Theorem pthon 23606

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 3074	. . . . 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 3074	. . . . . 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 6746	. . . . 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 6196	. . . . . . . . . 10 |- ( ( v = V /\ e = E ) -> ( v WalkOn e ) = ( V WalkOn E ) )
13	12	oveqd 6204	. . . . . . . . 9 |- ( ( v = V /\ e = E ) -> ( a ( v WalkOn e ) b ) = ( a ( V WalkOn E ) b ) )
14	13	breqd 4398	. . . . . . . 8 |- ( ( v = V /\ e = E ) -> ( f ( a ( v WalkOn e ) b ) p <-> f ( a ( V WalkOn E ) b ) p ) )
15		oveq12 6196	. . . . . . . . 9 |- ( ( v = V /\ e = E ) -> ( v Paths e ) = ( V Paths E ) )
16	15	breqd 4398	. . . . . . . 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 4450	. . . . . 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 6244	. . . . 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 23555	. . . . 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 6317	. . . 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 1219	. . 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 6204	. 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 4450	. . . . 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 23603	. . . . . . . 8 |- ( f ( V Paths E ) p -> f ( V Trails E ) p )
31		trliswlk 23570	. . . . . . . 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 23560	. . . . . 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 2537	. . . 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 6196	. . . . . . 7 |- ( ( a = A /\ b = B ) -> ( a ( V WalkOn E ) b ) = ( A ( V WalkOn E ) B ) )
38	37	breqd 4398	. . . . . 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 4450	. . . 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 2451	. . . 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 6317	. . 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 1219	. 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 2491	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 1370   e. wcel 1758   _Vcvv 3065   class class class wbr 4387   {copab 4444  (class class class)co 6187   |-> cmpt2 6189   Walks cwalk 23537   Trails ctrail 23538   Paths cpath 23539   WalkOn cwlkon 23541   PathOn cpthon 23543

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4498  ax-sep 4508  ax-nul 4516  ax-pow 4565  ax-pr 4626  ax-un 6469  ax-cnex 9436  ax-resscn 9437  ax-1cn 9438  ax-icn 9439  ax-addcl 9440  ax-addrcl 9441  ax-mulcl 9442  ax-mulrcl 9443  ax-mulcom 9444  ax-addass 9445  ax-mulass 9446  ax-distr 9447  ax-i2m1 9448  ax-1ne0 9449  ax-1rid 9450  ax-rnegex 9451  ax-rrecex 9452  ax-cnre 9453  ax-pre-lttri 9454  ax-pre-lttrn 9455  ax-pre-ltadd 9456  ax-pre-mulgt0 9457

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-ne 2644  df-nel 2645  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3067  df-sbc 3282  df-csb 3384  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-pss 3439  df-nul 3733  df-if 3887  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4187  df-int 4224  df-iun 4268  df-br 4388  df-opab 4446  df-mpt 4447  df-tr 4481  df-eprel 4727  df-id 4731  df-po 4736  df-so 4737  df-fr 4774  df-we 4776  df-ord 4817  df-on 4818  df-lim 4819  df-suc 4820  df-xp 4941  df-rel 4942  df-cnv 4943  df-co 4944  df-dm 4945  df-rn 4946  df-res 4947  df-ima 4948  df-iota 5476  df-fun 5515  df-fn 5516  df-f 5517  df-f1 5518  df-fo 5519  df-f1o 5520  df-fv 5521  df-riota 6148  df-ov 6190  df-oprab 6191  df-mpt2 6192  df-om 6574  df-1st 6674  df-2nd 6675  df-recs 6929  df-rdg 6963  df-1o 7017  df-oadd 7021  df-er 7198  df-map 7313  df-pm 7314  df-en 7408  df-dom 7409  df-sdom 7410  df-fin 7411  df-pnf 9518  df-mnf 9519  df-xr 9520  df-ltxr 9521  df-le 9522  df-sub 9695  df-neg 9696  df-nn 10421  df-n0 10678  df-z 10745  df-uz 10960  df-fz 11536  df-fzo 11647  df-word 12328  df-wlk 23547  df-trail 23548  df-pth 23549  df-pthon 23555

This theorem is referenced by:  ispthon  23607  pthonprop  23608