Theorem p1evtxdeqfi 16294

Description: If an edge E which does not contain vertex U is added to a graph G (yielding a graph F), the degree of U is the same in both graphs. (Contributed by AV, 2-Mar-2021.)

Hypotheses

Ref	Expression
p1evtxdeq.v	|- V = (Vtx ` G )
p1evtxdeq.i	|- I = (iEdg ` G )
p1evtxdeq.f	|- ( ph -> Fun I )
p1evtxdeq.fv	|- ( ph -> (Vtx ` F ) = V )
p1evtxdeq.fi	|- ( ph -> (iEdg ` F ) = ( I u. { <. K , E >. } ) )
p1evtxdeq.k	|- ( ph -> K e. X )
p1evtxdeq.d	|- ( ph -> K e/ dom I )
p1evtxdeq.u	|- ( ph -> U e. V )
p1evtxdeqfi.vfi	|- ( ph -> V e. Fin )
p1evtxdeqfi.u	|- ( ph -> G e. UPGraph )
p1evtxdeqfi.ifi	|- ( ph -> dom I e. Fin )
p1evtxdeqfi.e	|- ( ph -> E e. ~P V )
p1evtxdeqfi.2o	|- ( ph -> E ~~ 2o )
p1evtxdeq.e	|- ( ph -> E e. Y )
p1evtxdeq.n	|- ( ph -> U e/ E )

Assertion

Ref	Expression
p1evtxdeqfi	|- ( ph -> ( (VtxDeg ` F ) ` U ) = ( (VtxDeg ` G ) ` U ) )

Proof of Theorem p1evtxdeqfi

Step	Hyp	Ref	Expression
1		p1evtxdeq.v	. . 3 |- V = (Vtx ` G )
2		p1evtxdeq.i	. . 3 |- I = (iEdg ` G )
3		p1evtxdeq.f	. . 3 |- ( ph -> Fun I )
4		p1evtxdeq.fv	. . 3 |- ( ph -> (Vtx ` F ) = V )
5		p1evtxdeq.fi	. . 3 |- ( ph -> (iEdg ` F ) = ( I u. { <. K , E >. } ) )
6		p1evtxdeq.k	. . 3 |- ( ph -> K e. X )
7		p1evtxdeq.d	. . 3 |- ( ph -> K e/ dom I )
8		p1evtxdeq.u	. . 3 |- ( ph -> U e. V )
9		p1evtxdeqfi.vfi	. . 3 |- ( ph -> V e. Fin )
10		p1evtxdeqfi.u	. . 3 |- ( ph -> G e. UPGraph )
11		p1evtxdeqfi.ifi	. . 3 |- ( ph -> dom I e. Fin )
12		p1evtxdeqfi.e	. . 3 |- ( ph -> E e. ~P V )
13		p1evtxdeqfi.2o	. . 3 |- ( ph -> E ~~ 2o )
14		p1evtxdeq.e	. . 3 |- ( ph -> E e. Y )
15	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14	p1evtxdeqfilem 16293	. 2 |- ( ph -> ( (VtxDeg ` F ) ` U ) = ( ( (VtxDeg ` G ) ` U ) + ( (VtxDeg ` <. V , { <. K , E >. } >. ) ` U ) ) )
16	9	elexd 2826	. . . . 5 |- ( ph -> V e. _V )
17		opexg 4343	. . . . . . 7 |- ( ( K e. X /\ E e. ~P V ) -> <. K , E >. e. _V )
18	6, 12, 17	syl2anc 411	. . . . . 6 |- ( ph -> <. K , E >. e. _V )
19		snexg 4296	. . . . . 6 |- ( <. K , E >. e. _V -> { <. K , E >. } e. _V )
20	18, 19	syl 14	. . . . 5 |- ( ph -> { <. K , E >. } e. _V )
21		opiedgfv 16007	. . . . 5 |- ( ( V e. _V /\ { <. K , E >. } e. _V ) -> (iEdg ` <. V , { <. K , E >. } >. ) = { <. K , E >. } )
22	16, 20, 21	syl2anc 411	. . . 4 |- ( ph -> (iEdg ` <. V , { <. K , E >. } >. ) = { <. K , E >. } )
23		opvtxfv 16004	. . . . 5 |- ( ( V e. _V /\ { <. K , E >. } e. _V ) -> (Vtx ` <. V , { <. K , E >. } >. ) = V )
24	16, 20, 23	syl2anc 411	. . . 4 |- ( ph -> (Vtx ` <. V , { <. K , E >. } >. ) = V )
25	6, 9, 12, 13	upgr1een 16106	. . . 4 |- ( ph -> <. V , { <. K , E >. } >. e. UPGraph )
26		p1evtxdeq.n	. . . 4 |- ( ph -> U e/ E )
27	22, 24, 6, 8, 9, 25, 14, 26	1hevtxdg0fi 16289	. . 3 |- ( ph -> ( (VtxDeg ` <. V , { <. K , E >. } >. ) ` U ) = 0 )
28	27	oveq2d 6065	. 2 |- ( ph -> ( ( (VtxDeg ` G ) ` U ) + ( (VtxDeg ` <. V , { <. K , E >. } >. ) ` U ) ) = ( ( (VtxDeg ` G ) ` U ) + 0 ) )
29		eqid 2232	. . . . . 6 |- dom I = dom I
30	1, 2, 29, 11, 9, 10	vtxdgfif 16275	. . . . 5 |- ( ph -> (VtxDeg ` G ) : V --> NN0 )
31	30, 8	ffvelcdmd 5812	. . . 4 |- ( ph -> ( (VtxDeg ` G ) ` U ) e. NN0 )
32	31	nn0cnd 9551	. . 3 |- ( ph -> ( (VtxDeg ` G ) ` U ) e. CC )
33	32	addridd 8418	. 2 |- ( ph -> ( ( (VtxDeg ` G ) ` U ) + 0 ) = ( (VtxDeg ` G ) ` U ) )
34	15, 28, 33	3eqtrd 2269	1 |- ( ph -> ( (VtxDeg ` F ) ` U ) = ( (VtxDeg ` G ) ` U ) )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2203   e/ wnel 2507   _Vcvv 2812   u. cun 3208   ~Pcpw 3668   {csn 3688   <.cop 3691   class class class wbr 4108   dom cdm 4748   Fun wfun 5345   ` cfv 5351  (class class class)co 6049   2oc2o 6640   ~~ cen 6972   Fincfn 6974   0cc0 8123   + caddc 8126   NN0cn0 9492  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-ihash 11134  df-ndx 13204  df-slot 13205  df-base 13207  df-edgf 15987  df-vtx 15996  df-iedg 15997  df-upgren 16075  df-vtxdg 16269

This theorem is referenced by:  vdegp1aid  16296