Theorem trlsegvdeglem4 16458

Description: Lemma for trlsegvdeg . (Contributed by AV, 21-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
trlsegvdeglem4	|- ( ph -> dom (iEdg ` X ) = ( ( F " ( 0..^ N ) ) i^i dom I ) )

Proof of Theorem trlsegvdeglem4

Step	Hyp	Ref	Expression
1		trlsegvdeg.ix	. . 3 |- ( ph -> (iEdg ` X ) = ( I |` ( F " ( 0..^ N ) ) ) )
2	1	dmeqd 4958	. 2 |- ( ph -> dom (iEdg ` X ) = dom ( I |` ( F " ( 0..^ N ) ) ) )
3		dmres 5059	. 2 |- dom ( I |` ( F " ( 0..^ N ) ) ) = ( ( F " ( 0..^ N ) ) i^i dom I )
4	2, 3	eqtrdi 2281	1 |- ( ph -> dom (iEdg ` X ) = ( ( F " ( 0..^ N ) ) i^i dom I ) )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2203   i^i cin 3210   {csn 3689   <.cop 3692   class class class wbr 4109   dom cdm 4749   |` cres 4751   "cima 4752   Fun wfun 5346   ` cfv 5352  (class class class)co 6050   0cc0 8127   ...cfz 10342  ..^cfzo 10476  ♯chash 11138  Vtxcvtx 16007  iEdgciedg 16008  Trailsctrls 16375

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 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-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-br 4110  df-opab 4172  df-xp 4755  df-dm 4759  df-res 4761

This theorem is referenced by:  trlsegvdeglem6  16460  trlsegvdegfi  16462