Theorem cusgrafilem2 23356

Description: Lemma 2 for cusgrafi 23358. (Contributed by Alexander van der Vekens, 13-Jan-2018.)

Hypotheses

Ref	Expression
cusgrafi.p	|- P = { x e. ~P V | E. a e. V ( a =/= N /\ x = { a , N } ) }
cusgrafi.f	|- F = ( x e. ( V \ { N } ) |-> { x , N } )

Assertion

Ref	Expression
cusgrafilem2	|- ( ( V e. W /\ N e. V ) -> F : ( V \ { N } ) -1-1-onto-> P )

Distinct variable groups:   N, a, x   V, a, x   x, P   W, a, x

Allowed substitution hints:   P( a)   F( x, a)

Proof of Theorem cusgrafilem2

Dummy variables e v are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eldifsn 3995	. . . . . . 7 |- ( v e. ( V \ { N } ) <-> ( v e. V /\ v =/= N ) )
2		simpl 457	. . . . . . 7 |- ( ( v e. V /\ v =/= N ) -> v e. V )
3	1, 2	sylbi 195	. . . . . 6 |- ( v e. ( V \ { N } ) -> v e. V )
4		simpr 461	. . . . . 6 |- ( ( V e. W /\ N e. V ) -> N e. V )
5		prelpwi 4534	. . . . . 6 |- ( ( v e. V /\ N e. V ) -> { v , N } e. ~P V )
6	3, 4, 5	syl2anr 478	. . . . 5 |- ( ( ( V e. W /\ N e. V ) /\ v e. ( V \ { N } ) ) -> { v , N } e. ~P V )
7	1	biimpi 194	. . . . . . 7 |- ( v e. ( V \ { N } ) -> ( v e. V /\ v =/= N ) )
8	7	adantl 466	. . . . . 6 |- ( ( ( V e. W /\ N e. V ) /\ v e. ( V \ { N } ) ) -> ( v e. V /\ v =/= N ) )
9		simpr 461	. . . . . . . . 9 |- ( ( v e. V /\ v =/= N ) -> v =/= N )
10	1, 9	sylbi 195	. . . . . . . 8 |- ( v e. ( V \ { N } ) -> v =/= N )
11	10	adantl 466	. . . . . . 7 |- ( ( ( V e. W /\ N e. V ) /\ v e. ( V \ { N } ) ) -> v =/= N )
12		eqidd 2439	. . . . . . 7 |- ( ( ( V e. W /\ N e. V ) /\ v e. ( V \ { N } ) ) -> { v , N } = { v , N } )
13	11, 12	jca 532	. . . . . 6 |- ( ( ( V e. W /\ N e. V ) /\ v e. ( V \ { N } ) ) -> ( v =/= N /\ { v , N } = { v , N } ) )
14		neeq1 2611	. . . . . . . . 9 |- ( a = v -> ( a =/= N <-> v =/= N ) )
15		preq1 3949	. . . . . . . . . 10 |- ( a = v -> { a , N } = { v , N } )
16	15	eqeq2d 2449	. . . . . . . . 9 |- ( a = v -> ( { v , N } = { a , N } <-> { v , N } = { v , N } ) )
17	14, 16	anbi12d 710	. . . . . . . 8 |- ( a = v -> ( ( a =/= N /\ { v , N } = { a , N } ) <-> ( v =/= N /\ { v , N } = { v , N } ) ) )
18	17	adantl 466	. . . . . . 7 |- ( ( ( v e. V /\ v =/= N ) /\ a = v ) -> ( ( a =/= N /\ { v , N } = { a , N } ) <-> ( v =/= N /\ { v , N } = { v , N } ) ) )
19	2, 18	rspcedv 3072	. . . . . 6 |- ( ( v e. V /\ v =/= N ) -> ( ( v =/= N /\ { v , N } = { v , N } ) -> E. a e. V ( a =/= N /\ { v , N } = { a , N } ) ) )
20	8, 13, 19	sylc 60	. . . . 5 |- ( ( ( V e. W /\ N e. V ) /\ v e. ( V \ { N } ) ) -> E. a e. V ( a =/= N /\ { v , N } = { a , N } ) )
21		eqeq1 2444	. . . . . . . 8 |- ( x = { v , N } -> ( x = { a , N } <-> { v , N } = { a , N } ) )
22	21	anbi2d 703	. . . . . . 7 |- ( x = { v , N } -> ( ( a =/= N /\ x = { a , N } ) <-> ( a =/= N /\ { v , N } = { a , N } ) ) )
23	22	rexbidv 2731	. . . . . 6 |- ( x = { v , N } -> ( E. a e. V ( a =/= N /\ x = { a , N } ) <-> E. a e. V ( a =/= N /\ { v , N } = { a , N } ) ) )
24		cusgrafi.p	. . . . . 6 |- P = { x e. ~P V | E. a e. V ( a =/= N /\ x = { a , N } ) }
25	23, 24	elrab2 3114	. . . . 5 |- ( { v , N } e. P <-> ( { v , N } e. ~P V /\ E. a e. V ( a =/= N /\ { v , N } = { a , N } ) ) )
26	6, 20, 25	sylanbrc 664	. . . 4 |- ( ( ( V e. W /\ N e. V ) /\ v e. ( V \ { N } ) ) -> { v , N } e. P )
27	26	ralrimiva 2794	. . 3 |- ( ( V e. W /\ N e. V ) -> A. v e. ( V \ { N } ) { v , N } e. P )
28		preq1 3949	. . . . 5 |- ( x = v -> { x , N } = { v , N } )
29	28	eleq1d 2504	. . . 4 |- ( x = v -> ( { x , N } e. P <-> { v , N } e. P ) )
30	29	cbvralv 2942	. . 3 |- ( A. x e. ( V \ { N } ) { x , N } e. P <-> A. v e. ( V \ { N } ) { v , N } e. P )
31	27, 30	sylibr 212	. 2 |- ( ( V e. W /\ N e. V ) -> A. x e. ( V \ { N } ) { x , N } e. P )
32		simpl 457	. . . . . . . . . . 11 |- ( ( a =/= N /\ e = { a , N } ) -> a =/= N )
33	32	anim2i 569	. . . . . . . . . 10 |- ( ( a e. V /\ ( a =/= N /\ e = { a , N } ) ) -> ( a e. V /\ a =/= N ) )
34	33	adantl 466	. . . . . . . . 9 |- ( ( ( ( V e. W /\ N e. V ) /\ e e. ~P V ) /\ ( a e. V /\ ( a =/= N /\ e = { a , N } ) ) ) -> ( a e. V /\ a =/= N ) )
35		eldifsn 3995	. . . . . . . . 9 |- ( a e. ( V \ { N } ) <-> ( a e. V /\ a =/= N ) )
36	34, 35	sylibr 212	. . . . . . . 8 |- ( ( ( ( V e. W /\ N e. V ) /\ e e. ~P V ) /\ ( a e. V /\ ( a =/= N /\ e = { a , N } ) ) ) -> a e. ( V \ { N } ) )
37		eqeq1 2444	. . . . . . . . . . . . . 14 |- ( e = { a , N } -> ( e = { x , N } <-> { a , N } = { x , N } ) )
38	37	adantl 466	. . . . . . . . . . . . 13 |- ( ( a =/= N /\ e = { a , N } ) -> ( e = { x , N } <-> { a , N } = { x , N } ) )
39	38	ad2antlr 726	. . . . . . . . . . . 12 |- ( ( ( a e. V /\ ( a =/= N /\ e = { a , N } ) ) /\ x e. ( V \ { N } ) ) -> ( e = { x , N } <-> { a , N } = { x , N } ) )
40		vex 2970	. . . . . . . . . . . . . 14 |- a e. _V
41		vex 2970	. . . . . . . . . . . . . 14 |- x e. _V
42	40, 41	preqr1 4041	. . . . . . . . . . . . 13 |- ( { a , N } = { x , N } -> a = x )
43	42	eqcomd 2443	. . . . . . . . . . . 12 |- ( { a , N } = { x , N } -> x = a )
44	39, 43	syl6bi 228	. . . . . . . . . . 11 |- ( ( ( a e. V /\ ( a =/= N /\ e = { a , N } ) ) /\ x e. ( V \ { N } ) ) -> ( e = { x , N } -> x = a ) )
45	44	adantll 713	. . . . . . . . . 10 |- ( ( ( ( ( V e. W /\ N e. V ) /\ e e. ~P V ) /\ ( a e. V /\ ( a =/= N /\ e = { a , N } ) ) ) /\ x e. ( V \ { N } ) ) -> ( e = { x , N } -> x = a ) )
46		preq1 3949	. . . . . . . . . . . . . . . 16 |- ( a = x -> { a , N } = { x , N } )
47	46	equcoms 1733	. . . . . . . . . . . . . . 15 |- ( x = a -> { a , N } = { x , N } )
48	47	eqeq2d 2449	. . . . . . . . . . . . . 14 |- ( x = a -> ( e = { a , N } <-> e = { x , N } ) )
49	48	biimpcd 224	. . . . . . . . . . . . 13 |- ( e = { a , N } -> ( x = a -> e = { x , N } ) )
50	49	adantl 466	. . . . . . . . . . . 12 |- ( ( a =/= N /\ e = { a , N } ) -> ( x = a -> e = { x , N } ) )
51	50	adantl 466	. . . . . . . . . . 11 |- ( ( a e. V /\ ( a =/= N /\ e = { a , N } ) ) -> ( x = a -> e = { x , N } ) )
52	51	ad2antlr 726	. . . . . . . . . 10 |- ( ( ( ( ( V e. W /\ N e. V ) /\ e e. ~P V ) /\ ( a e. V /\ ( a =/= N /\ e = { a , N } ) ) ) /\ x e. ( V \ { N } ) ) -> ( x = a -> e = { x , N } ) )
53	45, 52	impbid 191	. . . . . . . . 9 |- ( ( ( ( ( V e. W /\ N e. V ) /\ e e. ~P V ) /\ ( a e. V /\ ( a =/= N /\ e = { a , N } ) ) ) /\ x e. ( V \ { N } ) ) -> ( e = { x , N } <-> x = a ) )
54	53	ralrimiva 2794	. . . . . . . 8 |- ( ( ( ( V e. W /\ N e. V ) /\ e e. ~P V ) /\ ( a e. V /\ ( a =/= N /\ e = { a , N } ) ) ) -> A. x e. ( V \ { N } ) ( e = { x , N } <-> x = a ) )
55	36, 54	jca 532	. . . . . . 7 |- ( ( ( ( V e. W /\ N e. V ) /\ e e. ~P V ) /\ ( a e. V /\ ( a =/= N /\ e = { a , N } ) ) ) -> ( a e. ( V \ { N } ) /\ A. x e. ( V \ { N } ) ( e = { x , N } <-> x = a ) ) )
56	55	ex 434	. . . . . 6 |- ( ( ( V e. W /\ N e. V ) /\ e e. ~P V ) -> ( ( a e. V /\ ( a =/= N /\ e = { a , N } ) ) -> ( a e. ( V \ { N } ) /\ A. x e. ( V \ { N } ) ( e = { x , N } <-> x = a ) ) ) )
57	56	reximdv2 2820	. . . . 5 |- ( ( ( V e. W /\ N e. V ) /\ e e. ~P V ) -> ( E. a e. V ( a =/= N /\ e = { a , N } ) -> E. a e. ( V \ { N } ) A. x e. ( V \ { N } ) ( e = { x , N } <-> x = a ) ) )
58	57	expimpd 603	. . . 4 |- ( ( V e. W /\ N e. V ) -> ( ( e e. ~P V /\ E. a e. V ( a =/= N /\ e = { a , N } ) ) -> E. a e. ( V \ { N } ) A. x e. ( V \ { N } ) ( e = { x , N } <-> x = a ) ) )
59		eqeq1 2444	. . . . . . 7 |- ( x = e -> ( x = { a , N } <-> e = { a , N } ) )
60	59	anbi2d 703	. . . . . 6 |- ( x = e -> ( ( a =/= N /\ x = { a , N } ) <-> ( a =/= N /\ e = { a , N } ) ) )
61	60	rexbidv 2731	. . . . 5 |- ( x = e -> ( E. a e. V ( a =/= N /\ x = { a , N } ) <-> E. a e. V ( a =/= N /\ e = { a , N } ) ) )
62	61, 24	elrab2 3114	. . . 4 |- ( e e. P <-> ( e e. ~P V /\ E. a e. V ( a =/= N /\ e = { a , N } ) ) )
63		reu6 3143	. . . 4 |- ( E! x e. ( V \ { N } ) e = { x , N } <-> E. a e. ( V \ { N } ) A. x e. ( V \ { N } ) ( e = { x , N } <-> x = a ) )
64	58, 62, 63	3imtr4g 270	. . 3 |- ( ( V e. W /\ N e. V ) -> ( e e. P -> E! x e. ( V \ { N } ) e = { x , N } ) )
65	64	ralrimiv 2793	. 2 |- ( ( V e. W /\ N e. V ) -> A. e e. P E! x e. ( V \ { N } ) e = { x , N } )
66		cusgrafi.f	. . 3 |- F = ( x e. ( V \ { N } ) |-> { x , N } )
67	66	f1ompt 5860	. 2 |- ( F : ( V \ { N } ) -1-1-onto-> P <-> ( A. x e. ( V \ { N } ) { x , N } e. P /\ A. e e. P E! x e. ( V \ { N } ) e = { x , N } ) )
68	31, 65, 67	sylanbrc 664	1 |- ( ( V e. W /\ N e. V ) -> F : ( V \ { N } ) -1-1-onto-> P )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1369   e. wcel 1756   =/= wne 2601   A.wral 2710   E.wrex 2711   E!wreu 2712   {crab 2714   \ cdif 3320   ~Pcpw 3855   {csn 3872   {cpr 3874   e. cmpt 4345   -1-1-onto->wf1o 5412

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pr 4526

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421

This theorem is referenced by:  cusgrafilem3  23357