Theorem edgnbusgreu 26269

Description: For each edge incident to a vertex there is exactly one neighbor of the vertex also incident to this edge in a simple graph. (Contributed by AV, 28-Oct-2020.)

Hypotheses

Ref	Expression
edgnbusgreu.v	|- V = (Vtx ` G )
edgnbusgreu.e	|- E = (Edg ` G )
edgnbusgreu.n	|- N = ( G NeighbVtx M )

Assertion

Ref	Expression
edgnbusgreu	|- ( ( ( G e. USGraph /\ M e. V ) /\ ( C e. E /\ M e. C ) ) -> E! n e. N C = { M , n } )

Distinct variable groups:   C, n   n, E   n, G   n, M   n, V

Allowed substitution hint:   N( n)

Proof of Theorem edgnbusgreu

Step	Hyp	Ref	Expression
1		simpl 473	. . . . . 6 |- ( ( G e. USGraph /\ M e. V ) -> G e. USGraph )
2	1	adantr 481	. . . . 5 |- ( ( ( G e. USGraph /\ M e. V ) /\ ( C e. E /\ M e. C ) ) -> G e. USGraph )
3		edgnbusgreu.e	. . . . . . . 8 |- E = (Edg ` G )
4	3	eleq2i 2693	. . . . . . 7 |- ( C e. E <-> C e. (Edg ` G ) )
5	4	biimpi 206	. . . . . 6 |- ( C e. E -> C e. (Edg ` G ) )
6	5	ad2antrl 764	. . . . 5 |- ( ( ( G e. USGraph /\ M e. V ) /\ ( C e. E /\ M e. C ) ) -> C e. (Edg ` G ) )
7		simprr 796	. . . . 5 |- ( ( ( G e. USGraph /\ M e. V ) /\ ( C e. E /\ M e. C ) ) -> M e. C )
8		usgredg2vtxeu 26113	. . . . 5 |- ( ( G e. USGraph /\ C e. (Edg ` G ) /\ M e. C ) -> E! n e. (Vtx ` G ) C = { M , n } )
9	2, 6, 7, 8	syl3anc 1326	. . . 4 |- ( ( ( G e. USGraph /\ M e. V ) /\ ( C e. E /\ M e. C ) ) -> E! n e. (Vtx ` G ) C = { M , n } )
10		df-reu 2919	. . . . 5 |- ( E! n e. (Vtx ` G ) C = { M , n } <-> E! n ( n e. (Vtx ` G ) /\ C = { M , n } ) )
11		prcom 4267	. . . . . . . . . . . . . . . 16 |- { M , n } = { n , M }
12	11	eqeq2i 2634	. . . . . . . . . . . . . . 15 |- ( C = { M , n } <-> C = { n , M } )
13	12	biimpi 206	. . . . . . . . . . . . . 14 |- ( C = { M , n } -> C = { n , M } )
14	13	eleq1d 2686	. . . . . . . . . . . . 13 |- ( C = { M , n } -> ( C e. E <-> { n , M } e. E ) )
15	14	biimpcd 239	. . . . . . . . . . . 12 |- ( C e. E -> ( C = { M , n } -> { n , M } e. E ) )
16	15	ad2antrl 764	. . . . . . . . . . 11 |- ( ( ( G e. USGraph /\ M e. V ) /\ ( C e. E /\ M e. C ) ) -> ( C = { M , n } -> { n , M } e. E ) )
17	16	adantld 483	. . . . . . . . . 10 |- ( ( ( G e. USGraph /\ M e. V ) /\ ( C e. E /\ M e. C ) ) -> ( ( n e. (Vtx ` G ) /\ C = { M , n } ) -> { n , M } e. E ) )
18	17	imp 445	. . . . . . . . 9 |- ( ( ( ( G e. USGraph /\ M e. V ) /\ ( C e. E /\ M e. C ) ) /\ ( n e. (Vtx ` G ) /\ C = { M , n } ) ) -> { n , M } e. E )
19		simprr 796	. . . . . . . . 9 |- ( ( ( ( G e. USGraph /\ M e. V ) /\ ( C e. E /\ M e. C ) ) /\ ( n e. (Vtx ` G ) /\ C = { M , n } ) ) -> C = { M , n } )
20	18, 19	jca 554	. . . . . . . 8 |- ( ( ( ( G e. USGraph /\ M e. V ) /\ ( C e. E /\ M e. C ) ) /\ ( n e. (Vtx ` G ) /\ C = { M , n } ) ) -> ( { n , M } e. E /\ C = { M , n } ) )
21		simpl 473	. . . . . . . . . 10 |- ( ( { n , M } e. E /\ C = { M , n } ) -> { n , M } e. E )
22		eqid 2622	. . . . . . . . . . . 12 |- (Vtx ` G ) = (Vtx ` G )
23	3, 22	usgrpredgv 26089	. . . . . . . . . . 11 |- ( ( G e. USGraph /\ { n , M } e. E ) -> ( n e. (Vtx ` G ) /\ M e. (Vtx ` G ) ) )
24	23	simpld 475	. . . . . . . . . 10 |- ( ( G e. USGraph /\ { n , M } e. E ) -> n e. (Vtx ` G ) )
25	2, 21, 24	syl2an 494	. . . . . . . . 9 |- ( ( ( ( G e. USGraph /\ M e. V ) /\ ( C e. E /\ M e. C ) ) /\ ( { n , M } e. E /\ C = { M , n } ) ) -> n e. (Vtx ` G ) )
26		simprr 796	. . . . . . . . 9 |- ( ( ( ( G e. USGraph /\ M e. V ) /\ ( C e. E /\ M e. C ) ) /\ ( { n , M } e. E /\ C = { M , n } ) ) -> C = { M , n } )
27	25, 26	jca 554	. . . . . . . 8 |- ( ( ( ( G e. USGraph /\ M e. V ) /\ ( C e. E /\ M e. C ) ) /\ ( { n , M } e. E /\ C = { M , n } ) ) -> ( n e. (Vtx ` G ) /\ C = { M , n } ) )
28	20, 27	impbida 877	. . . . . . 7 |- ( ( ( G e. USGraph /\ M e. V ) /\ ( C e. E /\ M e. C ) ) -> ( ( n e. (Vtx ` G ) /\ C = { M , n } ) <-> ( { n , M } e. E /\ C = { M , n } ) ) )
29	28	eubidv 2490	. . . . . 6 |- ( ( ( G e. USGraph /\ M e. V ) /\ ( C e. E /\ M e. C ) ) -> ( E! n ( n e. (Vtx ` G ) /\ C = { M , n } ) <-> E! n ( { n , M } e. E /\ C = { M , n } ) ) )
30	29	biimpd 219	. . . . 5 |- ( ( ( G e. USGraph /\ M e. V ) /\ ( C e. E /\ M e. C ) ) -> ( E! n ( n e. (Vtx ` G ) /\ C = { M , n } ) -> E! n ( { n , M } e. E /\ C = { M , n } ) ) )
31	10, 30	syl5bi 232	. . . 4 |- ( ( ( G e. USGraph /\ M e. V ) /\ ( C e. E /\ M e. C ) ) -> ( E! n e. (Vtx ` G ) C = { M , n } -> E! n ( { n , M } e. E /\ C = { M , n } ) ) )
32	9, 31	mpd 15	. . 3 |- ( ( ( G e. USGraph /\ M e. V ) /\ ( C e. E /\ M e. C ) ) -> E! n ( { n , M } e. E /\ C = { M , n } ) )
33		edgnbusgreu.n	. . . . . . . 8 |- N = ( G NeighbVtx M )
34	33	eleq2i 2693	. . . . . . 7 |- ( n e. N <-> n e. ( G NeighbVtx M ) )
35	3	nbusgreledg 26249	. . . . . . 7 |- ( G e. USGraph -> ( n e. ( G NeighbVtx M ) <-> { n , M } e. E ) )
36	34, 35	syl5bb 272	. . . . . 6 |- ( G e. USGraph -> ( n e. N <-> { n , M } e. E ) )
37	36	anbi1d 741	. . . . 5 |- ( G e. USGraph -> ( ( n e. N /\ C = { M , n } ) <-> ( { n , M } e. E /\ C = { M , n } ) ) )
38	37	ad2antrr 762	. . . 4 |- ( ( ( G e. USGraph /\ M e. V ) /\ ( C e. E /\ M e. C ) ) -> ( ( n e. N /\ C = { M , n } ) <-> ( { n , M } e. E /\ C = { M , n } ) ) )
39	38	eubidv 2490	. . 3 |- ( ( ( G e. USGraph /\ M e. V ) /\ ( C e. E /\ M e. C ) ) -> ( E! n ( n e. N /\ C = { M , n } ) <-> E! n ( { n , M } e. E /\ C = { M , n } ) ) )
40	32, 39	mpbird 247	. 2 |- ( ( ( G e. USGraph /\ M e. V ) /\ ( C e. E /\ M e. C ) ) -> E! n ( n e. N /\ C = { M , n } ) )
41		df-reu 2919	. 2 |- ( E! n e. N C = { M , n } <-> E! n ( n e. N /\ C = { M , n } ) )
42	40, 41	sylibr 224	1 |- ( ( ( G e. USGraph /\ M e. V ) /\ ( C e. E /\ M e. C ) ) -> E! n e. N C = { M , n } )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   E!weu 2470   E!wreu 2914   {cpr 4179   ` cfv 5888  (class class class)co 6650  Vtxcvtx 25874  Edgcedg 25939   USGraph cusgr 26044   NeighbVtx cnbgr 26224

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-uspgr 26045  df-usgr 26046  df-nbgr 26228

This theorem is referenced by:  nbusgrf1o0  26271