Theorem vdegp1aid 16772

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 11333	. . . . 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 5540	. . 3 |- ( ph -> Fun I )
7		vdegp1aid.vf	. . 3 |- ( ph -> (Vtx ` F ) = V )
8		vdegp1aid.f	. . . 4 |- ( ph -> (iEdg ` F ) = ( I ++ <" { X , Y } "> ) )
9		wrdv 11343	. . . . . 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 4351	. . . . . 6 |- ( ( X e. V /\ Y e. V ) -> { X , Y } e. _V )
14	11, 12, 13	syl2anc 415	. . . . 5 |- ( ph -> { X , Y } e. _V )
15		cats1un 11516	. . . . 5 |- ( ( I e. Word _V /\ { X , Y } e. _V ) -> ( I ++ <" { X , Y } "> ) = ( I u. { <. (♯ ` I ) , { X , Y } >. } ) )
16	10, 14, 15	syl2anc 415	. . . 4 |- ( ph -> ( I ++ <" { X , Y } "> ) = ( I u. { <. (♯ ` I ) , { X , Y } >. } ) )
17	8, 16	eqtrd 2271	. . 3 |- ( ph -> (iEdg ` F ) = ( I u. { <. (♯ ` I ) , { X , Y } >. } ) )
18		lencl 11331	. . . 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 11344	. . . 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 16478	. . . . . 6 |- ( X e. V -> G e. _V )
25	11, 24	syl 14	. . . . 5 |- ( ph -> G e. _V )
26	1, 2	wrdupgren 16554	. . . . 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 415	. . . 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 11346	. . . . 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 7256	. . . 4 |- ( ( I e. Fin /\ Fun I ) -> dom I e. Fin )
32	30, 6, 31	syl2anc 415	. . 3 |- ( ph -> dom I e. Fin )
33		prelpwi 4356	. . . 4 |- ( ( X e. V /\ Y e. V ) -> { X , Y } e. ~P V )
34	11, 12, 33	syl2anc 415	. . 3 |- ( ph -> { X , Y } e. ~P V )
35		vdegp1aid.xy	. . . 4 |- ( ph -> X =/= Y )
36		pr2ne 7545	. . . . 5 |- ( ( X e. V /\ Y e. V ) -> ( { X , Y } ~~ 2o <-> X =/= Y ) )
37	11, 12, 36	syl2anc 415	. . . 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 2441	. . . . . . 7 |- ( ph -> -. X = U )
41	40	neqcomd 2243	. . . . . 6 |- ( ph -> -. U = X )
42		vdegp1aid.yu	. . . . . . . 8 |- ( ph -> Y =/= U )
43	42	neneqd 2441	. . . . . . 7 |- ( ph -> -. Y = U )
44	43	neqcomd 2243	. . . . . 6 |- ( ph -> -. U = Y )
45		ioran 764	. . . . . 6 |- ( -. ( U = X \/ U = Y ) <-> ( -. U = X /\ -. U = Y ) )
46	41, 44, 45	sylanbrc 421	. . . . 5 |- ( ph -> -. ( U = X \/ U = Y ) )
47		elpri 3732	. . . . 5 |- ( U e. { X , Y } -> ( U = X \/ U = Y ) )
48	46, 47	nsyl 637	. . . 4 |- ( ph -> -. U e. { X , Y } )
49		df-nel 2516	. . . 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 16770	. 2 |- ( ph -> ( (VtxDeg ` F ) ` U ) = ( (VtxDeg ` G ) ` U ) )
52		vdegp1aid.d	. 2 |- ( ph -> ( (VtxDeg ` G ) ` U ) = P )
53	51, 52	eqtrd 2271	1 |- ( ph -> ( (VtxDeg ` F ) ` U ) = P )

This proof depends on syntax axioms:   -. wn 3   -> 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 3709   {cpr 3710   <.cop 3712   class class class wbr 4131   dom cdm 4776   Fun wfun 5374   -->wf 5376   ` cfv 5380  (class class class)co 6088   1oc1o 6684   2oc2o 6685   ~~ cen 7025   Fincfn 7027   0cc0 8186   NN0cn0 9575  ..^cfzo 10567  ♯chash 11237  Word cword 11327   ++ cconcat 11381   <"cs1 11406  Vtxcvtx 16470  iEdgciedg 16471  UPGraphcupgr 16549  VtxDegcvtxdg 16744

This proof depends on 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 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737  ax-cnex 8277  ax-resscn 8278  ax-1cn 8279  ax-1re 8280  ax-icn 8281  ax-addcl 8282  ax-addrcl 8283  ax-mulcl 8284  ax-addcom 8286  ax-mulcom 8287  ax-addass 8288  ax-mulass 8289  ax-distr 8290  ax-i2m1 8291  ax-0lt1 8292  ax-1rid 8293  ax-0id 8294  ax-rnegex 8295  ax-cnre 8297  ax-pre-ltirr 8298  ax-pre-ltwlin 8299  ax-pre-lttrn 8300  ax-pre-apti 8301  ax-pre-ltadd 8302

This proof 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 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-id 4440  df-iord 4513  df-on 4515  df-ilim 4516  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-recs 6580  df-irdg 6645  df-frec 6666  df-1o 6691  df-2o 6692  df-oadd 6695  df-er 6811  df-en 7028  df-dom 7029  df-fin 7030  df-pnf 8369  df-mnf 8370  df-xr 8371  df-ltxr 8372  df-le 8373  df-sub 8507  df-neg 8508  df-inn 9315  df-2 9373  df-3 9374  df-4 9375  df-5 9376  df-6 9377  df-7 9378  df-8 9379  df-9 9380  df-n0 9576  df-z 9657  df-dec 9790  df-uz 9939  df-xadd 10193  df-fz 10430  df-fzo 10568  df-ihash 11238  df-word 11328  df-concat 11382  df-s1 11407  df-ndx 13414  df-slot 13415  df-base 13417  df-edgf 16463  df-vtx 16472  df-iedg 16473  df-upgren 16551  df-vtxdg 16745

This theorem is used by:  konigsberglem1  16946  konigsberglem2  16947  konigsberglem3  16948