Theorem trlsegvdeglem7 27086

Description: Lemma for trlsegvdeg 27087. (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
trlsegvdeglem7	|- ( ph -> dom (iEdg ` Y ) e. Fin )

Proof of Theorem trlsegvdeglem7

Step	Hyp	Ref	Expression
1		trlsegvdeg.v	. . 3 |- V = (Vtx ` G )
2		trlsegvdeg.i	. . 3 |- I = (iEdg ` G )
3		trlsegvdeg.f	. . 3 |- ( ph -> Fun I )
4		trlsegvdeg.n	. . 3 |- ( ph -> N e. ( 0..^ ( # ` F ) ) )
5		trlsegvdeg.u	. . 3 |- ( ph -> U e. V )
6		trlsegvdeg.w	. . 3 |- ( ph -> F (Trails ` G ) P )
7		trlsegvdeg.vx	. . 3 |- ( ph -> (Vtx ` X ) = V )
8		trlsegvdeg.vy	. . 3 |- ( ph -> (Vtx ` Y ) = V )
9		trlsegvdeg.vz	. . 3 |- ( ph -> (Vtx ` Z ) = V )
10		trlsegvdeg.ix	. . 3 |- ( ph -> (iEdg ` X ) = ( I |` ( F " ( 0..^ N ) ) ) )
11		trlsegvdeg.iy	. . 3 |- ( ph -> (iEdg ` Y ) = { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } )
12		trlsegvdeg.iz	. . 3 |- ( ph -> (iEdg ` Z ) = ( I |` ( F " ( 0 ... N ) ) ) )
13	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12	trlsegvdeglem5 27084	. 2 |- ( ph -> dom (iEdg ` Y ) = { ( F ` N ) } )
14		snfi 8038	. 2 |- { ( F ` N ) } e. Fin
15	13, 14	syl6eqel 2709	1 |- ( ph -> dom (iEdg ` Y ) e. Fin )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   {csn 4177   <.cop 4183   class class class wbr 4653   dom cdm 5114   |` cres 5116   "cima 5117   Fun wfun 5882   ` cfv 5888  (class class class)co 6650   Fincfn 7955   0cc0 9936   ...cfz 12326  ..^cfzo 12465   #chash 13117  Vtxcvtx 25874  iEdgciedg 25875  Trailsctrls 26587

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-om 7066  df-1o 7560  df-en 7956  df-fin 7959

This theorem is referenced by:  trlsegvdeg  27087  eupth2lem3lem2  27089