Theorem vdegp1bid 16169

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 11120	. . . . 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 5486	. . 3 |- ( ph -> Fun I )
7		vdegp1aid.vf	. . 3 |- ( ph -> (Vtx ` F ) = V )
8		vdegp1bid.f	. . . 4 |- ( ph -> (iEdg ` F ) = ( I ++ <" { U , X } "> ) )
9		wrdv 11130	. . . . . 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 4301	. . . . . 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 11303	. . . . 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 2264	. . 3 |- ( ph -> (iEdg ` F ) = ( I u. { <. (♯ ` I ) , { U , X } >. } ) )
18		lencl 11118	. . . 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 11131	. . . 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 15874	. . . . . 6 |- ( U e. V -> G e. _V )
24	11, 23	syl 14	. . . . 5 |- ( ph -> G e. _V )
25	1, 2	wrdupgren 15950	. . . . 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 11122	. . . . 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 9490	. . . . 5 |- 0 e. ZZ
31	19	nn0zd 9600	. . . . 5 |- ( ph -> (♯ ` I ) e. ZZ )
32		fzofig 10695	. . . . 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 2308	. . 3 |- ( ph -> dom I e. Fin )
35		prelpwi 4306	. . . 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 2488	. . . 4 |- ( ph -> U =/= X )
39		pr2ne 7397	. . . . 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 3775	. . . 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 16167	. 2 |- ( ph -> ( (VtxDeg ` F ) ` U ) = ( ( (VtxDeg ` G ) ` U ) + 1 ) )
45		vdegp1aid.d	. . 3 |- ( ph -> ( (VtxDeg ` G ) ` U ) = P )
46	45	oveq1d 6033	. 2 |- ( ph -> ( ( (VtxDeg ` G ) ` U ) + 1 ) = ( P + 1 ) )
47	44, 46	eqtrd 2264	1 |- ( ph -> ( (VtxDeg ` F ) ` U ) = ( P + 1 ) )

Syntax hints:   -> wi 4   <-> wb 105   \/ wo 715   = wceq 1397   e. wcel 2202   =/= wne 2402   e/ wnel 2497   {crab 2514   _Vcvv 2802   u. cun 3198   ~Pcpw 3652   {csn 3669   {cpr 3670   <.cop 3672   class class class wbr 4088   dom cdm 4725   -->wf 5322   ` cfv 5326  (class class class)co 6018   1oc1o 6575   2oc2o 6576   ~~ cen 6907   Fincfn 6909   0cc0 8032   1c1 8033   + caddc 8035   NN0cn0 9402   ZZcz 9479  ..^cfzo 10377  ♯chash 11038  Word cword 11114   ++ cconcat 11168   <"cs1 11193  Vtxcvtx 15866  iEdgciedg 15867  UPGraphcupgr 15945  VtxDegcvtxdg 16140

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-addcom 8132  ax-mulcom 8133  ax-addass 8134  ax-mulass 8135  ax-distr 8136  ax-i2m1 8137  ax-0lt1 8138  ax-1rid 8139  ax-0id 8140  ax-rnegex 8141  ax-cnre 8143  ax-pre-ltirr 8144  ax-pre-ltwlin 8145  ax-pre-lttrn 8146  ax-pre-apti 8147  ax-pre-ltadd 8148

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-recs 6471  df-irdg 6536  df-frec 6557  df-1o 6582  df-2o 6583  df-oadd 6586  df-er 6702  df-en 6910  df-dom 6911  df-fin 6912  df-pnf 8216  df-mnf 8217  df-xr 8218  df-ltxr 8219  df-le 8220  df-sub 8352  df-neg 8353  df-inn 9144  df-2 9202  df-3 9203  df-4 9204  df-5 9205  df-6 9206  df-7 9207  df-8 9208  df-9 9209  df-n0 9403  df-z 9480  df-dec 9612  df-uz 9756  df-xadd 10008  df-fz 10244  df-fzo 10378  df-ihash 11039  df-word 11115  df-concat 11169  df-s1 11194  df-ndx 13087  df-slot 13088  df-base 13090  df-edgf 15859  df-vtx 15868  df-iedg 15869  df-upgren 15947  df-umgren 15948  df-vtxdg 16141

This theorem is referenced by:  vdegp1cid  16170