Theorem trlsegvdeglem3 27082

Description: Lemma for trlsegvdeg 27087. (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
trlsegvdeglem3	|- ( ph -> Fun (iEdg ` Y ) )

Proof of Theorem trlsegvdeglem3

Step	Hyp	Ref	Expression
1		fvex 6201	. . . 4 |- ( F ` N ) e. _V
2		fvex 6201	. . . 4 |- ( I ` ( F ` N ) ) e. _V
3	1, 2	pm3.2i 471	. . 3 |- ( ( F ` N ) e. _V /\ ( I ` ( F ` N ) ) e. _V )
4		funsng 5937	. . 3 |- ( ( ( F ` N ) e. _V /\ ( I ` ( F ` N ) ) e. _V ) -> Fun { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } )
5	3, 4	mp1i 13	. 2 |- ( ph -> Fun { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } )
6		trlsegvdeg.iy	. . 3 |- ( ph -> (iEdg ` Y ) = { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } )
7	6	funeqd 5910	. 2 |- ( ph -> ( Fun (iEdg ` Y ) <-> Fun { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } ) )
8	5, 7	mpbird 247	1 |- ( ph -> Fun (iEdg ` Y ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   {csn 4177   <.cop 4183   class class class wbr 4653   |` cres 5116   "cima 5117   Fun wfun 5882   ` cfv 5888  (class class class)co 6650   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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-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-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-iota 5851  df-fun 5890  df-fv 5896

This theorem is referenced by:  trlsegvdeg  27087