Theorem usgr2wlkspthlem1 26653

Description: Lemma 1 for usgr2wlkspth 26655. (Contributed by Alexander van der Vekens, 2-Mar-2018.) (Revised by AV, 26-Jan-2021.)

Assertion

Ref	Expression
usgr2wlkspthlem1	|- ( ( F (Walks ` G ) P /\ ( G e. USGraph /\ ( # ` F ) = 2 /\ ( P ` 0 ) =/= ( P ` ( # ` F ) ) ) ) -> Fun `' F )

Proof of Theorem usgr2wlkspthlem1

Step	Hyp	Ref	Expression
1		simp1 1061	. . . . . 6 |- ( ( G e. USGraph /\ ( # ` F ) = 2 /\ ( P ` 0 ) =/= ( P ` ( # ` F ) ) ) -> G e. USGraph )
2	1	anim2i 593	. . . . 5 |- ( ( F (Walks ` G ) P /\ ( G e. USGraph /\ ( # ` F ) = 2 /\ ( P ` 0 ) =/= ( P ` ( # ` F ) ) ) ) -> ( F (Walks ` G ) P /\ G e. USGraph ) )
3	2	ancomd 467	. . . 4 |- ( ( F (Walks ` G ) P /\ ( G e. USGraph /\ ( # ` F ) = 2 /\ ( P ` 0 ) =/= ( P ` ( # ` F ) ) ) ) -> ( G e. USGraph /\ F (Walks ` G ) P ) )
4		3simpc 1060	. . . . 5 |- ( ( G e. USGraph /\ ( # ` F ) = 2 /\ ( P ` 0 ) =/= ( P ` ( # ` F ) ) ) -> ( ( # ` F ) = 2 /\ ( P ` 0 ) =/= ( P ` ( # ` F ) ) ) )
5	4	adantl 482	. . . 4 |- ( ( F (Walks ` G ) P /\ ( G e. USGraph /\ ( # ` F ) = 2 /\ ( P ` 0 ) =/= ( P ` ( # ` F ) ) ) ) -> ( ( # ` F ) = 2 /\ ( P ` 0 ) =/= ( P ` ( # ` F ) ) ) )
6		usgr2wlkneq 26652	. . . 4 |- ( ( ( G e. USGraph /\ F (Walks ` G ) P ) /\ ( ( # ` F ) = 2 /\ ( P ` 0 ) =/= ( P ` ( # ` F ) ) ) ) -> ( ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 0 ) =/= ( P ` 2 ) /\ ( P ` 1 ) =/= ( P ` 2 ) ) /\ ( F ` 0 ) =/= ( F ` 1 ) ) )
7	3, 5, 6	syl2anc 693	. . 3 |- ( ( F (Walks ` G ) P /\ ( G e. USGraph /\ ( # ` F ) = 2 /\ ( P ` 0 ) =/= ( P ` ( # ` F ) ) ) ) -> ( ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 0 ) =/= ( P ` 2 ) /\ ( P ` 1 ) =/= ( P ` 2 ) ) /\ ( F ` 0 ) =/= ( F ` 1 ) ) )
8		fvexd 6203	. . . 4 |- ( ( ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 0 ) =/= ( P ` 2 ) /\ ( P ` 1 ) =/= ( P ` 2 ) ) /\ ( F ` 0 ) =/= ( F ` 1 ) ) -> ( F ` 0 ) e. _V )
9		fvexd 6203	. . . 4 |- ( ( ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 0 ) =/= ( P ` 2 ) /\ ( P ` 1 ) =/= ( P ` 2 ) ) /\ ( F ` 0 ) =/= ( F ` 1 ) ) -> ( F ` 1 ) e. _V )
10		simpr 477	. . . 4 |- ( ( ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 0 ) =/= ( P ` 2 ) /\ ( P ` 1 ) =/= ( P ` 2 ) ) /\ ( F ` 0 ) =/= ( F ` 1 ) ) -> ( F ` 0 ) =/= ( F ` 1 ) )
11	8, 9, 10	3jca 1242	. . 3 |- ( ( ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 0 ) =/= ( P ` 2 ) /\ ( P ` 1 ) =/= ( P ` 2 ) ) /\ ( F ` 0 ) =/= ( F ` 1 ) ) -> ( ( F ` 0 ) e. _V /\ ( F ` 1 ) e. _V /\ ( F ` 0 ) =/= ( F ` 1 ) ) )
12		funcnvs2 13658	. . 3 |- ( ( ( F ` 0 ) e. _V /\ ( F ` 1 ) e. _V /\ ( F ` 0 ) =/= ( F ` 1 ) ) -> Fun `' <" ( F ` 0 ) ( F ` 1 ) "> )
13	7, 11, 12	3syl 18	. 2 |- ( ( F (Walks ` G ) P /\ ( G e. USGraph /\ ( # ` F ) = 2 /\ ( P ` 0 ) =/= ( P ` ( # ` F ) ) ) ) -> Fun `' <" ( F ` 0 ) ( F ` 1 ) "> )
14		eqid 2622	. . . . . 6 |- (iEdg ` G ) = (iEdg ` G )
15	14	wlkf 26510	. . . . 5 |- ( F (Walks ` G ) P -> F e. Word dom (iEdg ` G ) )
16		simp2 1062	. . . . 5 |- ( ( G e. USGraph /\ ( # ` F ) = 2 /\ ( P ` 0 ) =/= ( P ` ( # ` F ) ) ) -> ( # ` F ) = 2 )
17		wrdlen2s2 13689	. . . . 5 |- ( ( F e. Word dom (iEdg ` G ) /\ ( # ` F ) = 2 ) -> F = <" ( F ` 0 ) ( F ` 1 ) "> )
18	15, 16, 17	syl2an 494	. . . 4 |- ( ( F (Walks ` G ) P /\ ( G e. USGraph /\ ( # ` F ) = 2 /\ ( P ` 0 ) =/= ( P ` ( # ` F ) ) ) ) -> F = <" ( F ` 0 ) ( F ` 1 ) "> )
19	18	cnveqd 5298	. . 3 |- ( ( F (Walks ` G ) P /\ ( G e. USGraph /\ ( # ` F ) = 2 /\ ( P ` 0 ) =/= ( P ` ( # ` F ) ) ) ) -> `' F = `' <" ( F ` 0 ) ( F ` 1 ) "> )
20	19	funeqd 5910	. 2 |- ( ( F (Walks ` G ) P /\ ( G e. USGraph /\ ( # ` F ) = 2 /\ ( P ` 0 ) =/= ( P ` ( # ` F ) ) ) ) -> ( Fun `' F <-> Fun `' <" ( F ` 0 ) ( F ` 1 ) "> ) )
21	13, 20	mpbird 247	1 |- ( ( F (Walks ` G ) P /\ ( G e. USGraph /\ ( # ` F ) = 2 /\ ( P ` 0 ) =/= ( P ` ( # ` F ) ) ) ) -> Fun `' F )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   _Vcvv 3200   class class class wbr 4653   `'ccnv 5113   dom cdm 5114   Fun wfun 5882   ` cfv 5888   0cc0 9936   1c1 9937   2c2 11070   #chash 13117  Word cword 13291   <"cs2 13586  iEdgciedg 25875   USGraph cusgr 26044  Walkscwlks 26492

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-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ifp 1013  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-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  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-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  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-pred 5680  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-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-concat 13301  df-s1 13302  df-s2 13593  df-edg 25940  df-uhgr 25953  df-upgr 25977  df-umgr 25978  df-uspgr 26045  df-usgr 26046  df-wlks 26495

This theorem is referenced by:  usgr2wlkspth  26655