Theorem uhgrwkspthlem2 26650

Description: Lemma 2 for uhgrwkspth 26651. (Contributed by AV, 25-Jan-2021.)

Assertion

Ref	Expression
uhgrwkspthlem2	|- ( ( F (Walks ` G ) P /\ ( ( # ` F ) = 1 /\ A =/= B ) /\ ( ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B ) ) -> Fun `' P )

Proof of Theorem uhgrwkspthlem2

Step	Hyp	Ref	Expression
1		eqid 2622	. . . 4 |- (Vtx ` G ) = (Vtx ` G )
2	1	wlkp 26512	. . 3 |- ( F (Walks ` G ) P -> P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) )
3		oveq2 6658	. . . . . . . . . . . . 13 |- ( ( # ` F ) = 1 -> ( 0 ... ( # ` F ) ) = ( 0 ... 1 ) )
4		1e0p1 11552	. . . . . . . . . . . . . . 15 |- 1 = ( 0 + 1 )
5	4	oveq2i 6661	. . . . . . . . . . . . . 14 |- ( 0 ... 1 ) = ( 0 ... ( 0 + 1 ) )
6		0z 11388	. . . . . . . . . . . . . . 15 |- 0 e. ZZ
7		fzpr 12396	. . . . . . . . . . . . . . 15 |- ( 0 e. ZZ -> ( 0 ... ( 0 + 1 ) ) = { 0 , ( 0 + 1 ) } )
8	6, 7	ax-mp 5	. . . . . . . . . . . . . 14 |- ( 0 ... ( 0 + 1 ) ) = { 0 , ( 0 + 1 ) }
9		0p1e1 11132	. . . . . . . . . . . . . . 15 |- ( 0 + 1 ) = 1
10	9	preq2i 4272	. . . . . . . . . . . . . 14 |- { 0 , ( 0 + 1 ) } = { 0 , 1 }
11	5, 8, 10	3eqtri 2648	. . . . . . . . . . . . 13 |- ( 0 ... 1 ) = { 0 , 1 }
12	3, 11	syl6eq 2672	. . . . . . . . . . . 12 |- ( ( # ` F ) = 1 -> ( 0 ... ( # ` F ) ) = { 0 , 1 } )
13	12	feq2d 6031	. . . . . . . . . . 11 |- ( ( # ` F ) = 1 -> ( P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) <-> P : { 0 , 1 } --> (Vtx ` G ) ) )
14	13	adantr 481	. . . . . . . . . 10 |- ( ( ( # ` F ) = 1 /\ ( ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B ) ) -> ( P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) <-> P : { 0 , 1 } --> (Vtx ` G ) ) )
15		simpl 473	. . . . . . . . . . . . 13 |- ( ( ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B ) -> ( P ` 0 ) = A )
16		simpr 477	. . . . . . . . . . . . 13 |- ( ( ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B ) -> ( P ` ( # ` F ) ) = B )
17	15, 16	neeq12d 2855	. . . . . . . . . . . 12 |- ( ( ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B ) -> ( ( P ` 0 ) =/= ( P ` ( # ` F ) ) <-> A =/= B ) )
18	17	bicomd 213	. . . . . . . . . . 11 |- ( ( ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B ) -> ( A =/= B <-> ( P ` 0 ) =/= ( P ` ( # ` F ) ) ) )
19		fveq2 6191	. . . . . . . . . . . 12 |- ( ( # ` F ) = 1 -> ( P ` ( # ` F ) ) = ( P ` 1 ) )
20	19	neeq2d 2854	. . . . . . . . . . 11 |- ( ( # ` F ) = 1 -> ( ( P ` 0 ) =/= ( P ` ( # ` F ) ) <-> ( P ` 0 ) =/= ( P ` 1 ) ) )
21	18, 20	sylan9bbr 737	. . . . . . . . . 10 |- ( ( ( # ` F ) = 1 /\ ( ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B ) ) -> ( A =/= B <-> ( P ` 0 ) =/= ( P ` 1 ) ) )
22	14, 21	anbi12d 747	. . . . . . . . 9 |- ( ( ( # ` F ) = 1 /\ ( ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B ) ) -> ( ( P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) /\ A =/= B ) <-> ( P : { 0 , 1 } --> (Vtx ` G ) /\ ( P ` 0 ) =/= ( P ` 1 ) ) ) )
23		1z 11407	. . . . . . . . . . . 12 |- 1 e. ZZ
24		fpr2g 6475	. . . . . . . . . . . 12 |- ( ( 0 e. ZZ /\ 1 e. ZZ ) -> ( P : { 0 , 1 } --> (Vtx ` G ) <-> ( ( P ` 0 ) e. (Vtx ` G ) /\ ( P ` 1 ) e. (Vtx ` G ) /\ P = { <. 0 , ( P ` 0 ) >. , <. 1 , ( P ` 1 ) >. } ) ) )
25	6, 23, 24	mp2an 708	. . . . . . . . . . 11 |- ( P : { 0 , 1 } --> (Vtx ` G ) <-> ( ( P ` 0 ) e. (Vtx ` G ) /\ ( P ` 1 ) e. (Vtx ` G ) /\ P = { <. 0 , ( P ` 0 ) >. , <. 1 , ( P ` 1 ) >. } ) )
26		funcnvs2 13658	. . . . . . . . . . . . . . . . 17 |- ( ( ( P ` 0 ) e. (Vtx ` G ) /\ ( P ` 1 ) e. (Vtx ` G ) /\ ( P ` 0 ) =/= ( P ` 1 ) ) -> Fun `' <" ( P ` 0 ) ( P ` 1 ) "> )
27	26	3expa 1265	. . . . . . . . . . . . . . . 16 |- ( ( ( ( P ` 0 ) e. (Vtx ` G ) /\ ( P ` 1 ) e. (Vtx ` G ) ) /\ ( P ` 0 ) =/= ( P ` 1 ) ) -> Fun `' <" ( P ` 0 ) ( P ` 1 ) "> )
28	27	adantl 482	. . . . . . . . . . . . . . 15 |- ( ( P = { <. 0 , ( P ` 0 ) >. , <. 1 , ( P ` 1 ) >. } /\ ( ( ( P ` 0 ) e. (Vtx ` G ) /\ ( P ` 1 ) e. (Vtx ` G ) ) /\ ( P ` 0 ) =/= ( P ` 1 ) ) ) -> Fun `' <" ( P ` 0 ) ( P ` 1 ) "> )
29		simpl 473	. . . . . . . . . . . . . . . . . 18 |- ( ( P = { <. 0 , ( P ` 0 ) >. , <. 1 , ( P ` 1 ) >. } /\ ( ( ( P ` 0 ) e. (Vtx ` G ) /\ ( P ` 1 ) e. (Vtx ` G ) ) /\ ( P ` 0 ) =/= ( P ` 1 ) ) ) -> P = { <. 0 , ( P ` 0 ) >. , <. 1 , ( P ` 1 ) >. } )
30		s2prop 13652	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( P ` 0 ) e. (Vtx ` G ) /\ ( P ` 1 ) e. (Vtx ` G ) ) -> <" ( P ` 0 ) ( P ` 1 ) "> = { <. 0 , ( P ` 0 ) >. , <. 1 , ( P ` 1 ) >. } )
31	30	eqcomd 2628	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( P ` 0 ) e. (Vtx ` G ) /\ ( P ` 1 ) e. (Vtx ` G ) ) -> { <. 0 , ( P ` 0 ) >. , <. 1 , ( P ` 1 ) >. } = <" ( P ` 0 ) ( P ` 1 ) "> )
32	31	adantr 481	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ( P ` 0 ) e. (Vtx ` G ) /\ ( P ` 1 ) e. (Vtx ` G ) ) /\ ( P ` 0 ) =/= ( P ` 1 ) ) -> { <. 0 , ( P ` 0 ) >. , <. 1 , ( P ` 1 ) >. } = <" ( P ` 0 ) ( P ` 1 ) "> )
33	32	adantl 482	. . . . . . . . . . . . . . . . . 18 |- ( ( P = { <. 0 , ( P ` 0 ) >. , <. 1 , ( P ` 1 ) >. } /\ ( ( ( P ` 0 ) e. (Vtx ` G ) /\ ( P ` 1 ) e. (Vtx ` G ) ) /\ ( P ` 0 ) =/= ( P ` 1 ) ) ) -> { <. 0 , ( P ` 0 ) >. , <. 1 , ( P ` 1 ) >. } = <" ( P ` 0 ) ( P ` 1 ) "> )
34	29, 33	eqtrd 2656	. . . . . . . . . . . . . . . . 17 |- ( ( P = { <. 0 , ( P ` 0 ) >. , <. 1 , ( P ` 1 ) >. } /\ ( ( ( P ` 0 ) e. (Vtx ` G ) /\ ( P ` 1 ) e. (Vtx ` G ) ) /\ ( P ` 0 ) =/= ( P ` 1 ) ) ) -> P = <" ( P ` 0 ) ( P ` 1 ) "> )
35	34	cnveqd 5298	. . . . . . . . . . . . . . . 16 |- ( ( P = { <. 0 , ( P ` 0 ) >. , <. 1 , ( P ` 1 ) >. } /\ ( ( ( P ` 0 ) e. (Vtx ` G ) /\ ( P ` 1 ) e. (Vtx ` G ) ) /\ ( P ` 0 ) =/= ( P ` 1 ) ) ) -> `' P = `' <" ( P ` 0 ) ( P ` 1 ) "> )
36	35	funeqd 5910	. . . . . . . . . . . . . . 15 |- ( ( P = { <. 0 , ( P ` 0 ) >. , <. 1 , ( P ` 1 ) >. } /\ ( ( ( P ` 0 ) e. (Vtx ` G ) /\ ( P ` 1 ) e. (Vtx ` G ) ) /\ ( P ` 0 ) =/= ( P ` 1 ) ) ) -> ( Fun `' P <-> Fun `' <" ( P ` 0 ) ( P ` 1 ) "> ) )
37	28, 36	mpbird 247	. . . . . . . . . . . . . 14 |- ( ( P = { <. 0 , ( P ` 0 ) >. , <. 1 , ( P ` 1 ) >. } /\ ( ( ( P ` 0 ) e. (Vtx ` G ) /\ ( P ` 1 ) e. (Vtx ` G ) ) /\ ( P ` 0 ) =/= ( P ` 1 ) ) ) -> Fun `' P )
38	37	exp32 631	. . . . . . . . . . . . 13 |- ( P = { <. 0 , ( P ` 0 ) >. , <. 1 , ( P ` 1 ) >. } -> ( ( ( P ` 0 ) e. (Vtx ` G ) /\ ( P ` 1 ) e. (Vtx ` G ) ) -> ( ( P ` 0 ) =/= ( P ` 1 ) -> Fun `' P ) ) )
39	38	impcom 446	. . . . . . . . . . . 12 |- ( ( ( ( P ` 0 ) e. (Vtx ` G ) /\ ( P ` 1 ) e. (Vtx ` G ) ) /\ P = { <. 0 , ( P ` 0 ) >. , <. 1 , ( P ` 1 ) >. } ) -> ( ( P ` 0 ) =/= ( P ` 1 ) -> Fun `' P ) )
40	39	3impa 1259	. . . . . . . . . . 11 |- ( ( ( P ` 0 ) e. (Vtx ` G ) /\ ( P ` 1 ) e. (Vtx ` G ) /\ P = { <. 0 , ( P ` 0 ) >. , <. 1 , ( P ` 1 ) >. } ) -> ( ( P ` 0 ) =/= ( P ` 1 ) -> Fun `' P ) )
41	25, 40	sylbi 207	. . . . . . . . . 10 |- ( P : { 0 , 1 } --> (Vtx ` G ) -> ( ( P ` 0 ) =/= ( P ` 1 ) -> Fun `' P ) )
42	41	imp 445	. . . . . . . . 9 |- ( ( P : { 0 , 1 } --> (Vtx ` G ) /\ ( P ` 0 ) =/= ( P ` 1 ) ) -> Fun `' P )
43	22, 42	syl6bi 243	. . . . . . . 8 |- ( ( ( # ` F ) = 1 /\ ( ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B ) ) -> ( ( P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) /\ A =/= B ) -> Fun `' P ) )
44	43	expd 452	. . . . . . 7 |- ( ( ( # ` F ) = 1 /\ ( ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B ) ) -> ( P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) -> ( A =/= B -> Fun `' P ) ) )
45	44	com12 32	. . . . . 6 |- ( P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) -> ( ( ( # ` F ) = 1 /\ ( ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B ) ) -> ( A =/= B -> Fun `' P ) ) )
46	45	expd 452	. . . . 5 |- ( P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) -> ( ( # ` F ) = 1 -> ( ( ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B ) -> ( A =/= B -> Fun `' P ) ) ) )
47	46	com34 91	. . . 4 |- ( P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) -> ( ( # ` F ) = 1 -> ( A =/= B -> ( ( ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B ) -> Fun `' P ) ) ) )
48	47	impd 447	. . 3 |- ( P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) -> ( ( ( # ` F ) = 1 /\ A =/= B ) -> ( ( ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B ) -> Fun `' P ) ) )
49	2, 48	syl 17	. 2 |- ( F (Walks ` G ) P -> ( ( ( # ` F ) = 1 /\ A =/= B ) -> ( ( ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B ) -> Fun `' P ) ) )
50	49	3imp 1256	1 |- ( ( F (Walks ` G ) P /\ ( ( # ` F ) = 1 /\ A =/= B ) /\ ( ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B ) ) -> Fun `' P )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   {cpr 4179   <.cop 4183   class class class wbr 4653   `'ccnv 5113   Fun wfun 5882   -->wf 5884   ` cfv 5888  (class class class)co 6650   0cc0 9936   1c1 9937   + caddc 9939   ZZcz 11377   ...cfz 12326   #chash 13117   <"cs2 13586  Vtxcvtx 25874  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-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-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-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  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-wlks 26495

This theorem is referenced by:  uhgrwkspth  26651