Theorem trlsegvdeglem2 16616

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 5414	. 2 |- ( ph -> Fun ( I |` ( F " ( 0..^ N ) ) ) )
3		trlsegvdeg.ix	. . 3 |- ( ph -> (iEdg ` X ) = ( I |` ( F " ( 0..^ N ) ) ) )
4	3	funeqd 5394	. 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 1402   e. wcel 2209   {csn 3705   <.cop 3708   class class class wbr 4125   |` cres 4771   "cima 4772   Fun wfun 5366   ` cfv 5372  (class class class)co 6075   0cc0 8169   ...cfz 10390  ..^cfzo 10527  ♯chash 11192  Vtxcvtx 16167  iEdgciedg 16168  Trailsctrls 16535

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-in 3226  df-ss 3233  df-br 4126  df-opab 4188  df-rel 4776  df-cnv 4777  df-co 4778  df-res 4781  df-fun 5374

This theorem is referenced by:  trlsegvdegfi  16622