Theorem trlsegvdeglem4 16584

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 4963	. 2 |- ( ph -> dom (iEdg ` X ) = dom ( I |` ( F " ( 0..^ N ) ) ) )
3		dmres 5064	. 2 |- dom ( I |` ( F " ( 0..^ N ) ) ) = ( ( F " ( 0..^ N ) ) i^i dom I )
4	2, 3	eqtrdi 2283	1 |- ( ph -> dom (iEdg ` X ) = ( ( F " ( 0..^ N ) ) i^i dom I ) )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2205   i^i cin 3213   {csn 3694   <.cop 3697   class class class wbr 4114   dom cdm 4754   |` cres 4756   "cima 4757   Fun wfun 5351   ` cfv 5357  (class class class)co 6058   0cc0 8143   ...cfz 10361  ..^cfzo 10498  ♯chash 11163  Vtxcvtx 16133  iEdgciedg 16134  Trailsctrls 16501

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 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-br 4115  df-opab 4177  df-xp 4760  df-dm 4764  df-res 4766

This theorem is referenced by:  trlsegvdeglem6  16586  trlsegvdegfi  16588