Theorem wksonproplem 26601

Description: Lemma for theorems for properties of walks between two vertices, e.g. trlsonprop 26604. (Contributed by AV, 16-Jan-2021.)

Hypotheses

Ref	Expression
wksonproplem.v	|- V = (Vtx ` G )
wksonproplem.b	|- ( ( ( G e. _V /\ A e. V /\ B e. V ) /\ ( F e. _V /\ P e. _V ) ) -> ( F ( A ( W ` G ) B ) P <-> ( F ( A ( O ` G ) B ) P /\ F ( Q ` G ) P ) ) )
wksonproplem.d	|- W = ( g e. _V |-> ( a e. (Vtx ` g ) , b e. (Vtx ` g ) |-> { <. f , p >. | ( f ( a ( O ` g ) b ) p /\ f ( Q ` g ) p ) } ) )
wksonproplem.w	|- ( ( ( G e. _V /\ A e. V /\ B e. V ) /\ f ( Q ` G ) p ) -> f (Walks ` G ) p )

Assertion

Ref	Expression
wksonproplem	|- ( F ( A ( W ` G ) B ) P -> ( ( G e. _V /\ A e. V /\ B e. V ) /\ ( F e. _V /\ P e. _V ) /\ ( F ( A ( O ` G ) B ) P /\ F ( Q ` G ) P ) ) )

Distinct variable groups:   A, a, b, f, g, p   B, a, b, f, g, p   G, a, b, f, g, p   O, a, b, g   Q, a, b, g   V, a, b, f, g, p

Allowed substitution hints:   P( f, g, p, a, b)   Q( f, p)   F( f, g, p, a, b)   O( f, p)   W( f, g, p, a, b)

Proof of Theorem wksonproplem

Step	Hyp	Ref	Expression
1		wksonproplem.v	. . . . . 6 |- V = (Vtx ` G )
2		fvex 6201	. . . . . 6 |- (Vtx ` G ) e. _V
3	1, 2	eqeltri 2697	. . . . 5 |- V e. _V
4		wksonproplem.d	. . . . . 6 |- W = ( g e. _V |-> ( a e. (Vtx ` g ) , b e. (Vtx ` g ) |-> { <. f , p >. | ( f ( a ( O ` g ) b ) p /\ f ( Q ` g ) p ) } ) )
5		simp1 1061	. . . . . . 7 |- ( ( G e. _V /\ A e. V /\ B e. V ) -> G e. _V )
6		simp2 1062	. . . . . . . 8 |- ( ( G e. _V /\ A e. V /\ B e. V ) -> A e. V )
7	6, 1	syl6eleq 2711	. . . . . . 7 |- ( ( G e. _V /\ A e. V /\ B e. V ) -> A e. (Vtx ` G ) )
8		simp3 1063	. . . . . . . 8 |- ( ( G e. _V /\ A e. V /\ B e. V ) -> B e. V )
9	8, 1	syl6eleq 2711	. . . . . . 7 |- ( ( G e. _V /\ A e. V /\ B e. V ) -> B e. (Vtx ` G ) )
10		wksv 26515	. . . . . . . 8 |- { <. f , p >. | f (Walks ` G ) p } e. _V
11	10	a1i 11	. . . . . . 7 |- ( ( G e. _V /\ A e. V /\ B e. V ) -> { <. f , p >. | f (Walks ` G ) p } e. _V )
12		wksonproplem.w	. . . . . . 7 |- ( ( ( G e. _V /\ A e. V /\ B e. V ) /\ f ( Q ` G ) p ) -> f (Walks ` G ) p )
13	5, 7, 9, 11, 12, 4	mptmpt2opabovd 7249	. . . . . 6 |- ( ( G e. _V /\ A e. V /\ B e. V ) -> ( A ( W ` G ) B ) = { <. f , p >. | ( f ( A ( O ` G ) B ) p /\ f ( Q ` G ) p ) } )
14		fveq2 6191	. . . . . . 7 |- ( g = G -> (Vtx ` g ) = (Vtx ` G ) )
15	14, 1	syl6eqr 2674	. . . . . 6 |- ( g = G -> (Vtx ` g ) = V )
16		fveq2 6191	. . . . . . . . 9 |- ( g = G -> ( O ` g ) = ( O ` G ) )
17	16	oveqd 6667	. . . . . . . 8 |- ( g = G -> ( a ( O ` g ) b ) = ( a ( O ` G ) b ) )
18	17	breqd 4664	. . . . . . 7 |- ( g = G -> ( f ( a ( O ` g ) b ) p <-> f ( a ( O ` G ) b ) p ) )
19		fveq2 6191	. . . . . . . 8 |- ( g = G -> ( Q ` g ) = ( Q ` G ) )
20	19	breqd 4664	. . . . . . 7 |- ( g = G -> ( f ( Q ` g ) p <-> f ( Q ` G ) p ) )
21	18, 20	anbi12d 747	. . . . . 6 |- ( g = G -> ( ( f ( a ( O ` g ) b ) p /\ f ( Q ` g ) p ) <-> ( f ( a ( O ` G ) b ) p /\ f ( Q ` G ) p ) ) )
22	4, 13, 15, 15, 21	bropfvvvv 7257	. . . . 5 |- ( ( V e. _V /\ V e. _V ) -> ( F ( A ( W ` G ) B ) P -> ( G e. _V /\ ( A e. V /\ B e. V ) /\ ( F e. _V /\ P e. _V ) ) ) )
23	3, 3, 22	mp2an 708	. . . 4 |- ( F ( A ( W ` G ) B ) P -> ( G e. _V /\ ( A e. V /\ B e. V ) /\ ( F e. _V /\ P e. _V ) ) )
24		3anass 1042	. . . . . 6 |- ( ( G e. _V /\ A e. V /\ B e. V ) <-> ( G e. _V /\ ( A e. V /\ B e. V ) ) )
25	24	anbi1i 731	. . . . 5 |- ( ( ( G e. _V /\ A e. V /\ B e. V ) /\ ( F e. _V /\ P e. _V ) ) <-> ( ( G e. _V /\ ( A e. V /\ B e. V ) ) /\ ( F e. _V /\ P e. _V ) ) )
26		df-3an 1039	. . . . 5 |- ( ( G e. _V /\ ( A e. V /\ B e. V ) /\ ( F e. _V /\ P e. _V ) ) <-> ( ( G e. _V /\ ( A e. V /\ B e. V ) ) /\ ( F e. _V /\ P e. _V ) ) )
27	25, 26	bitr4i 267	. . . 4 |- ( ( ( G e. _V /\ A e. V /\ B e. V ) /\ ( F e. _V /\ P e. _V ) ) <-> ( G e. _V /\ ( A e. V /\ B e. V ) /\ ( F e. _V /\ P e. _V ) ) )
28	23, 27	sylibr 224	. . 3 |- ( F ( A ( W ` G ) B ) P -> ( ( G e. _V /\ A e. V /\ B e. V ) /\ ( F e. _V /\ P e. _V ) ) )
29		wksonproplem.b	. . . . 5 |- ( ( ( G e. _V /\ A e. V /\ B e. V ) /\ ( F e. _V /\ P e. _V ) ) -> ( F ( A ( W ` G ) B ) P <-> ( F ( A ( O ` G ) B ) P /\ F ( Q ` G ) P ) ) )
30	29	biimpd 219	. . . 4 |- ( ( ( G e. _V /\ A e. V /\ B e. V ) /\ ( F e. _V /\ P e. _V ) ) -> ( F ( A ( W ` G ) B ) P -> ( F ( A ( O ` G ) B ) P /\ F ( Q ` G ) P ) ) )
31	30	imdistani 726	. . 3 |- ( ( ( ( G e. _V /\ A e. V /\ B e. V ) /\ ( F e. _V /\ P e. _V ) ) /\ F ( A ( W ` G ) B ) P ) -> ( ( ( G e. _V /\ A e. V /\ B e. V ) /\ ( F e. _V /\ P e. _V ) ) /\ ( F ( A ( O ` G ) B ) P /\ F ( Q ` G ) P ) ) )
32	28, 31	mpancom 703	. 2 |- ( F ( A ( W ` G ) B ) P -> ( ( ( G e. _V /\ A e. V /\ B e. V ) /\ ( F e. _V /\ P e. _V ) ) /\ ( F ( A ( O ` G ) B ) P /\ F ( Q ` G ) P ) ) )
33		df-3an 1039	. 2 |- ( ( ( G e. _V /\ A e. V /\ B e. V ) /\ ( F e. _V /\ P e. _V ) /\ ( F ( A ( O ` G ) B ) P /\ F ( Q ` G ) P ) ) <-> ( ( ( G e. _V /\ A e. V /\ B e. V ) /\ ( F e. _V /\ P e. _V ) ) /\ ( F ( A ( O ` G ) B ) P /\ F ( Q ` G ) P ) ) )
34	32, 33	sylibr 224	1 |- ( F ( A ( W ` G ) B ) P -> ( ( G e. _V /\ A e. V /\ B e. V ) /\ ( F e. _V /\ P e. _V ) /\ ( F ( A ( O ` G ) B ) P /\ F ( Q ` G ) P ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ 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  Vtxcvtx 25874  Walkscwlks 26492

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

This theorem is referenced by:  trlsonprop  26604  pthsonprop  26640  spthonprop  26641