Theorem cusgrafilem2 23211

Description: Lemma 2 for cusgrafi 23213. (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 3988	. . . . . . 7 |- ( v e. ( V \ { N } ) <-> ( v e. V /\ v =/= N ) )
2		simpl 454	. . . . . . 7 |- ( ( v e. V /\ v =/= N ) -> v e. V )
3	1, 2	sylbi 195	. . . . . 6 |- ( v e. ( V \ { N } ) -> v e. V )
4		simpr 458	. . . . . 6 |- ( ( V e. W /\ N e. V ) -> N e. V )
5		prelpwi 4527	. . . . . 6 |- ( ( v e. V /\ N e. V ) -> { v , N } e. ~P V )
6	3, 4, 5	syl2anr 475	. . . . 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 463	. . . . . 6 |- ( ( ( V e. W /\ N e. V ) /\ v e. ( V \ { N } ) ) -> ( v e. V /\ v =/= N ) )
9		simpr 458	. . . . . . . . 9 |- ( ( v e. V /\ v =/= N ) -> v =/= N )
10	1, 9	sylbi 195	. . . . . . . 8 |- ( v e. ( V \ { N } ) -> v =/= N )
11	10	adantl 463	. . . . . . 7 |- ( ( ( V e. W /\ N e. V ) /\ v e. ( V \ { N } ) ) -> v =/= N )
12		eqidd 2434	. . . . . . 7 |- ( ( ( V e. W /\ N e. V ) /\ v e. ( V \ { N } ) ) -> { v , N } = { v , N } )
13	11, 12	jca 529	. . . . . 6 |- ( ( ( V e. W /\ N e. V ) /\ v e. ( V \ { N } ) ) -> ( v =/= N /\ { v , N } = { v , N } ) )
14		neeq1 2606	. . . . . . . . 9 |- ( a = v -> ( a =/= N <-> v =/= N ) )
15		preq1 3942	. . . . . . . . . 10 |- ( a = v -> { a , N } = { v , N } )
16	15	eqeq2d 2444	. . . . . . . . 9 |- ( a = v -> ( { v , N } = { a , N } <-> { v , N } = { v , N } ) )
17	14, 16	anbi12d 703	. . . . . . . 8 |- ( a = v -> ( ( a =/= N /\ { v , N } = { a , N } ) <-> ( v =/= N /\ { v , N } = { v , N } ) ) )
18	17	adantl 463	. . . . . . 7 |- ( ( ( v e. V /\ v =/= N ) /\ a = v ) -> ( ( a =/= N /\ { v , N } = { a , N } ) <-> ( v =/= N /\ { v , N } = { v , N } ) ) )
19	2, 18	rspcedv 3066	. . . . . 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 2439	. . . . . . . 8 |- ( x = { v , N } -> ( x = { a , N } <-> { v , N } = { a , N } ) )
22	21	anbi2d 696	. . . . . . 7 |- ( x = { v , N } -> ( ( a =/= N /\ x = { a , N } ) <-> ( a =/= N /\ { v , N } = { a , N } ) ) )
23	22	rexbidv 2726	. . . . . 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 3108	. . . . 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 657	. . . 4 |- ( ( ( V e. W /\ N e. V ) /\ v e. ( V \ { N } ) ) -> { v , N } e. P )
27	26	ralrimiva 2789	. . 3 |- ( ( V e. W /\ N e. V ) -> A. v e. ( V \ { N } ) { v , N } e. P )
28		preq1 3942	. . . . 5 |- ( x = v -> { x , N } = { v , N } )
29	28	eleq1d 2499	. . . 4 |- ( x = v -> ( { x , N } e. P <-> { v , N } e. P ) )
30	29	cbvralv 2937	. . 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 454	. . . . . . . . . . 11 |- ( ( a =/= N /\ e = { a , N } ) -> a =/= N )
33	32	anim2i 564	. . . . . . . . . 10 |- ( ( a e. V /\ ( a =/= N /\ e = { a , N } ) ) -> ( a e. V /\ a =/= N ) )
34	33	adantl 463	. . . . . . . . 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 3988	. . . . . . . . 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 2439	. . . . . . . . . . . . . 14 |- ( e = { a , N } -> ( e = { x , N } <-> { a , N } = { x , N } ) )
38	37	adantl 463	. . . . . . . . . . . . 13 |- ( ( a =/= N /\ e = { a , N } ) -> ( e = { x , N } <-> { a , N } = { x , N } ) )
39	38	ad2antlr 719	. . . . . . . . . . . 12 |- ( ( ( a e. V /\ ( a =/= N /\ e = { a , N } ) ) /\ x e. ( V \ { N } ) ) -> ( e = { x , N } <-> { a , N } = { x , N } ) )
40		vex 2965	. . . . . . . . . . . . . 14 |- a e. _V
41		vex 2965	. . . . . . . . . . . . . 14 |- x e. _V
42	40, 41	preqr1 4034	. . . . . . . . . . . . 13 |- ( { a , N } = { x , N } -> a = x )
43	42	eqcomd 2438	. . . . . . . . . . . 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 706	. . . . . . . . . 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 3942	. . . . . . . . . . . . . . . 16 |- ( a = x -> { a , N } = { x , N } )
47	46	equcoms 1732	. . . . . . . . . . . . . . 15 |- ( x = a -> { a , N } = { x , N } )
48	47	eqeq2d 2444	. . . . . . . . . . . . . 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 463	. . . . . . . . . . . 12 |- ( ( a =/= N /\ e = { a , N } ) -> ( x = a -> e = { x , N } ) )
51	50	adantl 463	. . . . . . . . . . 11 |- ( ( a e. V /\ ( a =/= N /\ e = { a , N } ) ) -> ( x = a -> e = { x , N } ) )
52	51	ad2antlr 719	. . . . . . . . . 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 2789	. . . . . . . 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 529	. . . . . . 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 2815	. . . . 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 598	. . . 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 2439	. . . . . . 7 |- ( x = e -> ( x = { a , N } <-> e = { a , N } ) )
60	59	anbi2d 696	. . . . . 6 |- ( x = e -> ( ( a =/= N /\ x = { a , N } ) <-> ( a =/= N /\ e = { a , N } ) ) )
61	60	rexbidv 2726	. . . . 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 3108	. . . 4 |- ( e e. P <-> ( e e. ~P V /\ E. a e. V ( a =/= N /\ e = { a , N } ) ) )
63		reu6 3137	. . . 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 2788	. 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 5853	. 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 657	1 |- ( ( V e. W /\ N e. V ) -> F : ( V \ { N } ) -1-1-onto-> P )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1362   e. wcel 1755   =/= wne 2596   A.wral 2705   E.wrex 2706   E!wreu 2707   {crab 2709   \ cdif 3313   ~Pcpw 3848   {csn 3865   {cpr 3867   e. cmpt 4338   -1-1-onto->wf1o 5405

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-sep 4401  ax-nul 4409  ax-pr 4519

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2964  df-sbc 3176  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-op 3872  df-uni 4080  df-br 4281  df-opab 4339  df-mpt 4340  df-id 4623  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414

This theorem is referenced by:  cusgrafilem3  23212