Theorem vdegp1aid 16326

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 11234	. . . . 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 5514	. . 3 |- ( ph -> Fun I )
7		vdegp1aid.vf	. . 3 |- ( ph -> (Vtx ` F ) = V )
8		vdegp1aid.f	. . . 4 |- ( ph -> (iEdg ` F ) = ( I ++ <" { X , Y } "> ) )
9		wrdv 11244	. . . . . 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 4327	. . . . . 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 11417	. . . . 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 2267	. . 3 |- ( ph -> (iEdg ` F ) = ( I u. { <. (♯ ` I ) , { X , Y } >. } ) )
18		lencl 11232	. . . 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 11245	. . . 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 16032	. . . . . 6 |- ( X e. V -> G e. _V )
25	11, 24	syl 14	. . . . 5 |- ( ph -> G e. _V )
26	1, 2	wrdupgren 16108	. . . . 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 11247	. . . . 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 7206	. . . 4 |- ( ( I e. Fin /\ Fun I ) -> dom I e. Fin )
32	30, 6, 31	syl2anc 411	. . 3 |- ( ph -> dom I e. Fin )
33		prelpwi 4332	. . . 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 7491	. . . . 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 2435	. . . . . . 7 |- ( ph -> -. X = U )
41	40	neqcomd 2239	. . . . . 6 |- ( ph -> -. U = X )
42		vdegp1aid.yu	. . . . . . . 8 |- ( ph -> Y =/= U )
43	42	neneqd 2435	. . . . . . 7 |- ( ph -> -. Y = U )
44	43	neqcomd 2239	. . . . . 6 |- ( ph -> -. U = Y )
45		ioran 760	. . . . . 6 |- ( -. ( U = X \/ U = Y ) <-> ( -. U = X /\ -. U = Y ) )
46	41, 44, 45	sylanbrc 417	. . . . 5 |- ( ph -> -. ( U = X \/ U = Y ) )
47		elpri 3714	. . . . 5 |- ( U e. { X , Y } -> ( U = X \/ U = Y ) )
48	46, 47	nsyl 633	. . . 4 |- ( ph -> -. U e. { X , Y } )
49		df-nel 2510	. . . 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 16324	. 2 |- ( ph -> ( (VtxDeg ` F ) ` U ) = ( (VtxDeg ` G ) ` U ) )
52		vdegp1aid.d	. 2 |- ( ph -> ( (VtxDeg ` G ) ` U ) = P )
53	51, 52	eqtrd 2267	1 |- ( ph -> ( (VtxDeg ` F ) ` U ) = P )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 105   \/ wo 716   = wceq 1398   e. wcel 2205   =/= wne 2414   e/ wnel 2509   {crab 2526   _Vcvv 2815   u. cun 3211   ~Pcpw 3671   {csn 3691   {cpr 3692   <.cop 3694   class class class wbr 4111   dom cdm 4751   Fun wfun 5348   -->wf 5350   ` cfv 5354  (class class class)co 6052   1oc1o 6642   2oc2o 6643   ~~ cen 6975   Fincfn 6977   0cc0 8129   NN0cn0 9498  ..^cfzo 10480  ♯chash 11142  Word cword 11228   ++ cconcat 11282   <"cs1 11307  Vtxcvtx 16024  iEdgciedg 16025  UPGraphcupgr 16103  VtxDegcvtxdg 16298

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4227  ax-sep 4230  ax-nul 4238  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-iinf 4712  ax-cnex 8220  ax-resscn 8221  ax-1cn 8222  ax-1re 8223  ax-icn 8224  ax-addcl 8225  ax-addrcl 8226  ax-mulcl 8227  ax-addcom 8229  ax-mulcom 8230  ax-addass 8231  ax-mulass 8232  ax-distr 8233  ax-i2m1 8234  ax-0lt1 8235  ax-1rid 8236  ax-0id 8237  ax-rnegex 8238  ax-cnre 8240  ax-pre-ltirr 8241  ax-pre-ltwlin 8242  ax-pre-lttrn 8243  ax-pre-apti 8244  ax-pre-ltadd 8245

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-if 3623  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-tr 4211  df-id 4416  df-iord 4489  df-on 4491  df-ilim 4492  df-suc 4494  df-iom 4715  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-1st 6336  df-2nd 6337  df-recs 6538  df-irdg 6603  df-frec 6624  df-1o 6649  df-2o 6650  df-oadd 6653  df-er 6769  df-en 6978  df-dom 6979  df-fin 6980  df-pnf 8312  df-mnf 8313  df-xr 8314  df-ltxr 8315  df-le 8316  df-sub 8448  df-neg 8449  df-inn 9240  df-2 9298  df-3 9299  df-4 9300  df-5 9301  df-6 9302  df-7 9303  df-8 9304  df-9 9305  df-n0 9499  df-z 9580  df-dec 9713  df-uz 9857  df-xadd 10109  df-fz 10346  df-fzo 10481  df-ihash 11143  df-word 11229  df-concat 11283  df-s1 11308  df-ndx 13232  df-slot 13233  df-base 13235  df-edgf 16017  df-vtx 16026  df-iedg 16027  df-upgren 16105  df-vtxdg 16299

This theorem is referenced by:  konigsberglem1  16500  konigsberglem2  16501  konigsberglem3  16502