Theorem pthon 24250

Description: The set of paths between two vertices (in an undirected graph). (Contributed by Alexander van der Vekens, 8-Nov-2017.)

Assertion

Ref	Expression
pthon	|- ( ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ B e. V ) ) -> ( A ( V PathOn E ) B ) = { <. f , p >. | ( f ( A ( V WalkOn E ) B ) p /\ f ( V Paths E ) p ) } )

Distinct variable groups:   f, E, p   f, V, p   f, X, p   f, Y, p   A, f, p   B, f, p

Proof of Theorem pthon

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

Step	Hyp	Ref	Expression
1		elex 3122	. . . . 5 |- ( V e. X -> V e. _V )
2	1	ad2antrr 725	. . . 4 |- ( ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ B e. V ) ) -> V e. _V )
3		elex 3122	. . . . . 6 |- ( E e. Y -> E e. _V )
4	3	adantl 466	. . . . 5 |- ( ( V e. X /\ E e. Y ) -> E e. _V )
5	4	adantr 465	. . . 4 |- ( ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ B e. V ) ) -> E e. _V )
6		id 22	. . . . . . 7 |- ( V e. X -> V e. X )
7	6	ancli 551	. . . . . 6 |- ( V e. X -> ( V e. X /\ V e. X ) )
8	7	ad2antrr 725	. . . . 5 |- ( ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ B e. V ) ) -> ( V e. X /\ V e. X ) )
9		mpt2exga 6856	. . . . 5 |- ( ( V e. X /\ V e. X ) -> ( a e. V , b e. V |-> { <. f , p >. | ( f ( a ( V WalkOn E ) b ) p /\ f ( V Paths E ) p ) } ) e. _V )
10	8, 9	syl 16	. . . 4 |- ( ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ B e. V ) ) -> ( a e. V , b e. V |-> { <. f , p >. | ( f ( a ( V WalkOn E ) b ) p /\ f ( V Paths E ) p ) } ) e. _V )
11		simpl 457	. . . . . 6 |- ( ( v = V /\ e = E ) -> v = V )
12		oveq12 6291	. . . . . . . . . 10 |- ( ( v = V /\ e = E ) -> ( v WalkOn e ) = ( V WalkOn E ) )
13	12	oveqd 6299	. . . . . . . . 9 |- ( ( v = V /\ e = E ) -> ( a ( v WalkOn e ) b ) = ( a ( V WalkOn E ) b ) )
14	13	breqd 4458	. . . . . . . 8 |- ( ( v = V /\ e = E ) -> ( f ( a ( v WalkOn e ) b ) p <-> f ( a ( V WalkOn E ) b ) p ) )
15		oveq12 6291	. . . . . . . . 9 |- ( ( v = V /\ e = E ) -> ( v Paths e ) = ( V Paths E ) )
16	15	breqd 4458	. . . . . . . 8 |- ( ( v = V /\ e = E ) -> ( f ( v Paths e ) p <-> f ( V Paths E ) p ) )
17	14, 16	anbi12d 710	. . . . . . 7 |- ( ( v = V /\ e = E ) -> ( ( f ( a ( v WalkOn e ) b ) p /\ f ( v Paths e ) p ) <-> ( f ( a ( V WalkOn E ) b ) p /\ f ( V Paths E ) p ) ) )
18	17	opabbidv 4510	. . . . . 6 |- ( ( v = V /\ e = E ) -> { <. f , p >. | ( f ( a ( v WalkOn e ) b ) p /\ f ( v Paths e ) p ) } = { <. f , p >. | ( f ( a ( V WalkOn E ) b ) p /\ f ( V Paths E ) p ) } )
19	11, 11, 18	mpt2eq123dv 6341	. . . . 5 |- ( ( v = V /\ e = E ) -> ( a e. v , b e. v |-> { <. f , p >. | ( f ( a ( v WalkOn e ) b ) p /\ f ( v Paths e ) p ) } ) = ( a e. V , b e. V |-> { <. f , p >. | ( f ( a ( V WalkOn E ) b ) p /\ f ( V Paths E ) p ) } ) )
20		df-pthon 24189	. . . . 5 |- PathOn = ( v e. _V , e e. _V |-> ( a e. v , b e. v |-> { <. f , p >. | ( f ( a ( v WalkOn e ) b ) p /\ f ( v Paths e ) p ) } ) )
21	19, 20	ovmpt2ga 6414	. . . 4 |- ( ( V e. _V /\ E e. _V /\ ( a e. V , b e. V |-> { <. f , p >. | ( f ( a ( V WalkOn E ) b ) p /\ f ( V Paths E ) p ) } ) e. _V ) -> ( V PathOn E ) = ( a e. V , b e. V |-> { <. f , p >. | ( f ( a ( V WalkOn E ) b ) p /\ f ( V Paths E ) p ) } ) )
22	2, 5, 10, 21	syl3anc 1228	. . 3 |- ( ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ B e. V ) ) -> ( V PathOn E ) = ( a e. V , b e. V |-> { <. f , p >. | ( f ( a ( V WalkOn E ) b ) p /\ f ( V Paths E ) p ) } ) )
23	22	oveqd 6299	. 2 |- ( ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ B e. V ) ) -> ( A ( V PathOn E ) B ) = ( A ( a e. V , b e. V |-> { <. f , p >. | ( f ( a ( V WalkOn E ) b ) p /\ f ( V Paths E ) p ) } ) B ) )
24		simpl 457	. . . 4 |- ( ( A e. V /\ B e. V ) -> A e. V )
25	24	adantl 466	. . 3 |- ( ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ B e. V ) ) -> A e. V )
26		simprr 756	. . 3 |- ( ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ B e. V ) ) -> B e. V )
27		ancom 450	. . . . . . 7 |- ( ( f ( A ( V WalkOn E ) B ) p /\ f ( V Paths E ) p ) <-> ( f ( V Paths E ) p /\ f ( A ( V WalkOn E ) B ) p ) )
28	27	a1i 11	. . . . . 6 |- ( ( V e. X /\ E e. Y ) -> ( ( f ( A ( V WalkOn E ) B ) p /\ f ( V Paths E ) p ) <-> ( f ( V Paths E ) p /\ f ( A ( V WalkOn E ) B ) p ) ) )
29	28	opabbidv 4510	. . . . 5 |- ( ( V e. X /\ E e. Y ) -> { <. f , p >. | ( f ( A ( V WalkOn E ) B ) p /\ f ( V Paths E ) p ) } = { <. f , p >. | ( f ( V Paths E ) p /\ f ( A ( V WalkOn E ) B ) p ) } )
30		pthistrl 24247	. . . . . . . 8 |- ( f ( V Paths E ) p -> f ( V Trails E ) p )
31		trliswlk 24214	. . . . . . . 8 |- ( f ( V Trails E ) p -> f ( V Walks E ) p )
32	30, 31	syl 16	. . . . . . 7 |- ( f ( V Paths E ) p -> f ( V Walks E ) p )
33	32	wlkres 24195	. . . . . 6 |- ( ( V e. _V /\ E e. _V ) -> { <. f , p >. | ( f ( V Paths E ) p /\ f ( A ( V WalkOn E ) B ) p ) } e. _V )
34	1, 3, 33	syl2an 477	. . . . 5 |- ( ( V e. X /\ E e. Y ) -> { <. f , p >. | ( f ( V Paths E ) p /\ f ( A ( V WalkOn E ) B ) p ) } e. _V )
35	29, 34	eqeltrd 2555	. . . 4 |- ( ( V e. X /\ E e. Y ) -> { <. f , p >. | ( f ( A ( V WalkOn E ) B ) p /\ f ( V Paths E ) p ) } e. _V )
36	35	adantr 465	. . 3 |- ( ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ B e. V ) ) -> { <. f , p >. | ( f ( A ( V WalkOn E ) B ) p /\ f ( V Paths E ) p ) } e. _V )
37		oveq12 6291	. . . . . . 7 |- ( ( a = A /\ b = B ) -> ( a ( V WalkOn E ) b ) = ( A ( V WalkOn E ) B ) )
38	37	breqd 4458	. . . . . 6 |- ( ( a = A /\ b = B ) -> ( f ( a ( V WalkOn E ) b ) p <-> f ( A ( V WalkOn E ) B ) p ) )
39	38	anbi1d 704	. . . . 5 |- ( ( a = A /\ b = B ) -> ( ( f ( a ( V WalkOn E ) b ) p /\ f ( V Paths E ) p ) <-> ( f ( A ( V WalkOn E ) B ) p /\ f ( V Paths E ) p ) ) )
40	39	opabbidv 4510	. . . 4 |- ( ( a = A /\ b = B ) -> { <. f , p >. | ( f ( a ( V WalkOn E ) b ) p /\ f ( V Paths E ) p ) } = { <. f , p >. | ( f ( A ( V WalkOn E ) B ) p /\ f ( V Paths E ) p ) } )
41		eqid 2467	. . . 4 |- ( a e. V , b e. V |-> { <. f , p >. | ( f ( a ( V WalkOn E ) b ) p /\ f ( V Paths E ) p ) } ) = ( a e. V , b e. V |-> { <. f , p >. | ( f ( a ( V WalkOn E ) b ) p /\ f ( V Paths E ) p ) } )
42	40, 41	ovmpt2ga 6414	. . 3 |- ( ( A e. V /\ B e. V /\ { <. f , p >. | ( f ( A ( V WalkOn E ) B ) p /\ f ( V Paths E ) p ) } e. _V ) -> ( A ( a e. V , b e. V |-> { <. f , p >. | ( f ( a ( V WalkOn E ) b ) p /\ f ( V Paths E ) p ) } ) B ) = { <. f , p >. | ( f ( A ( V WalkOn E ) B ) p /\ f ( V Paths E ) p ) } )
43	25, 26, 36, 42	syl3anc 1228	. 2 |- ( ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ B e. V ) ) -> ( A ( a e. V , b e. V |-> { <. f , p >. | ( f ( a ( V WalkOn E ) b ) p /\ f ( V Paths E ) p ) } ) B ) = { <. f , p >. | ( f ( A ( V WalkOn E ) B ) p /\ f ( V Paths E ) p ) } )
44	23, 43	eqtrd 2508	1 |- ( ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ B e. V ) ) -> ( A ( V PathOn E ) B ) = { <. f , p >. | ( f ( A ( V WalkOn E ) B ) p /\ f ( V Paths E ) p ) } )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1379   e. wcel 1767   _Vcvv 3113   class class class wbr 4447   {copab 4504  (class class class)co 6282   |-> cmpt2 6284   Walks cwalk 24171   Trails ctrail 24172   Paths cpath 24173   WalkOn cwlkon 24175   PathOn cpthon 24177

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-er 7308  df-map 7419  df-pm 7420  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-n0 10792  df-z 10861  df-uz 11079  df-fz 11669  df-fzo 11789  df-word 12502  df-wlk 24181  df-trail 24182  df-pth 24183  df-pthon 24189

This theorem is referenced by:  ispthon  24251  pthonprop  24252