Theorem cusgrfilem2 39682

Description: Lemma 2 for cusgrfi 39684. (Contributed by Alexander van der Vekens, 13-Jan-2018.) (Revised by AV, 11-Nov-2020.)

Hypotheses

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

Assertion

Ref	Expression
cusgrfilem2	|- ( N e. V -> F : ( V \ { N } ) -1-1-onto-> P )

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

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

Proof of Theorem cusgrfilem2

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

Step	Hyp	Ref	Expression
1		eldifi 3544	. . . . 5 |- ( x e. ( V \ { N } ) -> x e. V )
2		id 22	. . . . 5 |- ( N e. V -> N e. V )
3		prelpwi 4647	. . . . 5 |- ( ( x e. V /\ N e. V ) -> { x , N } e. ~P V )
4	1, 2, 3	syl2anr 486	. . . 4 |- ( ( N e. V /\ x e. ( V \ { N } ) ) -> { x , N } e. ~P V )
5	1	adantl 473	. . . . 5 |- ( ( N e. V /\ x e. ( V \ { N } ) ) -> x e. V )
6		eldifsni 4089	. . . . . . 7 |- ( x e. ( V \ { N } ) -> x =/= N )
7	6	adantl 473	. . . . . 6 |- ( ( N e. V /\ x e. ( V \ { N } ) ) -> x =/= N )
8		eqidd 2472	. . . . . 6 |- ( ( N e. V /\ x e. ( V \ { N } ) ) -> { x , N } = { x , N } )
9	7, 8	jca 541	. . . . 5 |- ( ( N e. V /\ x e. ( V \ { N } ) ) -> ( x =/= N /\ { x , N } = { x , N } ) )
10		id 22	. . . . . 6 |- ( x e. V -> x e. V )
11		neeq1 2705	. . . . . . . 8 |- ( a = x -> ( a =/= N <-> x =/= N ) )
12		preq1 4042	. . . . . . . . 9 |- ( a = x -> { a , N } = { x , N } )
13	12	eqeq2d 2481	. . . . . . . 8 |- ( a = x -> ( { x , N } = { a , N } <-> { x , N } = { x , N } ) )
14	11, 13	anbi12d 725	. . . . . . 7 |- ( a = x -> ( ( a =/= N /\ { x , N } = { a , N } ) <-> ( x =/= N /\ { x , N } = { x , N } ) ) )
15	14	adantl 473	. . . . . 6 |- ( ( x e. V /\ a = x ) -> ( ( a =/= N /\ { x , N } = { a , N } ) <-> ( x =/= N /\ { x , N } = { x , N } ) ) )
16	10, 15	rspcedv 3140	. . . . 5 |- ( x e. V -> ( ( x =/= N /\ { x , N } = { x , N } ) -> E. a e. V ( a =/= N /\ { x , N } = { a , N } ) ) )
17	5, 9, 16	sylc 61	. . . 4 |- ( ( N e. V /\ x e. ( V \ { N } ) ) -> E. a e. V ( a =/= N /\ { x , N } = { a , N } ) )
18		cusgrfi.p	. . . . . 6 |- P = { x e. ~P V | E. a e. V ( a =/= N /\ x = { a , N } ) }
19	18	eleq2i 2541	. . . . 5 |- ( { x , N } e. P <-> { x , N } e. { x e. ~P V | E. a e. V ( a =/= N /\ x = { a , N } ) } )
20		eqeq1 2475	. . . . . . . 8 |- ( v = { x , N } -> ( v = { a , N } <-> { x , N } = { a , N } ) )
21	20	anbi2d 718	. . . . . . 7 |- ( v = { x , N } -> ( ( a =/= N /\ v = { a , N } ) <-> ( a =/= N /\ { x , N } = { a , N } ) ) )
22	21	rexbidv 2892	. . . . . 6 |- ( v = { x , N } -> ( E. a e. V ( a =/= N /\ v = { a , N } ) <-> E. a e. V ( a =/= N /\ { x , N } = { a , N } ) ) )
23		eqeq1 2475	. . . . . . . . 9 |- ( x = v -> ( x = { a , N } <-> v = { a , N } ) )
24	23	anbi2d 718	. . . . . . . 8 |- ( x = v -> ( ( a =/= N /\ x = { a , N } ) <-> ( a =/= N /\ v = { a , N } ) ) )
25	24	rexbidv 2892	. . . . . . 7 |- ( x = v -> ( E. a e. V ( a =/= N /\ x = { a , N } ) <-> E. a e. V ( a =/= N /\ v = { a , N } ) ) )
26	25	cbvrabv 3030	. . . . . 6 |- { x e. ~P V | E. a e. V ( a =/= N /\ x = { a , N } ) } = { v e. ~P V | E. a e. V ( a =/= N /\ v = { a , N } ) }
27	22, 26	elrab2 3186	. . . . 5 |- ( { x , N } e. { x e. ~P V | E. a e. V ( a =/= N /\ x = { a , N } ) } <-> ( { x , N } e. ~P V /\ E. a e. V ( a =/= N /\ { x , N } = { a , N } ) ) )
28	19, 27	bitri 257	. . . 4 |- ( { x , N } e. P <-> ( { x , N } e. ~P V /\ E. a e. V ( a =/= N /\ { x , N } = { a , N } ) ) )
29	4, 17, 28	sylanbrc 677	. . 3 |- ( ( N e. V /\ x e. ( V \ { N } ) ) -> { x , N } e. P )
30	29	ralrimiva 2809	. 2 |- ( N e. V -> A. x e. ( V \ { N } ) { x , N } e. P )
31		simpl 464	. . . . . . . . . . 11 |- ( ( a =/= N /\ e = { a , N } ) -> a =/= N )
32	31	anim2i 579	. . . . . . . . . 10 |- ( ( a e. V /\ ( a =/= N /\ e = { a , N } ) ) -> ( a e. V /\ a =/= N ) )
33	32	adantl 473	. . . . . . . . 9 |- ( ( ( N e. V /\ e e. ~P V ) /\ ( a e. V /\ ( a =/= N /\ e = { a , N } ) ) ) -> ( a e. V /\ a =/= N ) )
34		eldifsn 4088	. . . . . . . . 9 |- ( a e. ( V \ { N } ) <-> ( a e. V /\ a =/= N ) )
35	33, 34	sylibr 217	. . . . . . . 8 |- ( ( ( N e. V /\ e e. ~P V ) /\ ( a e. V /\ ( a =/= N /\ e = { a , N } ) ) ) -> a e. ( V \ { N } ) )
36		eqeq1 2475	. . . . . . . . . . . . . 14 |- ( e = { a , N } -> ( e = { x , N } <-> { a , N } = { x , N } ) )
37	36	adantl 473	. . . . . . . . . . . . 13 |- ( ( a =/= N /\ e = { a , N } ) -> ( e = { x , N } <-> { a , N } = { x , N } ) )
38	37	ad2antlr 741	. . . . . . . . . . . 12 |- ( ( ( a e. V /\ ( a =/= N /\ e = { a , N } ) ) /\ x e. ( V \ { N } ) ) -> ( e = { x , N } <-> { a , N } = { x , N } ) )
39		vex 3034	. . . . . . . . . . . . . 14 |- a e. _V
40		vex 3034	. . . . . . . . . . . . . 14 |- x e. _V
41	39, 40	preqr1 4139	. . . . . . . . . . . . 13 |- ( { a , N } = { x , N } -> a = x )
42	41	eqcomd 2477	. . . . . . . . . . . 12 |- ( { a , N } = { x , N } -> x = a )
43	38, 42	syl6bi 236	. . . . . . . . . . 11 |- ( ( ( a e. V /\ ( a =/= N /\ e = { a , N } ) ) /\ x e. ( V \ { N } ) ) -> ( e = { x , N } -> x = a ) )
44	43	adantll 728	. . . . . . . . . 10 |- ( ( ( ( N e. V /\ e e. ~P V ) /\ ( a e. V /\ ( a =/= N /\ e = { a , N } ) ) ) /\ x e. ( V \ { N } ) ) -> ( e = { x , N } -> x = a ) )
45	12	equcoms 1872	. . . . . . . . . . . . . . 15 |- ( x = a -> { a , N } = { x , N } )
46	45	eqeq2d 2481	. . . . . . . . . . . . . 14 |- ( x = a -> ( e = { a , N } <-> e = { x , N } ) )
47	46	biimpcd 232	. . . . . . . . . . . . 13 |- ( e = { a , N } -> ( x = a -> e = { x , N } ) )
48	47	adantl 473	. . . . . . . . . . . 12 |- ( ( a =/= N /\ e = { a , N } ) -> ( x = a -> e = { x , N } ) )
49	48	adantl 473	. . . . . . . . . . 11 |- ( ( a e. V /\ ( a =/= N /\ e = { a , N } ) ) -> ( x = a -> e = { x , N } ) )
50	49	ad2antlr 741	. . . . . . . . . 10 |- ( ( ( ( N e. V /\ e e. ~P V ) /\ ( a e. V /\ ( a =/= N /\ e = { a , N } ) ) ) /\ x e. ( V \ { N } ) ) -> ( x = a -> e = { x , N } ) )
51	44, 50	impbid 195	. . . . . . . . 9 |- ( ( ( ( N e. V /\ e e. ~P V ) /\ ( a e. V /\ ( a =/= N /\ e = { a , N } ) ) ) /\ x e. ( V \ { N } ) ) -> ( e = { x , N } <-> x = a ) )
52	51	ralrimiva 2809	. . . . . . . 8 |- ( ( ( 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 ) )
53	35, 52	jca 541	. . . . . . 7 |- ( ( ( 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 ) ) )
54	53	ex 441	. . . . . 6 |- ( ( 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 ) ) ) )
55	54	reximdv2 2855	. . . . 5 |- ( ( 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 ) ) )
56	55	expimpd 614	. . . 4 |- ( 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 ) ) )
57		eqeq1 2475	. . . . . . 7 |- ( x = e -> ( x = { a , N } <-> e = { a , N } ) )
58	57	anbi2d 718	. . . . . 6 |- ( x = e -> ( ( a =/= N /\ x = { a , N } ) <-> ( a =/= N /\ e = { a , N } ) ) )
59	58	rexbidv 2892	. . . . 5 |- ( x = e -> ( E. a e. V ( a =/= N /\ x = { a , N } ) <-> E. a e. V ( a =/= N /\ e = { a , N } ) ) )
60	59, 18	elrab2 3186	. . . 4 |- ( e e. P <-> ( e e. ~P V /\ E. a e. V ( a =/= N /\ e = { a , N } ) ) )
61		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 ) )
62	56, 60, 61	3imtr4g 278	. . 3 |- ( N e. V -> ( e e. P -> E! x e. ( V \ { N } ) e = { x , N } ) )
63	62	ralrimiv 2808	. 2 |- ( N e. V -> A. e e. P E! x e. ( V \ { N } ) e = { x , N } )
64		cusgrfi.f	. . 3 |- F = ( x e. ( V \ { N } ) |-> { x , N } )
65	64	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 } ) )
66	30, 63, 65	sylanbrc 677	1 |- ( 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   ` cfv 5589  Vtxcvtx 39251

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:  cusgrfilem3  39683