Theorem 1hevtxdg1 26402

Description: The vertex degree of vertex D in a graph G with only one hyperedge E (not being a loop) is 1 if D is incident with the edge E. (Contributed by AV, 2-Mar-2021.) (Proof shortened by AV, 17-Apr-2021.)

Hypotheses

Ref	Expression
1hevtxdg0.i	|- ( ph -> (iEdg ` G ) = { <. A , E >. } )
1hevtxdg0.v	|- ( ph -> (Vtx ` G ) = V )
1hevtxdg0.a	|- ( ph -> A e. X )
1hevtxdg0.d	|- ( ph -> D e. V )
1hevtxdg1.e	|- ( ph -> E e. ~P V )
1hevtxdg1.n	|- ( ph -> D e. E )
1hevtxdg1.l	|- ( ph -> 2 <_ ( # ` E ) )

Assertion

Ref	Expression
1hevtxdg1	|- ( ph -> ( (VtxDeg ` G ) ` D ) = 1 )

Proof of Theorem 1hevtxdg1

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		1hevtxdg0.i	. . . 4 |- ( ph -> (iEdg ` G ) = { <. A , E >. } )
2	1	dmeqd 5326	. . 3 |- ( ph -> dom (iEdg ` G ) = dom { <. A , E >. } )
3		1hevtxdg1.e	. . . 4 |- ( ph -> E e. ~P V )
4		dmsnopg 5606	. . . 4 |- ( E e. ~P V -> dom { <. A , E >. } = { A } )
5	3, 4	syl 17	. . 3 |- ( ph -> dom { <. A , E >. } = { A } )
6	2, 5	eqtrd 2656	. 2 |- ( ph -> dom (iEdg ` G ) = { A } )
7		1hevtxdg0.a	. . . . . . 7 |- ( ph -> A e. X )
8		1hevtxdg0.v	. . . . . . . . . 10 |- ( ph -> (Vtx ` G ) = V )
9	8	pweqd 4163	. . . . . . . . 9 |- ( ph -> ~P (Vtx ` G ) = ~P V )
10	3, 9	eleqtrrd 2704	. . . . . . . 8 |- ( ph -> E e. ~P (Vtx ` G ) )
11		1hevtxdg1.l	. . . . . . . 8 |- ( ph -> 2 <_ ( # ` E ) )
12		fveq2 6191	. . . . . . . . . 10 |- ( x = E -> ( # ` x ) = ( # ` E ) )
13	12	breq2d 4665	. . . . . . . . 9 |- ( x = E -> ( 2 <_ ( # ` x ) <-> 2 <_ ( # ` E ) ) )
14	13	elrab 3363	. . . . . . . 8 |- ( E e. { x e. ~P (Vtx ` G ) | 2 <_ ( # ` x ) } <-> ( E e. ~P (Vtx ` G ) /\ 2 <_ ( # ` E ) ) )
15	10, 11, 14	sylanbrc 698	. . . . . . 7 |- ( ph -> E e. { x e. ~P (Vtx ` G ) | 2 <_ ( # ` x ) } )
16	7, 15	fsnd 6179	. . . . . 6 |- ( ph -> { <. A , E >. } : { A } --> { x e. ~P (Vtx ` G ) | 2 <_ ( # ` x ) } )
17	16	adantr 481	. . . . 5 |- ( ( ph /\ dom (iEdg ` G ) = { A } ) -> { <. A , E >. } : { A } --> { x e. ~P (Vtx ` G ) | 2 <_ ( # ` x ) } )
18	1	adantr 481	. . . . . 6 |- ( ( ph /\ dom (iEdg ` G ) = { A } ) -> (iEdg ` G ) = { <. A , E >. } )
19		simpr 477	. . . . . 6 |- ( ( ph /\ dom (iEdg ` G ) = { A } ) -> dom (iEdg ` G ) = { A } )
20	18, 19	feq12d 6033	. . . . 5 |- ( ( ph /\ dom (iEdg ` G ) = { A } ) -> ( (iEdg ` G ) : dom (iEdg ` G ) --> { x e. ~P (Vtx ` G ) | 2 <_ ( # ` x ) } <-> { <. A , E >. } : { A } --> { x e. ~P (Vtx ` G ) | 2 <_ ( # ` x ) } ) )
21	17, 20	mpbird 247	. . . 4 |- ( ( ph /\ dom (iEdg ` G ) = { A } ) -> (iEdg ` G ) : dom (iEdg ` G ) --> { x e. ~P (Vtx ` G ) | 2 <_ ( # ` x ) } )
22		1hevtxdg0.d	. . . . . 6 |- ( ph -> D e. V )
23	22, 8	eleqtrrd 2704	. . . . 5 |- ( ph -> D e. (Vtx ` G ) )
24	23	adantr 481	. . . 4 |- ( ( ph /\ dom (iEdg ` G ) = { A } ) -> D e. (Vtx ` G ) )
25		eqid 2622	. . . . 5 |- (Vtx ` G ) = (Vtx ` G )
26		eqid 2622	. . . . 5 |- (iEdg ` G ) = (iEdg ` G )
27		eqid 2622	. . . . 5 |- dom (iEdg ` G ) = dom (iEdg ` G )
28		eqid 2622	. . . . 5 |- (VtxDeg ` G ) = (VtxDeg ` G )
29	25, 26, 27, 28	vtxdlfgrval 26381	. . . 4 |- ( ( (iEdg ` G ) : dom (iEdg ` G ) --> { x e. ~P (Vtx ` G ) | 2 <_ ( # ` x ) } /\ D e. (Vtx ` G ) ) -> ( (VtxDeg ` G ) ` D ) = ( # ` { x e. dom (iEdg ` G ) | D e. ( (iEdg ` G ) ` x ) } ) )
30	21, 24, 29	syl2anc 693	. . 3 |- ( ( ph /\ dom (iEdg ` G ) = { A } ) -> ( (VtxDeg ` G ) ` D ) = ( # ` { x e. dom (iEdg ` G ) | D e. ( (iEdg ` G ) ` x ) } ) )
31		rabeq 3192	. . . . 5 |- ( dom (iEdg ` G ) = { A } -> { x e. dom (iEdg ` G ) | D e. ( (iEdg ` G ) ` x ) } = { x e. { A } | D e. ( (iEdg ` G ) ` x ) } )
32	31	adantl 482	. . . 4 |- ( ( ph /\ dom (iEdg ` G ) = { A } ) -> { x e. dom (iEdg ` G ) | D e. ( (iEdg ` G ) ` x ) } = { x e. { A } | D e. ( (iEdg ` G ) ` x ) } )
33	32	fveq2d 6195	. . 3 |- ( ( ph /\ dom (iEdg ` G ) = { A } ) -> ( # ` { x e. dom (iEdg ` G ) | D e. ( (iEdg ` G ) ` x ) } ) = ( # ` { x e. { A } | D e. ( (iEdg ` G ) ` x ) } ) )
34		fveq2 6191	. . . . . . . . 9 |- ( x = A -> ( (iEdg ` G ) ` x ) = ( (iEdg ` G ) ` A ) )
35	34	eleq2d 2687	. . . . . . . 8 |- ( x = A -> ( D e. ( (iEdg ` G ) ` x ) <-> D e. ( (iEdg ` G ) ` A ) ) )
36	35	rabsnif 4258	. . . . . . 7 |- { x e. { A } | D e. ( (iEdg ` G ) ` x ) } = if ( D e. ( (iEdg ` G ) ` A ) , { A } , (/) )
37		1hevtxdg1.n	. . . . . . . . 9 |- ( ph -> D e. E )
38	1	fveq1d 6193	. . . . . . . . . 10 |- ( ph -> ( (iEdg ` G ) ` A ) = ( { <. A , E >. } ` A ) )
39		fvsng 6447	. . . . . . . . . . 11 |- ( ( A e. X /\ E e. ~P V ) -> ( { <. A , E >. } ` A ) = E )
40	7, 3, 39	syl2anc 693	. . . . . . . . . 10 |- ( ph -> ( { <. A , E >. } ` A ) = E )
41	38, 40	eqtrd 2656	. . . . . . . . 9 |- ( ph -> ( (iEdg ` G ) ` A ) = E )
42	37, 41	eleqtrrd 2704	. . . . . . . 8 |- ( ph -> D e. ( (iEdg ` G ) ` A ) )
43	42	iftrued 4094	. . . . . . 7 |- ( ph -> if ( D e. ( (iEdg ` G ) ` A ) , { A } , (/) ) = { A } )
44	36, 43	syl5eq 2668	. . . . . 6 |- ( ph -> { x e. { A } | D e. ( (iEdg ` G ) ` x ) } = { A } )
45	44	fveq2d 6195	. . . . 5 |- ( ph -> ( # ` { x e. { A } | D e. ( (iEdg ` G ) ` x ) } ) = ( # ` { A } ) )
46		hashsng 13159	. . . . . 6 |- ( A e. X -> ( # ` { A } ) = 1 )
47	7, 46	syl 17	. . . . 5 |- ( ph -> ( # ` { A } ) = 1 )
48	45, 47	eqtrd 2656	. . . 4 |- ( ph -> ( # ` { x e. { A } | D e. ( (iEdg ` G ) ` x ) } ) = 1 )
49	48	adantr 481	. . 3 |- ( ( ph /\ dom (iEdg ` G ) = { A } ) -> ( # ` { x e. { A } | D e. ( (iEdg ` G ) ` x ) } ) = 1 )
50	30, 33, 49	3eqtrd 2660	. 2 |- ( ( ph /\ dom (iEdg ` G ) = { A } ) -> ( (VtxDeg ` G ) ` D ) = 1 )
51	6, 50	mpdan 702	1 |- ( ph -> ( (VtxDeg ` G ) ` D ) = 1 )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {crab 2916   (/)c0 3915   ifcif 4086   ~Pcpw 4158   {csn 4177   <.cop 4183   class class class wbr 4653   dom cdm 5114   -->wf 5884   ` cfv 5888   1c1 9937   <_ cle 10075   2c2 11070   #chash 13117  Vtxcvtx 25874  iEdgciedg 25875  VtxDegcvtxdg 26361

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-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-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-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  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-xadd 11947  df-fz 12327  df-hash 13118  df-vtxdg 26362

This theorem is referenced by:  1hegrvtxdg1  26403  p1evtxdp1  26410