y}rS)
41		f1ofn 5289	. . . . . . . . . . . . . . . . . . . . . 22 |- (r:~P1 A-1-1-onto->~PA -> r Fn ~P1 A)
42	41	3ad2ant1 976	. . . . . . . . . . . . . . . . . . . . 21 |- ((r:~P1 A-1-1-onto->~PA /\ {y}rS /\ y e. A) -> r Fn ~P1 A)
43		snelpw1 4147	. . . . . . . . . . . . . . . . . . . . . . 23 |- ({y} e. ~P1 A <-> y e. A)
44	43	biimpri 197	. . . . . . . . . . . . . . . . . . . . . 22 |- (y e. A -> {y} e. ~P1 A)
45	44	3ad2ant3 978	. . . . . . . . . . . . . . . . . . . . 21 |- ((r:~P1 A-1-1-onto->~PA /\ {y}rS /\ y e. A) -> {y} e. ~P1 A)
46		fnbrfvb 5359	. . . . . . . . . . . . . . . . . . . . 21 |- ((r Fn ~P1 A /\ {y} e. ~P1 A) -> ((r` {y}) = S <-> {y}rS))
47	42, 45, 46	syl2anc 642	. . . . . . . . . . . . . . . . . . . 20 |- ((r:~P1 A-1-1-onto->~PA /\ {y}rS /\ y e. A) -> ((r` {y}) = S <-> {y}rS))
48	40, 47	mpbird 223	. . . . . . . . . . . . . . . . . . 19 |- ((r:~P1 A-1-1-onto->~PA /\ {y}rS /\ y e. A) -> (r` {y}) = S)
49	48	eleq2d 2420	. . . . . . . . . . . . . . . . . 18 |- ((r:~P1 A-1-1-onto->~PA /\ {y}rS /\ y e. A) -> (y e. (r` {y}) <-> y e. S))
50	49	notbid 285	. . . . . . . . . . . . . . . . 17 |- ((r:~P1 A-1-1-onto->~PA /\ {y}rS /\ y e. A) -> (-. y e. (r` {y}) <-> -. y e. S))
51	39, 50	bitr3d 246	. . . . . . . . . . . . . . . 16 |- ((r:~P1 A-1-1-onto->~PA /\ {y}rS /\ y e. A) -> ((y e. A /\ -. y e. (r` {y})) <-> -. y e. S))
52	37, 51	syl5bb 248	. . . . . . . . . . . . . . 15 |- ((r:~P1 A-1-1-onto->~PA /\ {y}rS /\ y e. A) -> (y e. S <-> -. y e. S))
53	31, 52	syl6bi 219	. . . . . . . . . . . . . 14 |- (u = {y} -> ((r:~P1 A-1-1-onto->~PA /\ urS /\ y e. A) -> (y e. S <-> -. y e. S)))
54	53	com12 27	. . . . . . . . . . . . 13 |- ((r:~P1 A-1-1-onto->~PA /\ urS /\ y e. A) -> (u = {y} -> (y e. S <-> -. y e. S)))
55	54	3expa 1151	. . . . . . . . . . . 12 |- (((r:~P1 A-1-1-onto->~PA /\ urS) /\ y e. A) -> (u = {y} -> (y e. S <-> -. y e. S)))
56	55	reximdva 2727	. . . . . . . . . . 11 |- ((r:~P1 A-1-1-onto->~PA /\ urS) -> (E.y e. A u = {y} -> E.y e. A (y e. S <-> -. y e. S)))
57	29, 56	syl5bi 208	. . . . . . . . . 10 |- ((r:~P1 A-1-1-onto->~PA /\ urS) -> (u e. ~P1 A -> E.y e. A (y e. S <-> -. y e. S)))
58	28, 57	mpd 14	. . . . . . . . 9 |- ((r:~P1 A-1-1-onto->~PA /\ urS) -> E.y e. A (y e. S <-> -. y e. S))
59	58	ex 423	. . . . . . . 8 |- (r:~P1 A-1-1-onto->~PA -> (urS -> E.y e. A (y e. S <-> -. y e. S)))
60	59	exlimdv 1636	. . . . . . 7 |- (r:~P1 A-1-1-onto->~PA -> (E.u urS -> E.y e. A (y e. S <-> -. y e. S)))
61	24, 60	syl5bi 208	. . . . . 6 |- (r:~P1 A-1-1-onto->~PA -> (S e. ran r -> E.y e. A (y e. S <-> -. y e. S)))
62	23, 61	sylbird 226	. . . . 5 |- (r:~P1 A-1-1-onto->~PA -> (S e. ~PA -> E.y e. A (y e. S <-> -. y e. S)))
63	19, 62	mpd 14	. . . 4 |- (r:~P1 A-1-1-onto->~PA -> E.y e. A (y e. S <-> -. y e. S))
64	63	eximi 1576	. . 3 |- (E.r r:~P1 A-1-1-onto->~PA -> E.rE.y e. A (y e. S <-> -. y e. S))
65	5, 64	sylbi 187	. 2 |- (~P1 A ~~ ~PA -> E.rE.y e. A (y e. S <-> -. y e. S))
66	4, 65	mto 167	1 |- -. ~P1 A ~~ ~PA

Syntax hints:  -. wn 3   -> wi 4   <-> wb 176   /\ wa 358   /\ w3a 934  E.wex 1541   = wceq 1642   e. wcel 1710  E.wrex 2616  {crab 2619  _Vcvv 2860   C_ wss 3258  ~Pcpw 3723  {csn 3738  ~P₁ cpw1 4136   class class class wbr 4640  dom cdm 4773  ran crn 4774   Fn wfn 4777  -onto->wfo 4780  -1-1-onto->wf1o 4781  ` cfv 4782   ~~ cen 6029

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-13 1712  ax-14 1714  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-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-1st 4724  df-swap 4725  df-sset 4726  df-co 4727  df-ima 4728  df-id 4768  df-xp 4785  df-cnv 4786  df-rn 4787  df-dm 4788  df-res 4789  df-fun 4790  df-fn 4791  df-f 4792  df-f1 4793  df-fo 4794  df-f1o 4795  df-fv 4796  df-2nd 4798  df-fullfun 5769  df-en 6030

This theorem is referenced by:  nenpw1pw  6087