Theorem cusgrafilem2 25287

Description: Lemma 2 for cusgrafi 25289. (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 4088	. . . . . . 7 |- ( v e. ( V \ { N } ) <-> ( v e. V /\ v =/= N ) )
2		simpl 464	. . . . . . 7 |- ( ( v e. V /\ v =/= N ) -> v e. V )
3	1, 2	sylbi 200	. . . . . 6 |- ( v e. ( V \ { N } ) -> v e. V )
4		simpr 468	. . . . . 6 |- ( ( V e. W /\ N e. V ) -> N e. V )
5		prelpwi 4647	. . . . . 6 |- ( ( v e. V /\ N e. V ) -> { v , N } e. ~P V )
6	3, 4, 5	syl2anr 486	. . . . 5 |- ( ( ( V e. W /\ N e. V ) /\ v e. ( V \ { N } ) ) -> { v , N } e. ~P V )
7	1	biimpi 199	. . . . . . 7 |- ( v e. ( V \ { N } ) -> ( v e. V /\ v =/= N ) )
8	7	adantl 473	. . . . . 6 |- ( ( ( V e. W /\ N e. V ) /\ v e. ( V \ { N } ) ) -> ( v e. V /\ v =/= N ) )
9		simpr 468	. . . . . . . . 9 |- ( ( v e. V /\ v =/= N ) -> v =/= N )
10	1, 9	sylbi 200	. . . . . . . 8 |- ( v e. ( V \ { N } ) -> v =/= N )
11	10	adantl 473	. . . . . . 7 |- ( ( ( V e. W /\ N e. V ) /\ v e. ( V \ { N } ) ) -> v =/= N )
12		eqidd 2472	. . . . . . 7 |- ( ( ( V e. W /\ N e. V ) /\ v e. ( V \ { N } ) ) -> { v , N } = { v , N } )
13	11, 12	jca 541	. . . . . 6 |- ( ( ( V e. W /\ N e. V ) /\ v e. ( V \ { N } ) ) -> ( v =/= N /\ { v , N } = { v , N } ) )
14		neeq1 2705	. . . . . . . . 9 |- ( a = v -> ( a =/= N <-> v =/= N ) )
15		preq1 4042	. . . . . . . . . 10 |- ( a = v -> { a , N } = { v , N } )
16	15	eqeq2d 2481	. . . . . . . . 9 |- ( a = v -> ( { v , N } = { a , N } <-> { v , N } = { v , N } ) )
17	14, 16	anbi12d 725	. . . . . . . 8 |- ( a = v -> ( ( a =/= N /\ { v , N } = { a , N } ) <-> ( v =/= N /\ { v , N } = { v , N } ) ) )
18	17	adantl 473	. . . . . . 7 |- ( ( ( v e. V /\ v =/= N ) /\ a = v ) -> ( ( a =/= N /\ { v , N } = { a , N } ) <-> ( v =/= N /\ { v , N } = { v , N } ) ) )
19	2, 18	rspcedv 3140	. . . . . 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 61	. . . . 5 |- ( ( ( V e. W /\ N e. V ) /\ v e. ( V \ { N } ) ) -> E. a e. V ( a =/= N /\ { v , N } = { a , N } ) )
21		eqeq1 2475	. . . . . . . 8 |- ( x = { v , N } -> ( x = { a , N } <-> { v , N } = { a , N } ) )
22	21	anbi2d 718	. . . . . . 7 |- ( x = { v , N } -> ( ( a =/= N /\ x = { a , N } ) <-> ( a =/= N /\ { v , N } = { a , N } ) ) )
23	22	rexbidv 2892	. . . . . 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 3186	. . . . 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 677	. . . 4 |- ( ( ( V e. W /\ N e. V ) /\ v e. ( V \ { N } ) ) -> { v , N } e. P )
27	26	ralrimiva 2809	. . 3 |- ( ( V e. W /\ N e. V ) -> A. v e. ( V \ { N } ) { v , N } e. P )
28		preq1 4042	. . . . 5 |- ( x = v -> { x , N } = { v , N } )
29	28	eleq1d 2533	. . . 4 |- ( x = v -> ( { x , N } e. P <-> { v , N } e. P ) )
30	29	cbvralv 3005	. . 3 |- ( A. x e. ( V \ { N } ) { x , N } e. P <-> A. v e. ( V \ { N } ) { v , N } e. P )
31	27, 30	sylibr 217	. 2 |- ( ( V e. W /\ N e. V ) -> A. x e. ( V \ { N } ) { x , N } e. P )
32		simpl 464	. . . . . . . . . . 11 |- ( ( a =/= N /\ e = { a , N } ) -> a =/= N )
33	32	anim2i 579	. . . . . . . . . 10 |- ( ( a e. V /\ ( a =/= N /\ e = { a , N } ) ) -> ( a e. V /\ a =/= N ) )
34	33	adantl 473	. . . . . . . . 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 4088	. . . . . . . . 9 |- ( a e. ( V \ { N } ) <-> ( a e. V /\ a =/= N ) )
36	34, 35	sylibr 217	. . . . . . . 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 2475	. . . . . . . . . . . . . 14 |- ( e = { a , N } -> ( e = { x , N } <-> { a , N } = { x , N } ) )
38	37	adantl 473	. . . . . . . . . . . . 13 |- ( ( a =/= N /\ e = { a , N } ) -> ( e = { x , N } <-> { a , N } = { x , N } ) )
39	38	ad2antlr 741	. . . . . . . . . . . 12 |- ( ( ( a e. V /\ ( a =/= N /\ e = { a , N } ) ) /\ x e. ( V \ { N } ) ) -> ( e = { x , N } <-> { a , N } = { x , N } ) )
40		vex 3034	. . . . . . . . . . . . . 14 |- a e. _V
41		vex 3034	. . . . . . . . . . . . . 14 |- x e. _V
42	40, 41	preqr1 4139	. . . . . . . . . . . . 13 |- ( { a , N } = { x , N } -> a = x )
43	42	eqcomd 2477	. . . . . . . . . . . 12 |- ( { a , N } = { x , N } -> x = a )
44	39, 43	syl6bi 236	. . . . . . . . . . 11 |- ( ( ( a e. V /\ ( a =/= N /\ e = { a , N } ) ) /\ x e. ( V \ { N } ) ) -> ( e = { x , N } -> x = a ) )
45	44	adantll 728	. . . . . . . . . 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 4042	. . . . . . . . . . . . . . . 16 |- ( a = x -> { a , N } = { x , N } )
47	46	equcoms 1872	. . . . . . . . . . . . . . 15 |- ( x = a -> { a , N } = { x , N } )
48	47	eqeq2d 2481	. . . . . . . . . . . . . 14 |- ( x = a -> ( e = { a , N } <-> e = { x , N } ) )
49	48	biimpcd 232	. . . . . . . . . . . . 13 |- ( e = { a , N } -> ( x = a -> e = { x , N } ) )
50	49	adantl 473	. . . . . . . . . . . 12 |- ( ( a =/= N /\ e = { a , N } ) -> ( x = a -> e = { x , N } ) )
51	50	adantl 473	. . . . . . . . . . 11 |- ( ( a e. V /\ ( a =/= N /\ e = { a , N } ) ) -> ( x = a -> e = { x , N } ) )
52	51	ad2antlr 741	. . . . . . . . . 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 195	. . . . . . . . 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 2809	. . . . . . . 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 541	. . . . . . 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 441	. . . . . 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 2855	. . . . 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 614	. . . 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 2475	. . . . . . 7 |- ( x = e -> ( x = { a , N } <-> e = { a , N } ) )
60	59	anbi2d 718	. . . . . 6 |- ( x = e -> ( ( a =/= N /\ x = { a , N } ) <-> ( a =/= N /\ e = { a , N } ) ) )
61	60	rexbidv 2892	. . . . 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 3186	. . . 4 |- ( e e. P <-> ( e e. ~P V /\ E. a e. V ( a =/= N /\ e = { a , N } ) ) )
63		reu6 3215	. . . 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 278	. . 3 |- ( ( V e. W /\ N e. V ) -> ( e e. P -> E! x e. ( V \ { N } ) e = { x , N } ) )
65	64	ralrimiv 2808	. 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 6059	. 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 677	1 |- ( ( V e. W /\ N e. V ) -> F : ( V \ { N } ) -1-1-onto-> P )

Syntax hints:   -> wi 4   <-> wb 189   /\ wa 376   = wceq 1452   e. wcel 1904   =/= wne 2641   A.wral 2756   E.wrex 2757   E!wreu 2758   {crab 2760   \ cdif 3387   ~Pcpw 3942   {csn 3959   {cpr 3961   |-> cmpt 4454   -1-1-onto->wf1o 5588

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597

This theorem is referenced by:  cusgrafilem3  25288