Theorem trlsegvdeglem4 16313

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 4933	. 2 |- ( ph -> dom (iEdg ` X ) = dom ( I |` ( F " ( 0..^ N ) ) ) )
3		dmres 5034	. 2 |- dom ( I |` ( F " ( 0..^ N ) ) ) = ( ( F " ( 0..^ N ) ) i^i dom I )
4	2, 3	eqtrdi 2280	1 |- ( ph -> dom (iEdg ` X ) = ( ( F " ( 0..^ N ) ) i^i dom I ) )

Syntax hints:   -> wi 4   = wceq 1397   e. wcel 2202   i^i cin 3199   {csn 3669   <.cop 3672   class class class wbr 4088   dom cdm 4725   |` 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-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089  df-opab 4151  df-xp 4731  df-dm 4735  df-res 4737

This theorem is referenced by:  trlsegvdeglem6  16315