Theorem pthon 24875

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 3067	. . . . 5 |- ( V e. X -> V e. _V )
2	1	ad2antrr 724	. . . 4 |- ( ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ B e. V ) ) -> V e. _V )
3		elex 3067	. . . . . 6 |- ( E e. Y -> E e. _V )
4	3	adantl 464	. . . . 5 |- ( ( V e. X /\ E e. Y ) -> E e. _V )
5	4	adantr 463	. . . 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 549	. . . . . 6 |- ( V e. X -> ( V e. X /\ V e. X ) )
8	7	ad2antrr 724	. . . . 5 |- ( ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ B e. V ) ) -> ( V e. X /\ V e. X ) )
9		mpt2exga 6814	. . . . 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 455	. . . . . 6 |- ( ( v = V /\ e = E ) -> v = V )
12		oveq12 6243	. . . . . . . . . 10 |- ( ( v = V /\ e = E ) -> ( v WalkOn e ) = ( V WalkOn E ) )
13	12	oveqd 6251	. . . . . . . . 9 |- ( ( v = V /\ e = E ) -> ( a ( v WalkOn e ) b ) = ( a ( V WalkOn E ) b ) )
14	13	breqd 4405	. . . . . . . 8 |- ( ( v = V /\ e = E ) -> ( f ( a ( v WalkOn e ) b ) p <-> f ( a ( V WalkOn E ) b ) p ) )
15		oveq12 6243	. . . . . . . . 9 |- ( ( v = V /\ e = E ) -> ( v Paths e ) = ( V Paths E ) )
16	15	breqd 4405	. . . . . . . 8 |- ( ( v = V /\ e = E ) -> ( f ( v Paths e ) p <-> f ( V Paths E ) p ) )
17	14, 16	anbi12d 709	. . . . . . 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 4457	. . . . . 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 6296	. . . . 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 24814	. . . . 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 6369	. . . 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 1230	. . 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 6251	. 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 455	. . . 4 |- ( ( A e. V /\ B e. V ) -> A e. V )
25	24	adantl 464	. . 3 |- ( ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ B e. V ) ) -> A e. V )
26		simprr 758	. . 3 |- ( ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ B e. V ) ) -> B e. V )
27		ancom 448	. . . . . . 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 4457	. . . . 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 24872	. . . . . . . 8 |- ( f ( V Paths E ) p -> f ( V Trails E ) p )
31		trliswlk 24839	. . . . . . . 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 24820	. . . . . 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 475	. . . . 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 2490	. . . 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 463	. . 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 6243	. . . . . . 7 |- ( ( a = A /\ b = B ) -> ( a ( V WalkOn E ) b ) = ( A ( V WalkOn E ) B ) )
38	37	breqd 4405	. . . . . 6 |- ( ( a = A /\ b = B ) -> ( f ( a ( V WalkOn E ) b ) p <-> f ( A ( V WalkOn E ) B ) p ) )
39	38	anbi1d 703	. . . . 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 4457	. . . 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 2402	. . . 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 6369	. . 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 1230	. 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 2443	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 367   = wceq 1405   e. wcel 1842   _Vcvv 3058   class class class wbr 4394   {copab 4451  (class class class)co 6234   |-> cmpt2 6236   Walks cwalk 24796   Trails ctrail 24797   Paths cpath 24798   WalkOn cwlkon 24800   PathOn cpthon 24802

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6530  ax-cnex 9498  ax-resscn 9499  ax-1cn 9500  ax-icn 9501  ax-addcl 9502  ax-addrcl 9503  ax-mulcl 9504  ax-mulrcl 9505  ax-mulcom 9506  ax-addass 9507  ax-mulass 9508  ax-distr 9509  ax-i2m1 9510  ax-1ne0 9511  ax-1rid 9512  ax-rnegex 9513  ax-rrecex 9514  ax-cnre 9515  ax-pre-lttri 9516  ax-pre-lttrn 9517  ax-pre-ltadd 9518  ax-pre-mulgt0 9519

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-int 4227  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-we 4783  df-ord 4824  df-on 4825  df-lim 4826  df-suc 4827  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  df-res 4954  df-ima 4955  df-iota 5489  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-riota 6196  df-ov 6237  df-oprab 6238  df-mpt2 6239  df-om 6639  df-1st 6738  df-2nd 6739  df-recs 6999  df-rdg 7033  df-1o 7087  df-oadd 7091  df-er 7268  df-map 7379  df-pm 7380  df-en 7475  df-dom 7476  df-sdom 7477  df-fin 7478  df-card 8272  df-cda 8500  df-pnf 9580  df-mnf 9581  df-xr 9582  df-ltxr 9583  df-le 9584  df-sub 9763  df-neg 9764  df-nn 10497  df-2 10555  df-n0 10757  df-z 10826  df-uz 11046  df-fz 11644  df-fzo 11768  df-hash 12360  df-word 12498  df-wlk 24806  df-trail 24807  df-pth 24808  df-pthon 24814

This theorem is referenced by:  ispthon  24876  pthonprop  24877