Theorem p1evtxdeqfi 16433

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 16432	. 2 |- ( ph -> ( (VtxDeg ` F ) ` U ) = ( ( (VtxDeg ` G ) ` U ) + ( (VtxDeg ` <. V , { <. K , E >. } >. ) ` U ) ) )
16	9	elexd 2829	. . . . 5 |- ( ph -> V e. _V )
17		opexg 4349	. . . . . . 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 4302	. . . . . 6 |- ( <. K , E >. e. _V -> { <. K , E >. } e. _V )
20	18, 19	syl 14	. . . . 5 |- ( ph -> { <. K , E >. } e. _V )
21		opiedgfv 16146	. . . . 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 16143	. . . . 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 16245	. . . 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 16428	. . 3 |- ( ph -> ( (VtxDeg ` <. V , { <. K , E >. } >. ) ` U ) = 0 )
28	27	oveq2d 6074	. 2 |- ( ph -> ( ( (VtxDeg ` G ) ` U ) + ( (VtxDeg ` <. V , { <. K , E >. } >. ) ` U ) ) = ( ( (VtxDeg ` G ) ` U ) + 0 ) )
29		eqid 2234	. . . . . 6 |- dom I = dom I
30	1, 2, 29, 11, 9, 10	vtxdgfif 16414	. . . . 5 |- ( ph -> (VtxDeg ` G ) : V --> NN0 )
31	30, 8	ffvelcdmd 5818	. . . 4 |- ( ph -> ( (VtxDeg ` G ) ` U ) e. NN0 )
32	31	nn0cnd 9572	. . 3 |- ( ph -> ( (VtxDeg ` G ) ` U ) e. CC )
33	32	addridd 8438	. 2 |- ( ph -> ( ( (VtxDeg ` G ) ` U ) + 0 ) = ( (VtxDeg ` G ) ` U ) )
34	15, 28, 33	3eqtrd 2271	1 |- ( ph -> ( (VtxDeg ` F ) ` U ) = ( (VtxDeg ` G ) ` U ) )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2205   e/ wnel 2509   _Vcvv 2815   u. cun 3212   ~Pcpw 3674   {csn 3694   <.cop 3697   class class class wbr 4114   dom cdm 4754   Fun wfun 5351   ` cfv 5357  (class class class)co 6058   2oc2o 6654   ~~ cen 6986   Fincfn 6988   0cc0 8143   + caddc 8146   NN0cn0 9513  Vtxcvtx 16133  iEdgciedg 16134  UPGraphcupgr 16212  VtxDegcvtxdg 16407

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 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259

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 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-irdg 6614  df-frec 6635  df-1o 6660  df-2o 6661  df-oadd 6664  df-er 6780  df-en 6989  df-dom 6990  df-fin 6991  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-inn 9255  df-2 9313  df-3 9314  df-4 9315  df-5 9316  df-6 9317  df-7 9318  df-8 9319  df-9 9320  df-n0 9514  df-z 9595  df-dec 9728  df-uz 9872  df-xadd 10125  df-fz 10362  df-ihash 11164  df-ndx 13299  df-slot 13300  df-base 13302  df-edgf 16126  df-vtx 16135  df-iedg 16136  df-upgren 16214  df-vtxdg 16408

This theorem is referenced by:  vdegp1aid  16435