Theorem frgrncvvdeqlem8 27170

Description: Lemma 8 for frgrncvvdeq 27173. This corresponds to statement 2 in [Huneke] p. 1: "The map is one-to-one since z in N(x) is uniquely determined as the common neighbor of x and a(x)". (Contributed by Alexander van der Vekens, 23-Dec-2017.) (Revised by AV, 10-May-2021.) (Revised 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
frgrncvvdeqlem8	|- ( ph -> A : D -1-1-> 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 frgrncvvdeqlem8

Dummy variables u w 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		simpr 477	. . 3 |- ( ( ph /\ A : D --> N ) -> A : D --> N )
13		ffvelrn 6357	. . . . . . . . 9 |- ( ( A : D --> N /\ u e. D ) -> ( A ` u ) e. N )
14	13	ad2ant2lr 784	. . . . . . . 8 |- ( ( ( ph /\ A : D --> N ) /\ ( u e. D /\ w e. D ) ) -> ( A ` u ) e. N )
15	14	adantr 481	. . . . . . 7 |- ( ( ( ( ph /\ A : D --> N ) /\ ( u e. D /\ w e. D ) ) /\ ( A ` u ) = ( A ` w ) ) -> ( A ` u ) e. N )
16	1, 2, 3, 4, 5, 6, 7, 8, 9, 10	frgrncvvdeqlem1 27163	. . . . . . . . . . 11 |- ( ph -> X e/ N )
17		preq1 4268	. . . . . . . . . . . . . . . . . . . . 21 |- ( x = u -> { x , y } = { u , y } )
18	17	eleq1d 2686	. . . . . . . . . . . . . . . . . . . 20 |- ( x = u -> ( { x , y } e. E <-> { u , y } e. E ) )
19	18	riotabidv 6613	. . . . . . . . . . . . . . . . . . 19 |- ( x = u -> ( iota_ y e. N { x , y } e. E ) = ( iota_ y e. N { u , y } e. E ) )
20	19	cbvmptv 4750	. . . . . . . . . . . . . . . . . 18 |- ( x e. D |-> ( iota_ y e. N { x , y } e. E ) ) = ( u e. D |-> ( iota_ y e. N { u , y } e. E ) )
21	10, 20	eqtri 2644	. . . . . . . . . . . . . . . . 17 |- A = ( u e. D |-> ( iota_ y e. N { u , y } e. E ) )
22	1, 2, 3, 4, 5, 6, 7, 8, 9, 21	frgrncvvdeqlem6 27168	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ u e. D ) -> { u , ( A ` u ) } e. E )
23		preq1 4268	. . . . . . . . . . . . . . . . . . . . 21 |- ( x = w -> { x , y } = { w , y } )
24	23	eleq1d 2686	. . . . . . . . . . . . . . . . . . . 20 |- ( x = w -> ( { x , y } e. E <-> { w , y } e. E ) )
25	24	riotabidv 6613	. . . . . . . . . . . . . . . . . . 19 |- ( x = w -> ( iota_ y e. N { x , y } e. E ) = ( iota_ y e. N { w , y } e. E ) )
26	25	cbvmptv 4750	. . . . . . . . . . . . . . . . . 18 |- ( x e. D |-> ( iota_ y e. N { x , y } e. E ) ) = ( w e. D |-> ( iota_ y e. N { w , y } e. E ) )
27	10, 26	eqtri 2644	. . . . . . . . . . . . . . . . 17 |- A = ( w e. D |-> ( iota_ y e. N { w , y } e. E ) )
28	1, 2, 3, 4, 5, 6, 7, 8, 9, 27	frgrncvvdeqlem6 27168	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ w e. D ) -> { w , ( A ` w ) } e. E )
29	22, 28	anim12dan 882	. . . . . . . . . . . . . . 15 |- ( ( ph /\ ( u e. D /\ w e. D ) ) -> ( { u , ( A ` u ) } e. E /\ { w , ( A ` w ) } e. E ) )
30		preq2 4269	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( A ` w ) = ( A ` u ) -> { w , ( A ` w ) } = { w , ( A ` u ) } )
31	30	eleq1d 2686	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( A ` w ) = ( A ` u ) -> ( { w , ( A ` w ) } e. E <-> { w , ( A ` u ) } e. E ) )
32	31	anbi2d 740	. . . . . . . . . . . . . . . . . . . 20 |- ( ( A ` w ) = ( A ` u ) -> ( ( { u , ( A ` u ) } e. E /\ { w , ( A ` w ) } e. E ) <-> ( { u , ( A ` u ) } e. E /\ { w , ( A ` u ) } e. E ) ) )
33	32	eqcoms 2630	. . . . . . . . . . . . . . . . . . 19 |- ( ( A ` u ) = ( A ` w ) -> ( ( { u , ( A ` u ) } e. E /\ { w , ( A ` w ) } e. E ) <-> ( { u , ( A ` u ) } e. E /\ { w , ( A ` u ) } e. E ) ) )
34	33	biimpa 501	. . . . . . . . . . . . . . . . . 18 |- ( ( ( A ` u ) = ( A ` w ) /\ ( { u , ( A ` u ) } e. E /\ { w , ( A ` w ) } e. E ) ) -> ( { u , ( A ` u ) } e. E /\ { w , ( A ` u ) } e. E ) )
35		df-ne 2795	. . . . . . . . . . . . . . . . . . 19 |- ( u =/= w <-> -. u = w )
36	2, 3	frgrnbnb 27157	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( G e. FriendGraph /\ ( u e. D /\ w e. D ) /\ u =/= w ) -> ( ( { u , ( A ` u ) } e. E /\ { w , ( A ` u ) } e. E ) -> ( A ` u ) = X ) )
37	9, 36	syl3an1 1359	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( ph /\ ( u e. D /\ w e. D ) /\ u =/= w ) -> ( ( { u , ( A ` u ) } e. E /\ { w , ( A ` u ) } e. E ) -> ( A ` u ) = X ) )
38	37	3expa 1265	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( ph /\ ( u e. D /\ w e. D ) ) /\ u =/= w ) -> ( ( { u , ( A ` u ) } e. E /\ { w , ( A ` u ) } e. E ) -> ( A ` u ) = X ) )
39		df-nel 2898	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( X e/ N <-> -. X e. N )
40		eleq1 2689	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ( ( A ` u ) = X -> ( ( A ` u ) e. N <-> X e. N ) )
41	40	biimpa 501	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( ( ( A ` u ) = X /\ ( A ` u ) e. N ) -> X e. N )
42	41	pm2.24d 147	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( ( ( A ` u ) = X /\ ( A ` u ) e. N ) -> ( -. X e. N -> u = w ) )
43	42	expcom 451	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( ( A ` u ) e. N -> ( ( A ` u ) = X -> ( -. X e. N -> u = w ) ) )
44	43	com13 88	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( -. X e. N -> ( ( A ` u ) = X -> ( ( A ` u ) e. N -> u = w ) ) )
45	39, 44	sylbi 207	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( X e/ N -> ( ( A ` u ) = X -> ( ( A ` u ) e. N -> u = w ) ) )
46	45	com12 32	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( A ` u ) = X -> ( X e/ N -> ( ( A ` u ) e. N -> u = w ) ) )
47	38, 46	syl6 35	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( ph /\ ( u e. D /\ w e. D ) ) /\ u =/= w ) -> ( ( { u , ( A ` u ) } e. E /\ { w , ( A ` u ) } e. E ) -> ( X e/ N -> ( ( A ` u ) e. N -> u = w ) ) ) )
48	47	expcom 451	. . . . . . . . . . . . . . . . . . . 20 |- ( u =/= w -> ( ( ph /\ ( u e. D /\ w e. D ) ) -> ( ( { u , ( A ` u ) } e. E /\ { w , ( A ` u ) } e. E ) -> ( X e/ N -> ( ( A ` u ) e. N -> u = w ) ) ) ) )
49	48	com23 86	. . . . . . . . . . . . . . . . . . 19 |- ( u =/= w -> ( ( { u , ( A ` u ) } e. E /\ { w , ( A ` u ) } e. E ) -> ( ( ph /\ ( u e. D /\ w e. D ) ) -> ( X e/ N -> ( ( A ` u ) e. N -> u = w ) ) ) ) )
50	35, 49	sylbir 225	. . . . . . . . . . . . . . . . . 18 |- ( -. u = w -> ( ( { u , ( A ` u ) } e. E /\ { w , ( A ` u ) } e. E ) -> ( ( ph /\ ( u e. D /\ w e. D ) ) -> ( X e/ N -> ( ( A ` u ) e. N -> u = w ) ) ) ) )
51	34, 50	syl5com 31	. . . . . . . . . . . . . . . . 17 |- ( ( ( A ` u ) = ( A ` w ) /\ ( { u , ( A ` u ) } e. E /\ { w , ( A ` w ) } e. E ) ) -> ( -. u = w -> ( ( ph /\ ( u e. D /\ w e. D ) ) -> ( X e/ N -> ( ( A ` u ) e. N -> u = w ) ) ) ) )
52	51	expcom 451	. . . . . . . . . . . . . . . 16 |- ( ( { u , ( A ` u ) } e. E /\ { w , ( A ` w ) } e. E ) -> ( ( A ` u ) = ( A ` w ) -> ( -. u = w -> ( ( ph /\ ( u e. D /\ w e. D ) ) -> ( X e/ N -> ( ( A ` u ) e. N -> u = w ) ) ) ) ) )
53	52	com24 95	. . . . . . . . . . . . . . 15 |- ( ( { u , ( A ` u ) } e. E /\ { w , ( A ` w ) } e. E ) -> ( ( ph /\ ( u e. D /\ w e. D ) ) -> ( -. u = w -> ( ( A ` u ) = ( A ` w ) -> ( X e/ N -> ( ( A ` u ) e. N -> u = w ) ) ) ) ) )
54	29, 53	mpcom 38	. . . . . . . . . . . . . 14 |- ( ( ph /\ ( u e. D /\ w e. D ) ) -> ( -. u = w -> ( ( A ` u ) = ( A ` w ) -> ( X e/ N -> ( ( A ` u ) e. N -> u = w ) ) ) ) )
55	54	ex 450	. . . . . . . . . . . . 13 |- ( ph -> ( ( u e. D /\ w e. D ) -> ( -. u = w -> ( ( A ` u ) = ( A ` w ) -> ( X e/ N -> ( ( A ` u ) e. N -> u = w ) ) ) ) ) )
56	55	com3r 87	. . . . . . . . . . . 12 |- ( -. u = w -> ( ph -> ( ( u e. D /\ w e. D ) -> ( ( A ` u ) = ( A ` w ) -> ( X e/ N -> ( ( A ` u ) e. N -> u = w ) ) ) ) ) )
57	56	com15 101	. . . . . . . . . . 11 |- ( X e/ N -> ( ph -> ( ( u e. D /\ w e. D ) -> ( ( A ` u ) = ( A ` w ) -> ( -. u = w -> ( ( A ` u ) e. N -> u = w ) ) ) ) ) )
58	16, 57	mpcom 38	. . . . . . . . . 10 |- ( ph -> ( ( u e. D /\ w e. D ) -> ( ( A ` u ) = ( A ` w ) -> ( -. u = w -> ( ( A ` u ) e. N -> u = w ) ) ) ) )
59	58	expd 452	. . . . . . . . 9 |- ( ph -> ( u e. D -> ( w e. D -> ( ( A ` u ) = ( A ` w ) -> ( -. u = w -> ( ( A ` u ) e. N -> u = w ) ) ) ) ) )
60	59	adantr 481	. . . . . . . 8 |- ( ( ph /\ A : D --> N ) -> ( u e. D -> ( w e. D -> ( ( A ` u ) = ( A ` w ) -> ( -. u = w -> ( ( A ` u ) e. N -> u = w ) ) ) ) ) )
61	60	imp42 620	. . . . . . 7 |- ( ( ( ( ph /\ A : D --> N ) /\ ( u e. D /\ w e. D ) ) /\ ( A ` u ) = ( A ` w ) ) -> ( -. u = w -> ( ( A ` u ) e. N -> u = w ) ) )
62	15, 61	mpid 44	. . . . . 6 |- ( ( ( ( ph /\ A : D --> N ) /\ ( u e. D /\ w e. D ) ) /\ ( A ` u ) = ( A ` w ) ) -> ( -. u = w -> u = w ) )
63	62	pm2.18d 124	. . . . 5 |- ( ( ( ( ph /\ A : D --> N ) /\ ( u e. D /\ w e. D ) ) /\ ( A ` u ) = ( A ` w ) ) -> u = w )
64	63	ex 450	. . . 4 |- ( ( ( ph /\ A : D --> N ) /\ ( u e. D /\ w e. D ) ) -> ( ( A ` u ) = ( A ` w ) -> u = w ) )
65	64	ralrimivva 2971	. . 3 |- ( ( ph /\ A : D --> N ) -> A. u e. D A. w e. D ( ( A ` u ) = ( A ` w ) -> u = w ) )
66		dff13 6512	. . 3 |- ( A : D -1-1-> N <-> ( A : D --> N /\ A. u e. D A. w e. D ( ( A ` u ) = ( A ` w ) -> u = w ) ) )
67	12, 65, 66	sylanbrc 698	. 2 |- ( ( ph /\ A : D --> N ) -> A : D -1-1-> N )
68	11, 67	mpdan 702	1 |- ( ph -> A : D -1-1-> N )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   e/ wnel 2897   A.wral 2912   {cpr 4179   |-> cmpt 4729   -->wf 5884   -1-1->wf1 5885   ` cfv 5888   iota_crio 6610  (class class class)co 6650  Vtxcvtx 25874  Edgcedg 25939   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