Theorem cusgrfilem2 26352

Description: Lemma 2 for cusgrfi 26354. (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 3732	. . . . 5 |- ( x e. ( V \ { N } ) -> x e. V )
2		id 22	. . . . 5 |- ( N e. V -> N e. V )
3		prelpwi 4915	. . . . 5 |- ( ( x e. V /\ N e. V ) -> { x , N } e. ~P V )
4	1, 2, 3	syl2anr 495	. . . 4 |- ( ( N e. V /\ x e. ( V \ { N } ) ) -> { x , N } e. ~P V )
5	1	adantl 482	. . . . 5 |- ( ( N e. V /\ x e. ( V \ { N } ) ) -> x e. V )
6		eldifsni 4320	. . . . . . 7 |- ( x e. ( V \ { N } ) -> x =/= N )
7	6	adantl 482	. . . . . 6 |- ( ( N e. V /\ x e. ( V \ { N } ) ) -> x =/= N )
8		eqidd 2623	. . . . . 6 |- ( ( N e. V /\ x e. ( V \ { N } ) ) -> { x , N } = { x , N } )
9	7, 8	jca 554	. . . . 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 2856	. . . . . . . 8 |- ( a = x -> ( a =/= N <-> x =/= N ) )
12		preq1 4268	. . . . . . . . 9 |- ( a = x -> { a , N } = { x , N } )
13	12	eqeq2d 2632	. . . . . . . 8 |- ( a = x -> ( { x , N } = { a , N } <-> { x , N } = { x , N } ) )
14	11, 13	anbi12d 747	. . . . . . 7 |- ( a = x -> ( ( a =/= N /\ { x , N } = { a , N } ) <-> ( x =/= N /\ { x , N } = { x , N } ) ) )
15	14	adantl 482	. . . . . 6 |- ( ( x e. V /\ a = x ) -> ( ( a =/= N /\ { x , N } = { a , N } ) <-> ( x =/= N /\ { x , N } = { x , N } ) ) )
16	10, 15	rspcedv 3313	. . . . 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 65	. . . 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 2693	. . . . 5 |- ( { x , N } e. P <-> { x , N } e. { x e. ~P V | E. a e. V ( a =/= N /\ x = { a , N } ) } )
20		eqeq1 2626	. . . . . . . 8 |- ( v = { x , N } -> ( v = { a , N } <-> { x , N } = { a , N } ) )
21	20	anbi2d 740	. . . . . . 7 |- ( v = { x , N } -> ( ( a =/= N /\ v = { a , N } ) <-> ( a =/= N /\ { x , N } = { a , N } ) ) )
22	21	rexbidv 3052	. . . . . 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 2626	. . . . . . . . 9 |- ( x = v -> ( x = { a , N } <-> v = { a , N } ) )
24	23	anbi2d 740	. . . . . . . 8 |- ( x = v -> ( ( a =/= N /\ x = { a , N } ) <-> ( a =/= N /\ v = { a , N } ) ) )
25	24	rexbidv 3052	. . . . . . 7 |- ( x = v -> ( E. a e. V ( a =/= N /\ x = { a , N } ) <-> E. a e. V ( a =/= N /\ v = { a , N } ) ) )
26	25	cbvrabv 3199	. . . . . 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 3366	. . . . 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 264	. . . 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 698	. . 3 |- ( ( N e. V /\ x e. ( V \ { N } ) ) -> { x , N } e. P )
30	29	ralrimiva 2966	. 2 |- ( N e. V -> A. x e. ( V \ { N } ) { x , N } e. P )
31		simpl 473	. . . . . . . . . . 11 |- ( ( a =/= N /\ e = { a , N } ) -> a =/= N )
32	31	anim2i 593	. . . . . . . . . 10 |- ( ( a e. V /\ ( a =/= N /\ e = { a , N } ) ) -> ( a e. V /\ a =/= N ) )
33	32	adantl 482	. . . . . . . . 9 |- ( ( ( N e. V /\ e e. ~P V ) /\ ( a e. V /\ ( a =/= N /\ e = { a , N } ) ) ) -> ( a e. V /\ a =/= N ) )
34		eldifsn 4317	. . . . . . . . 9 |- ( a e. ( V \ { N } ) <-> ( a e. V /\ a =/= N ) )
35	33, 34	sylibr 224	. . . . . . . 8 |- ( ( ( N e. V /\ e e. ~P V ) /\ ( a e. V /\ ( a =/= N /\ e = { a , N } ) ) ) -> a e. ( V \ { N } ) )
36		eqeq1 2626	. . . . . . . . . . . . . 14 |- ( e = { a , N } -> ( e = { x , N } <-> { a , N } = { x , N } ) )
37	36	adantl 482	. . . . . . . . . . . . 13 |- ( ( a =/= N /\ e = { a , N } ) -> ( e = { x , N } <-> { a , N } = { x , N } ) )
38	37	ad2antlr 763	. . . . . . . . . . . 12 |- ( ( ( a e. V /\ ( a =/= N /\ e = { a , N } ) ) /\ x e. ( V \ { N } ) ) -> ( e = { x , N } <-> { a , N } = { x , N } ) )
39		vex 3203	. . . . . . . . . . . . . 14 |- a e. _V
40		vex 3203	. . . . . . . . . . . . . 14 |- x e. _V
41	39, 40	preqr1 4379	. . . . . . . . . . . . 13 |- ( { a , N } = { x , N } -> a = x )
42	41	equcomd 1946	. . . . . . . . . . . 12 |- ( { a , N } = { x , N } -> x = a )
43	38, 42	syl6bi 243	. . . . . . . . . . 11 |- ( ( ( a e. V /\ ( a =/= N /\ e = { a , N } ) ) /\ x e. ( V \ { N } ) ) -> ( e = { x , N } -> x = a ) )
44	43	adantll 750	. . . . . . . . . 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 1947	. . . . . . . . . . . . . . 15 |- ( x = a -> { a , N } = { x , N } )
46	45	eqeq2d 2632	. . . . . . . . . . . . . 14 |- ( x = a -> ( e = { a , N } <-> e = { x , N } ) )
47	46	biimpcd 239	. . . . . . . . . . . . 13 |- ( e = { a , N } -> ( x = a -> e = { x , N } ) )
48	47	adantl 482	. . . . . . . . . . . 12 |- ( ( a =/= N /\ e = { a , N } ) -> ( x = a -> e = { x , N } ) )
49	48	adantl 482	. . . . . . . . . . 11 |- ( ( a e. V /\ ( a =/= N /\ e = { a , N } ) ) -> ( x = a -> e = { x , N } ) )
50	49	ad2antlr 763	. . . . . . . . . 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 202	. . . . . . . . 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 2966	. . . . . . . 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 554	. . . . . . 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 450	. . . . . 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 3014	. . . . 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 629	. . . 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 2626	. . . . . . 7 |- ( x = e -> ( x = { a , N } <-> e = { a , N } ) )
58	57	anbi2d 740	. . . . . 6 |- ( x = e -> ( ( a =/= N /\ x = { a , N } ) <-> ( a =/= N /\ e = { a , N } ) ) )
59	58	rexbidv 3052	. . . . 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 3366	. . . 4 |- ( e e. P <-> ( e e. ~P V /\ E. a e. V ( a =/= N /\ e = { a , N } ) ) )
61		reu6 3395	. . . 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 285	. . 3 |- ( N e. V -> ( e e. P -> E! x e. ( V \ { N } ) e = { x , N } ) )
63	62	ralrimiv 2965	. 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 6382	. 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 698	1 |- ( N e. V -> F : ( V \ { N } ) -1-1-onto-> P )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   E!wreu 2914   {crab 2916   \ cdif 3571   ~Pcpw 4158   {csn 4177   {cpr 4179   |-> cmpt 4729   -1-1-onto->wf1o 5887   ` cfv 5888  Vtxcvtx 25874

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896

This theorem is referenced by:  cusgrfilem3  26353