Theorem p1evtxdeqfi 16166

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 16165	. 2 |- ( ph -> ( (VtxDeg ` F ) ` U ) = ( ( (VtxDeg ` G ) ` U ) + ( (VtxDeg ` <. V , { <. K , E >. } >. ) ` U ) ) )
16	9	elexd 2816	. . . . 5 |- ( ph -> V e. _V )
17		opexg 4320	. . . . . . 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 4274	. . . . . 6 |- ( <. K , E >. e. _V -> { <. K , E >. } e. _V )
20	18, 19	syl 14	. . . . 5 |- ( ph -> { <. K , E >. } e. _V )
21		opiedgfv 15879	. . . . 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 15876	. . . . 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 15978	. . . 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 16161	. . 3 |- ( ph -> ( (VtxDeg ` <. V , { <. K , E >. } >. ) ` U ) = 0 )
28	27	oveq2d 6034	. 2 |- ( ph -> ( ( (VtxDeg ` G ) ` U ) + ( (VtxDeg ` <. V , { <. K , E >. } >. ) ` U ) ) = ( ( (VtxDeg ` G ) ` U ) + 0 ) )
29		eqid 2231	. . . . . 6 |- dom I = dom I
30	1, 2, 29, 11, 9, 10	vtxdgfif 16147	. . . . 5 |- ( ph -> (VtxDeg ` G ) : V --> NN0 )
31	30, 8	ffvelcdmd 5783	. . . 4 |- ( ph -> ( (VtxDeg ` G ) ` U ) e. NN0 )
32	31	nn0cnd 9457	. . 3 |- ( ph -> ( (VtxDeg ` G ) ` U ) e. CC )
33	32	addridd 8328	. 2 |- ( ph -> ( ( (VtxDeg ` G ) ` U ) + 0 ) = ( (VtxDeg ` G ) ` U ) )
34	15, 28, 33	3eqtrd 2268	1 |- ( ph -> ( (VtxDeg ` F ) ` U ) = ( (VtxDeg ` G ) ` U ) )

Syntax hints:   -> wi 4   = wceq 1397   e. wcel 2202   e/ wnel 2497   _Vcvv 2802   u. cun 3198   ~Pcpw 3652   {csn 3669   <.cop 3672   class class class wbr 4088   dom cdm 4725   Fun wfun 5320   ` cfv 5326  (class class class)co 6018   2oc2o 6576   ~~ cen 6907   Fincfn 6909   0cc0 8032   + caddc 8035   NN0cn0 9402  Vtxcvtx 15866  iEdgciedg 15867  UPGraphcupgr 15945  VtxDegcvtxdg 16140

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-addcom 8132  ax-mulcom 8133  ax-addass 8134  ax-mulass 8135  ax-distr 8136  ax-i2m1 8137  ax-0lt1 8138  ax-1rid 8139  ax-0id 8140  ax-rnegex 8141  ax-cnre 8143  ax-pre-ltirr 8144  ax-pre-ltwlin 8145  ax-pre-lttrn 8146  ax-pre-apti 8147  ax-pre-ltadd 8148

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-recs 6471  df-irdg 6536  df-frec 6557  df-1o 6582  df-2o 6583  df-oadd 6586  df-er 6702  df-en 6910  df-dom 6911  df-fin 6912  df-pnf 8216  df-mnf 8217  df-xr 8218  df-ltxr 8219  df-le 8220  df-sub 8352  df-neg 8353  df-inn 9144  df-2 9202  df-3 9203  df-4 9204  df-5 9205  df-6 9206  df-7 9207  df-8 9208  df-9 9209  df-n0 9403  df-z 9480  df-dec 9612  df-uz 9756  df-xadd 10008  df-fz 10244  df-ihash 11039  df-ndx 13087  df-slot 13088  df-base 13090  df-edgf 15859  df-vtx 15868  df-iedg 15869  df-upgren 15947  df-vtxdg 16141

This theorem is referenced by:  vdegp1aid  16168