Theorem vdegp1bid 16539

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 11293	. . . . 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 5535	. . 3 |- ( ph -> Fun I )
7		vdegp1aid.vf	. . 3 |- ( ph -> (Vtx ` F ) = V )
8		vdegp1bid.f	. . . 4 |- ( ph -> (iEdg ` F ) = ( I ++ <" { U , X } "> ) )
9		wrdv 11303	. . . . . 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 4347	. . . . . 6 |- ( ( U e. V /\ X e. V ) -> { U , X } e. _V )
14	11, 12, 13	syl2anc 415	. . . . 5 |- ( ph -> { U , X } e. _V )
15		cats1un 11476	. . . . 5 |- ( ( I e. Word _V /\ { U , X } e. _V ) -> ( I ++ <" { U , X } "> ) = ( I u. { <. (♯ ` I ) , { U , X } >. } ) )
16	10, 14, 15	syl2anc 415	. . . 4 |- ( ph -> ( I ++ <" { U , X } "> ) = ( I u. { <. (♯ ` I ) , { U , X } >. } ) )
17	8, 16	eqtrd 2271	. . 3 |- ( ph -> (iEdg ` F ) = ( I u. { <. (♯ ` I ) , { U , X } >. } ) )
18		lencl 11291	. . . 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 11304	. . . 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 16244	. . . . . 6 |- ( U e. V -> G e. _V )
24	11, 23	syl 14	. . . . 5 |- ( ph -> G e. _V )
25	1, 2	wrdupgren 16320	. . . . 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 415	. . . 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 11295	. . . . 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 9638	. . . . 5 |- 0 e. ZZ
31	19	nn0zd 9749	. . . . 5 |- ( ph -> (♯ ` I ) e. ZZ )
32		fzofig 10852	. . . . 5 |- ( ( 0 e. ZZ /\ (♯ ` I ) e. ZZ ) -> ( 0..^ (♯ ` I ) ) e. Fin )
33	30, 31, 32	sylancr 418	. . . 4 |- ( ph -> ( 0..^ (♯ ` I ) ) e. Fin )
34	29, 33	eqeltrd 2315	. . 3 |- ( ph -> dom I e. Fin )
35		prelpwi 4352	. . . 4 |- ( ( U e. V /\ X e. V ) -> { U , X } e. ~P V )
36	11, 12, 35	syl2anc 415	. . 3 |- ( ph -> { U , X } e. ~P V )
37		vdegp1bid.xu	. . . . 5 |- ( ph -> X =/= U )
38	37	necomd 2506	. . . 4 |- ( ph -> U =/= X )
39		pr2ne 7532	. . . . 5 |- ( ( U e. V /\ X e. V ) -> ( { U , X } ~~ 2o <-> U =/= X ) )
40	11, 12, 39	syl2anc 415	. . . 4 |- ( ph -> ( { U , X } ~~ 2o <-> U =/= X ) )
41	38, 40	mpbird 167	. . 3 |- ( ph -> { U , X } ~~ 2o )
42		prid1g 3814	. . . 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 16537	. 2 |- ( ph -> ( (VtxDeg ` F ) ` U ) = ( ( (VtxDeg ` G ) ` U ) + 1 ) )
45		vdegp1aid.d	. . 3 |- ( ph -> ( (VtxDeg ` G ) ` U ) = P )
46	45	oveq1d 6094	. 2 |- ( ph -> ( ( (VtxDeg ` G ) ` U ) + 1 ) = ( P + 1 ) )
47	44, 46	eqtrd 2271	1 |- ( ph -> ( (VtxDeg ` F ) ` U ) = ( P + 1 ) )

Syntax hints:   -> wi 4   <-> wb 105   \/ wo 720   = wceq 1402   e. wcel 2209   =/= wne 2420   e/ wnel 2515   {crab 2532   _Vcvv 2821   u. cun 3218   ~Pcpw 3688   {csn 3708   {cpr 3709   <.cop 3711   class class class wbr 4128   dom cdm 4772   -->wf 5371   ` cfv 5375  (class class class)co 6079   1oc1o 6674   2oc2o 6675   ~~ cen 7014   Fincfn 7016   0cc0 8173   1c1 8174   + caddc 8176   NN0cn0 9546   ZZcz 9627  ..^cfzo 10532  ♯chash 11197  Word cword 11287   ++ cconcat 11341   <"cs1 11366  Vtxcvtx 16236  iEdgciedg 16237  UPGraphcupgr 16315  VtxDegcvtxdg 16510

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4244  ax-sep 4247  ax-nul 4257  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-iinf 4733  ax-cnex 8264  ax-resscn 8265  ax-1cn 8266  ax-1re 8267  ax-icn 8268  ax-addcl 8269  ax-addrcl 8270  ax-mulcl 8271  ax-addcom 8273  ax-mulcom 8274  ax-addass 8275  ax-mulass 8276  ax-distr 8277  ax-i2m1 8278  ax-0lt1 8279  ax-1rid 8280  ax-0id 8281  ax-rnegex 8282  ax-cnre 8284  ax-pre-ltirr 8285  ax-pre-ltwlin 8286  ax-pre-lttrn 8287  ax-pre-apti 8288  ax-pre-ltadd 8289

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-tr 4228  df-id 4436  df-iord 4509  df-on 4511  df-ilim 4512  df-suc 4514  df-iom 4736  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-riota 6032  df-ov 6082  df-oprab 6083  df-mpo 6084  df-1st 6368  df-2nd 6369  df-recs 6570  df-irdg 6635  df-frec 6656  df-1o 6681  df-2o 6682  df-oadd 6685  df-er 6801  df-en 7017  df-dom 7018  df-fin 7019  df-pnf 8356  df-mnf 8357  df-xr 8358  df-ltxr 8359  df-le 8360  df-sub 8493  df-neg 8494  df-inn 9288  df-2 9346  df-3 9347  df-4 9348  df-5 9349  df-6 9350  df-7 9351  df-8 9352  df-9 9353  df-n0 9547  df-z 9628  df-dec 9761  df-uz 9905  df-xadd 10158  df-fz 10395  df-fzo 10533  df-ihash 11198  df-word 11288  df-concat 11342  df-s1 11367  df-ndx 13338  df-slot 13339  df-base 13341  df-edgf 16229  df-vtx 16238  df-iedg 16239  df-upgren 16317  df-umgren 16318  df-vtxdg 16511

This theorem is referenced by:  vdegp1cid  16540  konigsberglem1  16712  konigsberglem2  16713