Theorem vdegp1bid 16297

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 11223	. . . . 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 5511	. . 3 |- ( ph -> Fun I )
7		vdegp1aid.vf	. . 3 |- ( ph -> (Vtx ` F ) = V )
8		vdegp1bid.f	. . . 4 |- ( ph -> (iEdg ` F ) = ( I ++ <" { U , X } "> ) )
9		wrdv 11233	. . . . . 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 4324	. . . . . 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 11406	. . . . 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 2265	. . 3 |- ( ph -> (iEdg ` F ) = ( I u. { <. (♯ ` I ) , { U , X } >. } ) )
18		lencl 11221	. . . 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 11234	. . . 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 16002	. . . . . 6 |- ( U e. V -> G e. _V )
24	11, 23	syl 14	. . . . 5 |- ( ph -> G e. _V )
25	1, 2	wrdupgren 16078	. . . . 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 11225	. . . . 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 9584	. . . . 5 |- 0 e. ZZ
31	19	nn0zd 9694	. . . . 5 |- ( ph -> (♯ ` I ) e. ZZ )
32		fzofig 10790	. . . . 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 2309	. . 3 |- ( ph -> dom I e. Fin )
35		prelpwi 4329	. . . 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 2498	. . . 4 |- ( ph -> U =/= X )
39		pr2ne 7488	. . . . 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 3794	. . . 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 16295	. 2 |- ( ph -> ( (VtxDeg ` F ) ` U ) = ( ( (VtxDeg ` G ) ` U ) + 1 ) )
45		vdegp1aid.d	. . 3 |- ( ph -> ( (VtxDeg ` G ) ` U ) = P )
46	45	oveq1d 6064	. 2 |- ( ph -> ( ( (VtxDeg ` G ) ` U ) + 1 ) = ( P + 1 ) )
47	44, 46	eqtrd 2265	1 |- ( ph -> ( (VtxDeg ` F ) ` U ) = ( P + 1 ) )

Syntax hints:   -> wi 4   <-> wb 105   \/ wo 716   = wceq 1398   e. wcel 2203   =/= wne 2412   e/ wnel 2507   {crab 2524   _Vcvv 2812   u. cun 3208   ~Pcpw 3668   {csn 3688   {cpr 3689   <.cop 3691   class class class wbr 4108   dom cdm 4748   -->wf 5347   ` cfv 5351  (class class class)co 6049   1oc1o 6639   2oc2o 6640   ~~ cen 6972   Fincfn 6974   0cc0 8123   1c1 8124   + caddc 8126   NN0cn0 9492   ZZcz 9573  ..^cfzo 10472  ♯chash 11133  Word cword 11217   ++ cconcat 11271   <"cs1 11296  Vtxcvtx 15994  iEdgciedg 15995  UPGraphcupgr 16073  VtxDegcvtxdg 16268

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 2205  ax-14 2206  ax-ext 2214  ax-coll 4224  ax-sep 4227  ax-nul 4235  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-iinf 4709  ax-cnex 8214  ax-resscn 8215  ax-1cn 8216  ax-1re 8217  ax-icn 8218  ax-addcl 8219  ax-addrcl 8220  ax-mulcl 8221  ax-addcom 8223  ax-mulcom 8224  ax-addass 8225  ax-mulass 8226  ax-distr 8227  ax-i2m1 8228  ax-0lt1 8229  ax-1rid 8230  ax-0id 8231  ax-rnegex 8232  ax-cnre 8234  ax-pre-ltirr 8235  ax-pre-ltwlin 8236  ax-pre-lttrn 8237  ax-pre-apti 8238  ax-pre-ltadd 8239

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 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-if 3620  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-tr 4208  df-id 4413  df-iord 4486  df-on 4488  df-ilim 4489  df-suc 4491  df-iom 4712  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-riota 6002  df-ov 6052  df-oprab 6053  df-mpo 6054  df-1st 6333  df-2nd 6334  df-recs 6535  df-irdg 6600  df-frec 6621  df-1o 6646  df-2o 6647  df-oadd 6650  df-er 6766  df-en 6975  df-dom 6976  df-fin 6977  df-pnf 8306  df-mnf 8307  df-xr 8308  df-ltxr 8309  df-le 8310  df-sub 8442  df-neg 8443  df-inn 9234  df-2 9292  df-3 9293  df-4 9294  df-5 9295  df-6 9296  df-7 9297  df-8 9298  df-9 9299  df-n0 9493  df-z 9574  df-dec 9706  df-uz 9850  df-xadd 10102  df-fz 10339  df-fzo 10473  df-ihash 11134  df-word 11218  df-concat 11272  df-s1 11297  df-ndx 13204  df-slot 13205  df-base 13207  df-edgf 15987  df-vtx 15996  df-iedg 15997  df-upgren 16075  df-umgren 16076  df-vtxdg 16269

This theorem is referenced by:  vdegp1cid  16298  konigsberglem1  16470  konigsberglem2  16471