Theorem vdegp1bid 16327

Description: The induction step for a vertex degree calculation, for example in the Königsberg graph. If the degree of U in the edge set E is P, then adding { U , X } to the edge set, where X =/= U, yields degree P + 1. (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 )
vdegp1bid.x	|- ( ph -> X e. V )
vdegp1bid.xu	|- ( ph -> X =/= U )
vdegp1bid.f	|- ( ph -> (iEdg ` F ) = ( I ++ <" { U , X } "> ) )

Assertion

Ref	Expression
vdegp1bid	|- ( ph -> ( (VtxDeg ` F ) ` U ) = ( P + 1 ) )

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

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

Proof of Theorem vdegp1bid

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		vdegp1bid.f	. . . 4 |- ( ph -> (iEdg ` F ) = ( I ++ <" { U , X } "> ) )
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.u	. . . . . 6 |- ( ph -> U e. V )
12		vdegp1bid.x	. . . . . 6 |- ( ph -> X e. V )
13		prexg 4327	. . . . . 6 |- ( ( U e. V /\ X e. V ) -> { U , X } e. _V )
14	11, 12, 13	syl2anc 411	. . . . 5 |- ( ph -> { U , X } e. _V )
15		cats1un 11417	. . . . 5 |- ( ( I e. Word _V /\ { U , X } e. _V ) -> ( I ++ <" { U , X } "> ) = ( I u. { <. (♯ ` I ) , { U , X } >. } ) )
16	10, 14, 15	syl2anc 411	. . . 4 |- ( ph -> ( I ++ <" { U , X } "> ) = ( I u. { <. (♯ ` I ) , { U , X } >. } ) )
17	8, 16	eqtrd 2267	. . 3 |- ( ph -> (iEdg ` F ) = ( I u. { <. (♯ ` I ) , { U , X } >. } ) )
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.fi	. . 3 |- ( ph -> V e. Fin )
23	1	1vgrex 16032	. . . . . 6 |- ( U e. V -> G e. _V )
24	11, 23	syl 14	. . . . 5 |- ( ph -> G e. _V )
25	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 ) } ) )
26	24, 3, 25	syl2anc 411	. . . 4 |- ( ph -> ( G e. UPGraph <-> I e. Word { x e. ~P V | ( x ~~ 1o \/ x ~~ 2o ) } ) )
27	3, 26	mpbird 167	. . 3 |- ( ph -> G e. UPGraph )
28		wrddm 11236	. . . . 5 |- ( I e. Word { x e. ~P V | ( x ~~ 1o \/ x ~~ 2o ) } -> dom I = ( 0..^ (♯ ` I ) ) )
29	3, 28	syl 14	. . . 4 |- ( ph -> dom I = ( 0..^ (♯ ` I ) ) )
30		0z 9590	. . . . 5 |- 0 e. ZZ
31	19	nn0zd 9701	. . . . 5 |- ( ph -> (♯ ` I ) e. ZZ )
32		fzofig 10798	. . . . 5 |- ( ( 0 e. ZZ /\ (♯ ` I ) e. ZZ ) -> ( 0..^ (♯ ` I ) ) e. Fin )
33	30, 31, 32	sylancr 414	. . . 4 |- ( ph -> ( 0..^ (♯ ` I ) ) e. Fin )
34	29, 33	eqeltrd 2311	. . 3 |- ( ph -> dom I e. Fin )
35		prelpwi 4332	. . . 4 |- ( ( U e. V /\ X e. V ) -> { U , X } e. ~P V )
36	11, 12, 35	syl2anc 411	. . 3 |- ( ph -> { U , X } e. ~P V )
37		vdegp1bid.xu	. . . . 5 |- ( ph -> X =/= U )
38	37	necomd 2500	. . . 4 |- ( ph -> U =/= X )
39		pr2ne 7491	. . . . 5 |- ( ( U e. V /\ X e. V ) -> ( { U , X } ~~ 2o <-> U =/= X ) )
40	11, 12, 39	syl2anc 411	. . . 4 |- ( ph -> ( { U , X } ~~ 2o <-> U =/= X ) )
41	38, 40	mpbird 167	. . 3 |- ( ph -> { U , X } ~~ 2o )
42		prid1g 3797	. . . 4 |- ( U e. V -> U e. { U , X } )
43	11, 42	syl 14	. . 3 |- ( ph -> U e. { U , X } )
44	1, 2, 6, 7, 17, 19, 21, 11, 22, 27, 34, 36, 41, 43	p1evtxdp1fi 16325	. 2 |- ( ph -> ( (VtxDeg ` F ) ` U ) = ( ( (VtxDeg ` G ) ` U ) + 1 ) )
45		vdegp1aid.d	. . 3 |- ( ph -> ( (VtxDeg ` G ) ` U ) = P )
46	45	oveq1d 6067	. 2 |- ( ph -> ( ( (VtxDeg ` G ) ` U ) + 1 ) = ( P + 1 ) )
47	44, 46	eqtrd 2267	1 |- ( ph -> ( (VtxDeg ` F ) ` U ) = ( P + 1 ) )

Syntax hints:   -> 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   -->wf 5350   ` cfv 5354  (class class class)co 6052   1oc1o 6642   2oc2o 6643   ~~ cen 6975   Fincfn 6977   0cc0 8129   1c1 8130   + caddc 8132   NN0cn0 9498   ZZcz 9579  ..^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-umgren 16106  df-vtxdg 16299

This theorem is referenced by:  vdegp1cid  16328  konigsberglem1  16500  konigsberglem2  16501