Theorem p1evtxdeqfi 16123

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 16122	. 2 |- ( ph -> ( (VtxDeg ` F ) ` U ) = ( ( (VtxDeg ` G ) ` U ) + ( (VtxDeg ` <. V , { <. K , E >. } >. ) ` U ) ) )
16	9	elexd 2814	. . . . 5 |- ( ph -> V e. _V )
17		opexg 4318	. . . . . . 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 4272	. . . . . 6 |- ( <. K , E >. e. _V -> { <. K , E >. } e. _V )
20	18, 19	syl 14	. . . . 5 |- ( ph -> { <. K , E >. } e. _V )
21		opiedgfv 15869	. . . . 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 15866	. . . . 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 15968	. . . 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 16118	. . 3 |- ( ph -> ( (VtxDeg ` <. V , { <. K , E >. } >. ) ` U ) = 0 )
28	27	oveq2d 6029	. 2 |- ( ph -> ( ( (VtxDeg ` G ) ` U ) + ( (VtxDeg ` <. V , { <. K , E >. } >. ) ` U ) ) = ( ( (VtxDeg ` G ) ` U ) + 0 ) )
29		eqid 2229	. . . . . 6 |- dom I = dom I
30	1, 2, 29, 11, 9, 10	vtxdgfif 16104	. . . . 5 |- ( ph -> (VtxDeg ` G ) : V --> NN0 )
31	30, 8	ffvelcdmd 5779	. . . 4 |- ( ph -> ( (VtxDeg ` G ) ` U ) e. NN0 )
32	31	nn0cnd 9450	. . 3 |- ( ph -> ( (VtxDeg ` G ) ` U ) e. CC )
33	32	addridd 8321	. 2 |- ( ph -> ( ( (VtxDeg ` G ) ` U ) + 0 ) = ( (VtxDeg ` G ) ` U ) )
34	15, 28, 33	3eqtrd 2266	1 |- ( ph -> ( (VtxDeg ` F ) ` U ) = ( (VtxDeg ` G ) ` U ) )

Syntax hints:   -> wi 4   = wceq 1395   e. wcel 2200   e/ wnel 2495   _Vcvv 2800   u. cun 3196   ~Pcpw 3650   {csn 3667   <.cop 3670   class class class wbr 4086   dom cdm 4723   Fun wfun 5318   ` cfv 5324  (class class class)co 6013   2oc2o 6571   ~~ cen 6902   Fincfn 6904   0cc0 8025   + caddc 8028   NN0cn0 9395  Vtxcvtx 15856  iEdgciedg 15857  UPGraphcupgr 15935  VtxDegcvtxdg 16097

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684  ax-cnex 8116  ax-resscn 8117  ax-1cn 8118  ax-1re 8119  ax-icn 8120  ax-addcl 8121  ax-addrcl 8122  ax-mulcl 8123  ax-addcom 8125  ax-mulcom 8126  ax-addass 8127  ax-mulass 8128  ax-distr 8129  ax-i2m1 8130  ax-0lt1 8131  ax-1rid 8132  ax-0id 8133  ax-rnegex 8134  ax-cnre 8136  ax-pre-ltirr 8137  ax-pre-ltwlin 8138  ax-pre-lttrn 8139  ax-pre-apti 8140  ax-pre-ltadd 8141

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-if 3604  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-id 4388  df-iord 4461  df-on 4463  df-ilim 4464  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-recs 6466  df-irdg 6531  df-frec 6552  df-1o 6577  df-2o 6578  df-oadd 6581  df-er 6697  df-en 6905  df-dom 6906  df-fin 6907  df-pnf 8209  df-mnf 8210  df-xr 8211  df-ltxr 8212  df-le 8213  df-sub 8345  df-neg 8346  df-inn 9137  df-2 9195  df-3 9196  df-4 9197  df-5 9198  df-6 9199  df-7 9200  df-8 9201  df-9 9202  df-n0 9396  df-z 9473  df-dec 9605  df-uz 9749  df-xadd 10001  df-fz 10237  df-ihash 11031  df-ndx 13078  df-slot 13079  df-base 13081  df-edgf 15849  df-vtx 15858  df-iedg 15859  df-upgren 15937  df-vtxdg 16098

This theorem is referenced by:  vdegp1aid  16125