Theorem wlkson 26552

Description: The set of walks between two vertices. (Contributed by Alexander van der Vekens, 12-Dec-2017.) (Revised by AV, 30-Dec-2020.) (Revised by AV, 22-Mar-2021.)

Hypothesis

Ref	Expression
wlkson.v	|- V = (Vtx ` G )

Assertion

Ref	Expression
wlkson	|- ( ( A e. V /\ B e. V ) -> ( A (WalksOn ` G ) B ) = { <. f , p >. | ( f (Walks ` G ) p /\ ( p ` 0 ) = A /\ ( p ` ( # ` f ) ) = B ) } )

Distinct variable groups:   A, f, p   B, f, p   f, G, p   f, V, p

Proof of Theorem wlkson

Dummy variables a b g are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		wlkson.v	. . . . 5 |- V = (Vtx ` G )
2	1	1vgrex 25882	. . . 4 |- ( A e. V -> G e. _V )
3	2	adantr 481	. . 3 |- ( ( A e. V /\ B e. V ) -> G e. _V )
4		simpl 473	. . . 4 |- ( ( A e. V /\ B e. V ) -> A e. V )
5	4, 1	syl6eleq 2711	. . 3 |- ( ( A e. V /\ B e. V ) -> A e. (Vtx ` G ) )
6		simpr 477	. . . 4 |- ( ( A e. V /\ B e. V ) -> B e. V )
7	6, 1	syl6eleq 2711	. . 3 |- ( ( A e. V /\ B e. V ) -> B e. (Vtx ` G ) )
8		wksv 26515	. . . 4 |- { <. f , p >. | f (Walks ` G ) p } e. _V
9	8	a1i 11	. . 3 |- ( ( A e. V /\ B e. V ) -> { <. f , p >. | f (Walks ` G ) p } e. _V )
10		simpr 477	. . 3 |- ( ( ( A e. V /\ B e. V ) /\ f (Walks ` G ) p ) -> f (Walks ` G ) p )
11		eqeq2 2633	. . . 4 |- ( a = A -> ( ( p ` 0 ) = a <-> ( p ` 0 ) = A ) )
12		eqeq2 2633	. . . 4 |- ( b = B -> ( ( p ` ( # ` f ) ) = b <-> ( p ` ( # ` f ) ) = B ) )
13	11, 12	bi2anan9 917	. . 3 |- ( ( a = A /\ b = B ) -> ( ( ( p ` 0 ) = a /\ ( p ` ( # ` f ) ) = b ) <-> ( ( p ` 0 ) = A /\ ( p ` ( # ` f ) ) = B ) ) )
14		biidd 252	. . 3 |- ( g = G -> ( ( ( p ` 0 ) = a /\ ( p ` ( # ` f ) ) = b ) <-> ( ( p ` 0 ) = a /\ ( p ` ( # ` f ) ) = b ) ) )
15		df-wlkson 26496	. . . 4 |- WalksOn = ( g e. _V |-> ( a e. (Vtx ` g ) , b e. (Vtx ` g ) |-> { <. f , p >. | ( f (Walks ` g ) p /\ ( p ` 0 ) = a /\ ( p ` ( # ` f ) ) = b ) } ) )
16		eqid 2622	. . . . . 6 |- (Vtx ` g ) = (Vtx ` g )
17		3anass 1042	. . . . . . . 8 |- ( ( f (Walks ` g ) p /\ ( p ` 0 ) = a /\ ( p ` ( # ` f ) ) = b ) <-> ( f (Walks ` g ) p /\ ( ( p ` 0 ) = a /\ ( p ` ( # ` f ) ) = b ) ) )
18		ancom 466	. . . . . . . 8 |- ( ( f (Walks ` g ) p /\ ( ( p ` 0 ) = a /\ ( p ` ( # ` f ) ) = b ) ) <-> ( ( ( p ` 0 ) = a /\ ( p ` ( # ` f ) ) = b ) /\ f (Walks ` g ) p ) )
19	17, 18	bitri 264	. . . . . . 7 |- ( ( f (Walks ` g ) p /\ ( p ` 0 ) = a /\ ( p ` ( # ` f ) ) = b ) <-> ( ( ( p ` 0 ) = a /\ ( p ` ( # ` f ) ) = b ) /\ f (Walks ` g ) p ) )
20	19	opabbii 4717	. . . . . 6 |- { <. f , p >. | ( f (Walks ` g ) p /\ ( p ` 0 ) = a /\ ( p ` ( # ` f ) ) = b ) } = { <. f , p >. | ( ( ( p ` 0 ) = a /\ ( p ` ( # ` f ) ) = b ) /\ f (Walks ` g ) p ) }
21	16, 16, 20	mpt2eq123i 6718	. . . . 5 |- ( a e. (Vtx ` g ) , b e. (Vtx ` g ) |-> { <. f , p >. | ( f (Walks ` g ) p /\ ( p ` 0 ) = a /\ ( p ` ( # ` f ) ) = b ) } ) = ( a e. (Vtx ` g ) , b e. (Vtx ` g ) |-> { <. f , p >. | ( ( ( p ` 0 ) = a /\ ( p ` ( # ` f ) ) = b ) /\ f (Walks ` g ) p ) } )
22	21	mpteq2i 4741	. . . 4 |- ( g e. _V |-> ( a e. (Vtx ` g ) , b e. (Vtx ` g ) |-> { <. f , p >. | ( f (Walks ` g ) p /\ ( p ` 0 ) = a /\ ( p ` ( # ` f ) ) = b ) } ) ) = ( g e. _V |-> ( a e. (Vtx ` g ) , b e. (Vtx ` g ) |-> { <. f , p >. | ( ( ( p ` 0 ) = a /\ ( p ` ( # ` f ) ) = b ) /\ f (Walks ` g ) p ) } ) )
23	15, 22	eqtri 2644	. . 3 |- WalksOn = ( g e. _V |-> ( a e. (Vtx ` g ) , b e. (Vtx ` g ) |-> { <. f , p >. | ( ( ( p ` 0 ) = a /\ ( p ` ( # ` f ) ) = b ) /\ f (Walks ` g ) p ) } ) )
24	3, 5, 7, 9, 10, 13, 14, 23	mptmpt2opabbrd 7248	. 2 |- ( ( A e. V /\ B e. V ) -> ( A (WalksOn ` G ) B ) = { <. f , p >. | ( ( ( p ` 0 ) = A /\ ( p ` ( # ` f ) ) = B ) /\ f (Walks ` G ) p ) } )
25		ancom 466	. . . 4 |- ( ( ( ( p ` 0 ) = A /\ ( p ` ( # ` f ) ) = B ) /\ f (Walks ` G ) p ) <-> ( f (Walks ` G ) p /\ ( ( p ` 0 ) = A /\ ( p ` ( # ` f ) ) = B ) ) )
26		3anass 1042	. . . 4 |- ( ( f (Walks ` G ) p /\ ( p ` 0 ) = A /\ ( p ` ( # ` f ) ) = B ) <-> ( f (Walks ` G ) p /\ ( ( p ` 0 ) = A /\ ( p ` ( # ` f ) ) = B ) ) )
27	25, 26	bitr4i 267	. . 3 |- ( ( ( ( p ` 0 ) = A /\ ( p ` ( # ` f ) ) = B ) /\ f (Walks ` G ) p ) <-> ( f (Walks ` G ) p /\ ( p ` 0 ) = A /\ ( p ` ( # ` f ) ) = B ) )
28	27	opabbii 4717	. 2 |- { <. f , p >. | ( ( ( p ` 0 ) = A /\ ( p ` ( # ` f ) ) = B ) /\ f (Walks ` G ) p ) } = { <. f , p >. | ( f (Walks ` G ) p /\ ( p ` 0 ) = A /\ ( p ` ( # ` f ) ) = B ) }
29	24, 28	syl6eq 2672	1 |- ( ( A e. V /\ B e. V ) -> ( A (WalksOn ` G ) B ) = { <. f , p >. | ( f (Walks ` G ) p /\ ( p ` 0 ) = A /\ ( p ` ( # ` f ) ) = B ) } )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   _Vcvv 3200   class class class wbr 4653   {copab 4712   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   0cc0 9936   #chash 13117  Vtxcvtx 25874  Walkscwlks 26492  WalksOncwlkson 26493

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-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-wlks 26495  df-wlkson 26496

This theorem is referenced by:  iswlkon  26553  wlkonprop  26554