Theorem pthon 25305

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 3054	. . . . 5 |- ( V e. X -> V e. _V )
2	1	ad2antrr 732	. . . 4 |- ( ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ B e. V ) ) -> V e. _V )
3		elex 3054	. . . . . 6 |- ( E e. Y -> E e. _V )
4	3	adantl 468	. . . . 5 |- ( ( V e. X /\ E e. Y ) -> E e. _V )
5	4	adantr 467	. . . 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 554	. . . . . 6 |- ( V e. X -> ( V e. X /\ V e. X ) )
8	7	ad2antrr 732	. . . . 5 |- ( ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ B e. V ) ) -> ( V e. X /\ V e. X ) )
9		mpt2exga 6869	. . . . 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 17	. . . 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 459	. . . . . 6 |- ( ( v = V /\ e = E ) -> v = V )
12		oveq12 6299	. . . . . . . . . 10 |- ( ( v = V /\ e = E ) -> ( v WalkOn e ) = ( V WalkOn E ) )
13	12	oveqd 6307	. . . . . . . . 9 |- ( ( v = V /\ e = E ) -> ( a ( v WalkOn e ) b ) = ( a ( V WalkOn E ) b ) )
14	13	breqd 4413	. . . . . . . 8 |- ( ( v = V /\ e = E ) -> ( f ( a ( v WalkOn e ) b ) p <-> f ( a ( V WalkOn E ) b ) p ) )
15		oveq12 6299	. . . . . . . . 9 |- ( ( v = V /\ e = E ) -> ( v Paths e ) = ( V Paths E ) )
16	15	breqd 4413	. . . . . . . 8 |- ( ( v = V /\ e = E ) -> ( f ( v Paths e ) p <-> f ( V Paths E ) p ) )
17	14, 16	anbi12d 717	. . . . . . 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 4466	. . . . . 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 6353	. . . . 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 25244	. . . . 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 6426	. . . 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 1268	. . 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 6307	. 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 459	. . . 4 |- ( ( A e. V /\ B e. V ) -> A e. V )
25	24	adantl 468	. . 3 |- ( ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ B e. V ) ) -> A e. V )
26		simprr 766	. . 3 |- ( ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ B e. V ) ) -> B e. V )
27		ancom 452	. . . . . . 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 4466	. . . . 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 25302	. . . . . . . 8 |- ( f ( V Paths E ) p -> f ( V Trails E ) p )
31		trliswlk 25269	. . . . . . . 8 |- ( f ( V Trails E ) p -> f ( V Walks E ) p )
32	30, 31	syl 17	. . . . . . 7 |- ( f ( V Paths E ) p -> f ( V Walks E ) p )
33	32	wlkres 25250	. . . . . 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 480	. . . . 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 2529	. . . 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 467	. . 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 6299	. . . . . . 7 |- ( ( a = A /\ b = B ) -> ( a ( V WalkOn E ) b ) = ( A ( V WalkOn E ) B ) )
38	37	breqd 4413	. . . . . 6 |- ( ( a = A /\ b = B ) -> ( f ( a ( V WalkOn E ) b ) p <-> f ( A ( V WalkOn E ) B ) p ) )
39	38	anbi1d 711	. . . . 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 4466	. . . 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 6426	. . 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 1268	. 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 2485	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 188   /\ wa 371   = wceq 1444   e. wcel 1887   _Vcvv 3045   class class class wbr 4402   {copab 4460  (class class class)co 6290   |-> cmpt2 6292   Walks cwalk 25226   Trails ctrail 25227   Paths cpath 25228   WalkOn cwlkon 25230   PathOn cpthon 25232

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-oadd 7186  df-er 7363  df-map 7474  df-pm 7475  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-card 8373  df-cda 8598  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-nn 10610  df-2 10668  df-n0 10870  df-z 10938  df-uz 11160  df-fz 11785  df-fzo 11916  df-hash 12516  df-word 12664  df-wlk 25236  df-trail 25237  df-pth 25238  df-pthon 25244

This theorem is referenced by:  ispthon  25306  pthonprop  25307