Theorem vdegp1aid 16125

Description: The induction step for a vertex degree calculation. If the degree of U in the edge set E is P, then adding { X , Y } to the edge set, where X =/= U =/= Y, yields degree P as well. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 28-Feb-2016.) (Revised by AV, 3-Mar-2021.)

Hypotheses

Ref	Expression
vdegp1ai.vg	|- V = (Vtx ` G )
vdegp1aid.u	|- ( ph -> U e. V )
vdegp1ai.i	|- I = (iEdg ` G )
vdegp1aid.w	|- ( ph -> I e. Word { x e. ~P V | ( x ~~ 1o \/ x ~~ 2o ) } )
vdegp1aid.d	|- ( ph -> ( (VtxDeg ` G ) ` U ) = P )
vdegp1aid.vf	|- ( ph -> (Vtx ` F ) = V )
vdegp1aid.fi	|- ( ph -> V e. Fin )
vdegp1aid.x	|- ( ph -> X e. V )
vdegp1aid.xu	|- ( ph -> X =/= U )
vdegp1aid.y	|- ( ph -> Y e. V )
vdegp1aid.yu	|- ( ph -> Y =/= U )
vdegp1aid.xy	|- ( ph -> X =/= Y )
vdegp1aid.f	|- ( ph -> (iEdg ` F ) = ( I ++ <" { X , Y } "> ) )

Assertion

Ref	Expression
vdegp1aid	|- ( ph -> ( (VtxDeg ` F ) ` U ) = P )

Distinct variable groups:   x, U   x, V   x, X   x, Y   x, G

Allowed substitution hints:   ph( x)   P( x)   F( x)   I( x)

Proof of Theorem vdegp1aid

Step	Hyp	Ref	Expression
1		vdegp1ai.vg	. . 3 |- V = (Vtx ` G )
2		vdegp1ai.i	. . 3 |- I = (iEdg ` G )
3		vdegp1aid.w	. . . . 5 |- ( ph -> I e. Word { x e. ~P V | ( x ~~ 1o \/ x ~~ 2o ) } )
4		wrdf 11112	. . . . 5 |- ( I e. Word { x e. ~P V | ( x ~~ 1o \/ x ~~ 2o ) } -> I : ( 0..^ (♯ ` I ) ) --> { x e. ~P V | ( x ~~ 1o \/ x ~~ 2o ) } )
5	3, 4	syl 14	. . . 4 |- ( ph -> I : ( 0..^ (♯ ` I ) ) --> { x e. ~P V | ( x ~~ 1o \/ x ~~ 2o ) } )
6	5	ffund 5483	. . 3 |- ( ph -> Fun I )
7		vdegp1aid.vf	. . 3 |- ( ph -> (Vtx ` F ) = V )
8		vdegp1aid.f	. . . 4 |- ( ph -> (iEdg ` F ) = ( I ++ <" { X , Y } "> ) )
9		wrdv 11122	. . . . . 6 |- ( I e. Word { x e. ~P V | ( x ~~ 1o \/ x ~~ 2o ) } -> I e. Word _V )
10	3, 9	syl 14	. . . . 5 |- ( ph -> I e. Word _V )
11		vdegp1aid.x	. . . . . 6 |- ( ph -> X e. V )
12		vdegp1aid.y	. . . . . 6 |- ( ph -> Y e. V )
13		prexg 4299	. . . . . 6 |- ( ( X e. V /\ Y e. V ) -> { X , Y } e. _V )
14	11, 12, 13	syl2anc 411	. . . . 5 |- ( ph -> { X , Y } e. _V )
15		cats1un 11295	. . . . 5 |- ( ( I e. Word _V /\ { X , Y } e. _V ) -> ( I ++ <" { X , Y } "> ) = ( I u. { <. (♯ ` I ) , { X , Y } >. } ) )
16	10, 14, 15	syl2anc 411	. . . 4 |- ( ph -> ( I ++ <" { X , Y } "> ) = ( I u. { <. (♯ ` I ) , { X , Y } >. } ) )
17	8, 16	eqtrd 2262	. . 3 |- ( ph -> (iEdg ` F ) = ( I u. { <. (♯ ` I ) , { X , Y } >. } ) )
18		lencl 11110	. . . 4 |- ( I e. Word { x e. ~P V | ( x ~~ 1o \/ x ~~ 2o ) } -> (♯ ` I ) e. NN0 )
19	3, 18	syl 14	. . 3 |- ( ph -> (♯ ` I ) e. NN0 )
20		wrdlndm 11123	. . . 4 |- ( I e. Word { x e. ~P V | ( x ~~ 1o \/ x ~~ 2o ) } -> (♯ ` I ) e/ dom I )
21	3, 20	syl 14	. . 3 |- ( ph -> (♯ ` I ) e/ dom I )
22		vdegp1aid.u	. . 3 |- ( ph -> U e. V )
23		vdegp1aid.fi	. . 3 |- ( ph -> V e. Fin )
24	1	1vgrex 15864	. . . . . 6 |- ( X e. V -> G e. _V )
25	11, 24	syl 14	. . . . 5 |- ( ph -> G e. _V )
26	1, 2	wrdupgren 15940	. . . . 5 |- ( ( G e. _V /\ I e. Word { x e. ~P V | ( x ~~ 1o \/ x ~~ 2o ) } ) -> ( G e. UPGraph <-> I e. Word { x e. ~P V | ( x ~~ 1o \/ x ~~ 2o ) } ) )
27	25, 3, 26	syl2anc 411	. . . 4 |- ( ph -> ( G e. UPGraph <-> I e. Word { x e. ~P V | ( x ~~ 1o \/ x ~~ 2o ) } ) )
28	3, 27	mpbird 167	. . 3 |- ( ph -> G e. UPGraph )
29		wrdfin 11125	. . . . 5 |- ( I e. Word { x e. ~P V | ( x ~~ 1o \/ x ~~ 2o ) } -> I e. Fin )
30	3, 29	syl 14	. . . 4 |- ( ph -> I e. Fin )
31		fundmfi 7130	. . . 4 |- ( ( I e. Fin /\ Fun I ) -> dom I e. Fin )
32	30, 6, 31	syl2anc 411	. . 3 |- ( ph -> dom I e. Fin )
33		prelpwi 4304	. . . 4 |- ( ( X e. V /\ Y e. V ) -> { X , Y } e. ~P V )
34	11, 12, 33	syl2anc 411	. . 3 |- ( ph -> { X , Y } e. ~P V )
35		vdegp1aid.xy	. . . 4 |- ( ph -> X =/= Y )
36		pr2ne 7391	. . . . 5 |- ( ( X e. V /\ Y e. V ) -> ( { X , Y } ~~ 2o <-> X =/= Y ) )
37	11, 12, 36	syl2anc 411	. . . 4 |- ( ph -> ( { X , Y } ~~ 2o <-> X =/= Y ) )
38	35, 37	mpbird 167	. . 3 |- ( ph -> { X , Y } ~~ 2o )
39		vdegp1aid.xu	. . . . . . . 8 |- ( ph -> X =/= U )
40	39	neneqd 2421	. . . . . . 7 |- ( ph -> -. X = U )
41	40	neqcomd 2234	. . . . . 6 |- ( ph -> -. U = X )
42		vdegp1aid.yu	. . . . . . . 8 |- ( ph -> Y =/= U )
43	42	neneqd 2421	. . . . . . 7 |- ( ph -> -. Y = U )
44	43	neqcomd 2234	. . . . . 6 |- ( ph -> -. U = Y )
45		ioran 757	. . . . . 6 |- ( -. ( U = X \/ U = Y ) <-> ( -. U = X /\ -. U = Y ) )
46	41, 44, 45	sylanbrc 417	. . . . 5 |- ( ph -> -. ( U = X \/ U = Y ) )
47		elpri 3690	. . . . 5 |- ( U e. { X , Y } -> ( U = X \/ U = Y ) )
48	46, 47	nsyl 631	. . . 4 |- ( ph -> -. U e. { X , Y } )
49		df-nel 2496	. . . 4 |- ( U e/ { X , Y } <-> -. U e. { X , Y } )
50	48, 49	sylibr 134	. . 3 |- ( ph -> U e/ { X , Y } )
51	1, 2, 6, 7, 17, 19, 21, 22, 23, 28, 32, 34, 38, 14, 50	p1evtxdeqfi 16123	. 2 |- ( ph -> ( (VtxDeg ` F ) ` U ) = ( (VtxDeg ` G ) ` U ) )
52		vdegp1aid.d	. 2 |- ( ph -> ( (VtxDeg ` G ) ` U ) = P )
53	51, 52	eqtrd 2262	1 |- ( ph -> ( (VtxDeg ` F ) ` U ) = P )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 105   \/ wo 713   = wceq 1395   e. wcel 2200   =/= wne 2400   e/ wnel 2495   {crab 2512   _Vcvv 2800   u. cun 3196   ~Pcpw 3650   {csn 3667   {cpr 3668   <.cop 3670   class class class wbr 4086   dom cdm 4723   Fun wfun 5318   -->wf 5320   ` cfv 5324  (class class class)co 6013   1oc1o 6570   2oc2o 6571   ~~ cen 6902   Fincfn 6904   0cc0 8025   NN0cn0 9395  ..^cfzo 10370  ♯chash 11030  Word cword 11106   ++ cconcat 11160   <"cs1 11185  Vtxcvtx 15856  iEdgciedg 15857  UPGraphcupgr 15935  VtxDegcvtxdg 16097

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684  ax-cnex 8116  ax-resscn 8117  ax-1cn 8118  ax-1re 8119  ax-icn 8120  ax-addcl 8121  ax-addrcl 8122  ax-mulcl 8123  ax-addcom 8125  ax-mulcom 8126  ax-addass 8127  ax-mulass 8128  ax-distr 8129  ax-i2m1 8130  ax-0lt1 8131  ax-1rid 8132  ax-0id 8133  ax-rnegex 8134  ax-cnre 8136  ax-pre-ltirr 8137  ax-pre-ltwlin 8138  ax-pre-lttrn 8139  ax-pre-apti 8140  ax-pre-ltadd 8141

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-if 3604  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-id 4388  df-iord 4461  df-on 4463  df-ilim 4464  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-recs 6466  df-irdg 6531  df-frec 6552  df-1o 6577  df-2o 6578  df-oadd 6581  df-er 6697  df-en 6905  df-dom 6906  df-fin 6907  df-pnf 8209  df-mnf 8210  df-xr 8211  df-ltxr 8212  df-le 8213  df-sub 8345  df-neg 8346  df-inn 9137  df-2 9195  df-3 9196  df-4 9197  df-5 9198  df-6 9199  df-7 9200  df-8 9201  df-9 9202  df-n0 9396  df-z 9473  df-dec 9605  df-uz 9749  df-xadd 10001  df-fz 10237  df-fzo 10371  df-ihash 11031  df-word 11107  df-concat 11161  df-s1 11186  df-ndx 13078  df-slot 13079  df-base 13081  df-edgf 15849  df-vtx 15858  df-iedg 15859  df-upgren 15937  df-vtxdg 16098

This theorem is referenced by: (None)