Theorem frgrncvvdeqlem9 27171

Description: Lemma 9 for frgrncvvdeq 27173. This corresponds to statement 3 in [Huneke] p. 1: "By symmetry the map is onto". (Contributed by Alexander van der Vekens, 24-Dec-2017.) (Revised by AV, 10-May-2021.) (Proof shortened by AV, 30-Dec-2021.)

Hypotheses

Ref	Expression
frgrncvvdeq.v1	|- V = (Vtx ` G )
frgrncvvdeq.e	|- E = (Edg ` G )
frgrncvvdeq.nx	|- D = ( G NeighbVtx X )
frgrncvvdeq.ny	|- N = ( G NeighbVtx Y )
frgrncvvdeq.x	|- ( ph -> X e. V )
frgrncvvdeq.y	|- ( ph -> Y e. V )
frgrncvvdeq.ne	|- ( ph -> X =/= Y )
frgrncvvdeq.xy	|- ( ph -> Y e/ D )
frgrncvvdeq.f	|- ( ph -> G e. FriendGraph )
frgrncvvdeq.a	|- A = ( x e. D |-> ( iota_ y e. N { x , y } e. E ) )

Assertion

Ref	Expression
frgrncvvdeqlem9	|- ( ph -> A : D -onto-> N )

Distinct variable groups:   y, D   y, G   y, V   y, Y   ph, y, x   y, E   y, N   x, D   x, N   ph, x   x, E

Allowed substitution hints:   A( x, y)   G( x)   V( x)   X( x, y)   Y( x)

Proof of Theorem frgrncvvdeqlem9

Dummy variables n m are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		frgrncvvdeq.v1	. . 3 |- V = (Vtx ` G )
2		frgrncvvdeq.e	. . 3 |- E = (Edg ` G )
3		frgrncvvdeq.nx	. . 3 |- D = ( G NeighbVtx X )
4		frgrncvvdeq.ny	. . 3 |- N = ( G NeighbVtx Y )
5		frgrncvvdeq.x	. . 3 |- ( ph -> X e. V )
6		frgrncvvdeq.y	. . 3 |- ( ph -> Y e. V )
7		frgrncvvdeq.ne	. . 3 |- ( ph -> X =/= Y )
8		frgrncvvdeq.xy	. . 3 |- ( ph -> Y e/ D )
9		frgrncvvdeq.f	. . 3 |- ( ph -> G e. FriendGraph )
10		frgrncvvdeq.a	. . 3 |- A = ( x e. D |-> ( iota_ y e. N { x , y } e. E ) )
11	1, 2, 3, 4, 5, 6, 7, 8, 9, 10	frgrncvvdeqlem4 27166	. 2 |- ( ph -> A : D --> N )
12	9	adantr 481	. . . . . . 7 |- ( ( ph /\ n e. N ) -> G e. FriendGraph )
13	4	eleq2i 2693	. . . . . . . . . 10 |- ( n e. N <-> n e. ( G NeighbVtx Y ) )
14		frgrusgr 27124	. . . . . . . . . . 11 |- ( G e. FriendGraph -> G e. USGraph )
15	1	nbgrisvtx 26255	. . . . . . . . . . . 12 |- ( ( G e. USGraph /\ n e. ( G NeighbVtx Y ) ) -> n e. V )
16	15	ex 450	. . . . . . . . . . 11 |- ( G e. USGraph -> ( n e. ( G NeighbVtx Y ) -> n e. V ) )
17	9, 14, 16	3syl 18	. . . . . . . . . 10 |- ( ph -> ( n e. ( G NeighbVtx Y ) -> n e. V ) )
18	13, 17	syl5bi 232	. . . . . . . . 9 |- ( ph -> ( n e. N -> n e. V ) )
19	18	imp 445	. . . . . . . 8 |- ( ( ph /\ n e. N ) -> n e. V )
20	5	adantr 481	. . . . . . . 8 |- ( ( ph /\ n e. N ) -> X e. V )
21	1, 2, 3, 4, 5, 6, 7, 8, 9, 10	frgrncvvdeqlem1 27163	. . . . . . . . . 10 |- ( ph -> X e/ N )
22		df-nel 2898	. . . . . . . . . . 11 |- ( X e/ N <-> -. X e. N )
23		nelelne 2892	. . . . . . . . . . 11 |- ( -. X e. N -> ( n e. N -> n =/= X ) )
24	22, 23	sylbi 207	. . . . . . . . . 10 |- ( X e/ N -> ( n e. N -> n =/= X ) )
25	21, 24	syl 17	. . . . . . . . 9 |- ( ph -> ( n e. N -> n =/= X ) )
26	25	imp 445	. . . . . . . 8 |- ( ( ph /\ n e. N ) -> n =/= X )
27	19, 20, 26	3jca 1242	. . . . . . 7 |- ( ( ph /\ n e. N ) -> ( n e. V /\ X e. V /\ n =/= X ) )
28	12, 27	jca 554	. . . . . 6 |- ( ( ph /\ n e. N ) -> ( G e. FriendGraph /\ ( n e. V /\ X e. V /\ n =/= X ) ) )
29	1, 2	frcond2 27131	. . . . . . 7 |- ( G e. FriendGraph -> ( ( n e. V /\ X e. V /\ n =/= X ) -> E! m e. V ( { n , m } e. E /\ { m , X } e. E ) ) )
30	29	imp 445	. . . . . 6 |- ( ( G e. FriendGraph /\ ( n e. V /\ X e. V /\ n =/= X ) ) -> E! m e. V ( { n , m } e. E /\ { m , X } e. E ) )
31		reurex 3160	. . . . . . 7 |- ( E! m e. V ( { n , m } e. E /\ { m , X } e. E ) -> E. m e. V ( { n , m } e. E /\ { m , X } e. E ) )
32		df-rex 2918	. . . . . . 7 |- ( E. m e. V ( { n , m } e. E /\ { m , X } e. E ) <-> E. m ( m e. V /\ ( { n , m } e. E /\ { m , X } e. E ) ) )
33	31, 32	sylib 208	. . . . . 6 |- ( E! m e. V ( { n , m } e. E /\ { m , X } e. E ) -> E. m ( m e. V /\ ( { n , m } e. E /\ { m , X } e. E ) ) )
34	28, 30, 33	3syl 18	. . . . 5 |- ( ( ph /\ n e. N ) -> E. m ( m e. V /\ ( { n , m } e. E /\ { m , X } e. E ) ) )
35	2	nbusgreledg 26249	. . . . . . . . . . . . . 14 |- ( G e. USGraph -> ( m e. ( G NeighbVtx X ) <-> { m , X } e. E ) )
36	35	bicomd 213	. . . . . . . . . . . . 13 |- ( G e. USGraph -> ( { m , X } e. E <-> m e. ( G NeighbVtx X ) ) )
37	9, 14, 36	3syl 18	. . . . . . . . . . . 12 |- ( ph -> ( { m , X } e. E <-> m e. ( G NeighbVtx X ) ) )
38	37	biimpa 501	. . . . . . . . . . 11 |- ( ( ph /\ { m , X } e. E ) -> m e. ( G NeighbVtx X ) )
39	3	eleq2i 2693	. . . . . . . . . . 11 |- ( m e. D <-> m e. ( G NeighbVtx X ) )
40	38, 39	sylibr 224	. . . . . . . . . 10 |- ( ( ph /\ { m , X } e. E ) -> m e. D )
41	40	ad2ant2rl 785	. . . . . . . . 9 |- ( ( ( ph /\ n e. N ) /\ ( { n , m } e. E /\ { m , X } e. E ) ) -> m e. D )
42	2	nbusgreledg 26249	. . . . . . . . . . . . . . . 16 |- ( G e. USGraph -> ( n e. ( G NeighbVtx m ) <-> { n , m } e. E ) )
43	42	biimpar 502	. . . . . . . . . . . . . . 15 |- ( ( G e. USGraph /\ { n , m } e. E ) -> n e. ( G NeighbVtx m ) )
44	43	a1d 25	. . . . . . . . . . . . . 14 |- ( ( G e. USGraph /\ { n , m } e. E ) -> ( { m , X } e. E -> n e. ( G NeighbVtx m ) ) )
45	44	expimpd 629	. . . . . . . . . . . . 13 |- ( G e. USGraph -> ( ( { n , m } e. E /\ { m , X } e. E ) -> n e. ( G NeighbVtx m ) ) )
46	9, 14, 45	3syl 18	. . . . . . . . . . . 12 |- ( ph -> ( ( { n , m } e. E /\ { m , X } e. E ) -> n e. ( G NeighbVtx m ) ) )
47	46	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ n e. N ) -> ( ( { n , m } e. E /\ { m , X } e. E ) -> n e. ( G NeighbVtx m ) ) )
48	47	imp 445	. . . . . . . . . 10 |- ( ( ( ph /\ n e. N ) /\ ( { n , m } e. E /\ { m , X } e. E ) ) -> n e. ( G NeighbVtx m ) )
49		elin 3796	. . . . . . . . . . . . . . . 16 |- ( n e. ( ( G NeighbVtx m ) i^i N ) <-> ( n e. ( G NeighbVtx m ) /\ n e. N ) )
50		simpl 473	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ph /\ { m , X } e. E ) -> ph )
51	50, 40	jca 554	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ { m , X } e. E ) -> ( ph /\ m e. D ) )
52		preq1 4268	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( x = m -> { x , y } = { m , y } )
53	52	eleq1d 2686	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( x = m -> ( { x , y } e. E <-> { m , y } e. E ) )
54	53	riotabidv 6613	. . . . . . . . . . . . . . . . . . . . . 22 |- ( x = m -> ( iota_ y e. N { x , y } e. E ) = ( iota_ y e. N { m , y } e. E ) )
55	54	cbvmptv 4750	. . . . . . . . . . . . . . . . . . . . 21 |- ( x e. D |-> ( iota_ y e. N { x , y } e. E ) ) = ( m e. D |-> ( iota_ y e. N { m , y } e. E ) )
56	10, 55	eqtri 2644	. . . . . . . . . . . . . . . . . . . 20 |- A = ( m e. D |-> ( iota_ y e. N { m , y } e. E ) )
57	1, 2, 3, 4, 5, 6, 7, 8, 9, 56	frgrncvvdeqlem5 27167	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ m e. D ) -> { ( A ` m ) } = ( ( G NeighbVtx m ) i^i N ) )
58		eleq2 2690	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( G NeighbVtx m ) i^i N ) = { ( A ` m ) } -> ( n e. ( ( G NeighbVtx m ) i^i N ) <-> n e. { ( A ` m ) } ) )
59	58	eqcoms 2630	. . . . . . . . . . . . . . . . . . . 20 |- ( { ( A ` m ) } = ( ( G NeighbVtx m ) i^i N ) -> ( n e. ( ( G NeighbVtx m ) i^i N ) <-> n e. { ( A ` m ) } ) )
60		elsni 4194	. . . . . . . . . . . . . . . . . . . 20 |- ( n e. { ( A ` m ) } -> n = ( A ` m ) )
61	59, 60	syl6bi 243	. . . . . . . . . . . . . . . . . . 19 |- ( { ( A ` m ) } = ( ( G NeighbVtx m ) i^i N ) -> ( n e. ( ( G NeighbVtx m ) i^i N ) -> n = ( A ` m ) ) )
62	51, 57, 61	3syl 18	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ { m , X } e. E ) -> ( n e. ( ( G NeighbVtx m ) i^i N ) -> n = ( A ` m ) ) )
63	62	expcom 451	. . . . . . . . . . . . . . . . 17 |- ( { m , X } e. E -> ( ph -> ( n e. ( ( G NeighbVtx m ) i^i N ) -> n = ( A ` m ) ) ) )
64	63	com3r 87	. . . . . . . . . . . . . . . 16 |- ( n e. ( ( G NeighbVtx m ) i^i N ) -> ( { m , X } e. E -> ( ph -> n = ( A ` m ) ) ) )
65	49, 64	sylbir 225	. . . . . . . . . . . . . . 15 |- ( ( n e. ( G NeighbVtx m ) /\ n e. N ) -> ( { m , X } e. E -> ( ph -> n = ( A ` m ) ) ) )
66	65	ex 450	. . . . . . . . . . . . . 14 |- ( n e. ( G NeighbVtx m ) -> ( n e. N -> ( { m , X } e. E -> ( ph -> n = ( A ` m ) ) ) ) )
67	66	com14 96	. . . . . . . . . . . . 13 |- ( ph -> ( n e. N -> ( { m , X } e. E -> ( n e. ( G NeighbVtx m ) -> n = ( A ` m ) ) ) ) )
68	67	imp 445	. . . . . . . . . . . 12 |- ( ( ph /\ n e. N ) -> ( { m , X } e. E -> ( n e. ( G NeighbVtx m ) -> n = ( A ` m ) ) ) )
69	68	adantld 483	. . . . . . . . . . 11 |- ( ( ph /\ n e. N ) -> ( ( { n , m } e. E /\ { m , X } e. E ) -> ( n e. ( G NeighbVtx m ) -> n = ( A ` m ) ) ) )
70	69	imp 445	. . . . . . . . . 10 |- ( ( ( ph /\ n e. N ) /\ ( { n , m } e. E /\ { m , X } e. E ) ) -> ( n e. ( G NeighbVtx m ) -> n = ( A ` m ) ) )
71	48, 70	mpd 15	. . . . . . . . 9 |- ( ( ( ph /\ n e. N ) /\ ( { n , m } e. E /\ { m , X } e. E ) ) -> n = ( A ` m ) )
72	41, 71	jca 554	. . . . . . . 8 |- ( ( ( ph /\ n e. N ) /\ ( { n , m } e. E /\ { m , X } e. E ) ) -> ( m e. D /\ n = ( A ` m ) ) )
73	72	ex 450	. . . . . . 7 |- ( ( ph /\ n e. N ) -> ( ( { n , m } e. E /\ { m , X } e. E ) -> ( m e. D /\ n = ( A ` m ) ) ) )
74	73	adantld 483	. . . . . 6 |- ( ( ph /\ n e. N ) -> ( ( m e. V /\ ( { n , m } e. E /\ { m , X } e. E ) ) -> ( m e. D /\ n = ( A ` m ) ) ) )
75	74	eximdv 1846	. . . . 5 |- ( ( ph /\ n e. N ) -> ( E. m ( m e. V /\ ( { n , m } e. E /\ { m , X } e. E ) ) -> E. m ( m e. D /\ n = ( A ` m ) ) ) )
76	34, 75	mpd 15	. . . 4 |- ( ( ph /\ n e. N ) -> E. m ( m e. D /\ n = ( A ` m ) ) )
77		df-rex 2918	. . . 4 |- ( E. m e. D n = ( A ` m ) <-> E. m ( m e. D /\ n = ( A ` m ) ) )
78	76, 77	sylibr 224	. . 3 |- ( ( ph /\ n e. N ) -> E. m e. D n = ( A ` m ) )
79	78	ralrimiva 2966	. 2 |- ( ph -> A. n e. N E. m e. D n = ( A ` m ) )
80		dffo3 6374	. 2 |- ( A : D -onto-> N <-> ( A : D --> N /\ A. n e. N E. m e. D n = ( A ` m ) ) )
81	11, 79, 80	sylanbrc 698	1 |- ( ph -> A : D -onto-> N )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   e/ wnel 2897   A.wral 2912   E.wrex 2913   E!wreu 2914   i^i cin 3573   {csn 4177   {cpr 4179   |-> cmpt 4729   -->wf 5884   -onto->wfo 5886   ` cfv 5888   iota_crio 6610  (class class class)co 6650  Vtxcvtx 25874  Edgcedg 25939   USGraph cusgr 26044   NeighbVtx cnbgr 26224   FriendGraph cfrgr 27120

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-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  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-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-fz 12327  df-hash 13118  df-edg 25940  df-upgr 25977  df-umgr 25978  df-usgr 26046  df-nbgr 26228  df-frgr 27121

This theorem is referenced by:  frgrncvvdeqlem10  27172