Theorem trlsegvdeglem2 16311

Description: Lemma for trlsegvdeg . (Contributed by AV, 20-Feb-2021.)

Hypotheses

Ref	Expression
trlsegvdeg.v	|- V = (Vtx ` G )
trlsegvdeg.i	|- I = (iEdg ` G )
trlsegvdeg.f	|- ( ph -> Fun I )
trlsegvdeg.n	|- ( ph -> N e. ( 0..^ (♯ ` F ) ) )
trlsegvdeg.u	|- ( ph -> U e. V )
trlsegvdeg.w	|- ( ph -> F (Trails ` G ) P )
trlsegvdeg.vx	|- ( ph -> (Vtx ` X ) = V )
trlsegvdeg.vy	|- ( ph -> (Vtx ` Y ) = V )
trlsegvdeg.vz	|- ( ph -> (Vtx ` Z ) = V )
trlsegvdeg.ix	|- ( ph -> (iEdg ` X ) = ( I |` ( F " ( 0..^ N ) ) ) )
trlsegvdeg.iy	|- ( ph -> (iEdg ` Y ) = { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } )
trlsegvdeg.iz	|- ( ph -> (iEdg ` Z ) = ( I |` ( F " ( 0 ... N ) ) ) )

Assertion

Ref	Expression
trlsegvdeglem2	|- ( ph -> Fun (iEdg ` X ) )

Proof of Theorem trlsegvdeglem2

Step	Hyp	Ref	Expression
1		trlsegvdeg.f	. . 3 |- ( ph -> Fun I )
2	1	funresd 5368	. 2 |- ( ph -> Fun ( I |` ( F " ( 0..^ N ) ) ) )
3		trlsegvdeg.ix	. . 3 |- ( ph -> (iEdg ` X ) = ( I |` ( F " ( 0..^ N ) ) ) )
4	3	funeqd 5348	. 2 |- ( ph -> ( Fun (iEdg ` X ) <-> Fun ( I |` ( F " ( 0..^ N ) ) ) ) )
5	2, 4	mpbird 167	1 |- ( ph -> Fun (iEdg ` X ) )

Syntax hints:   -> wi 4   = wceq 1397   e. wcel 2202   {csn 3669   <.cop 3672   class class class wbr 4088   |` cres 4727   "cima 4728   Fun wfun 5320   ` cfv 5326  (class class class)co 6017   0cc0 8031   ...cfz 10242  ..^cfzo 10376  ♯chash 11036  Vtxcvtx 15862  iEdgciedg 15863  Trailsctrls 16230

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-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-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-in 3206  df-ss 3213  df-br 4089  df-opab 4151  df-rel 4732  df-cnv 4733  df-co 4734  df-res 4737  df-fun 5328

This theorem is referenced by: (None)