Theorem wspthneq1eq2 26745

Description: Two simple paths with identical sequences of vertices start and end at the same vertices. (Contributed by AV, 14-May-2021.)

Assertion

Ref	Expression
wspthneq1eq2	|- ( ( P e. ( A ( N WSPathsNOn G ) B ) /\ P e. ( C ( N WSPathsNOn G ) D ) ) -> ( A = C /\ B = D ) )

Proof of Theorem wspthneq1eq2

Dummy variables f h are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2622	. . 3 |- (Vtx ` G ) = (Vtx ` G )
2	1	wspthnonp 26744	. 2 |- ( P e. ( A ( N WSPathsNOn G ) B ) -> ( ( N e. NN0 /\ G e. _V ) /\ ( A e. (Vtx ` G ) /\ B e. (Vtx ` G ) ) /\ ( P e. ( A ( N WWalksNOn G ) B ) /\ E. f f ( A (SPathsOn ` G ) B ) P ) ) )
3	1	wspthnonp 26744	. 2 |- ( P e. ( C ( N WSPathsNOn G ) D ) -> ( ( N e. NN0 /\ G e. _V ) /\ ( C e. (Vtx ` G ) /\ D e. (Vtx ` G ) ) /\ ( P e. ( C ( N WWalksNOn G ) D ) /\ E. h h ( C (SPathsOn ` G ) D ) P ) ) )
4		simp3r 1090	. . 3 |- ( ( ( N e. NN0 /\ G e. _V ) /\ ( A e. (Vtx ` G ) /\ B e. (Vtx ` G ) ) /\ ( P e. ( A ( N WWalksNOn G ) B ) /\ E. f f ( A (SPathsOn ` G ) B ) P ) ) -> E. f f ( A (SPathsOn ` G ) B ) P )
5		simp3r 1090	. . 3 |- ( ( ( N e. NN0 /\ G e. _V ) /\ ( C e. (Vtx ` G ) /\ D e. (Vtx ` G ) ) /\ ( P e. ( C ( N WWalksNOn G ) D ) /\ E. h h ( C (SPathsOn ` G ) D ) P ) ) -> E. h h ( C (SPathsOn ` G ) D ) P )
6		spthonpthon 26647	. . . . . . . . . 10 |- ( f ( A (SPathsOn ` G ) B ) P -> f ( A (PathsOn ` G ) B ) P )
7		spthonpthon 26647	. . . . . . . . . 10 |- ( h ( C (SPathsOn ` G ) D ) P -> h ( C (PathsOn ` G ) D ) P )
8	6, 7	anim12i 590	. . . . . . . . 9 |- ( ( f ( A (SPathsOn ` G ) B ) P /\ h ( C (SPathsOn ` G ) D ) P ) -> ( f ( A (PathsOn ` G ) B ) P /\ h ( C (PathsOn ` G ) D ) P ) )
9		pthontrlon 26643	. . . . . . . . . 10 |- ( f ( A (PathsOn ` G ) B ) P -> f ( A (TrailsOn ` G ) B ) P )
10		pthontrlon 26643	. . . . . . . . . 10 |- ( h ( C (PathsOn ` G ) D ) P -> h ( C (TrailsOn ` G ) D ) P )
11		trlsonwlkon 26606	. . . . . . . . . . 11 |- ( f ( A (TrailsOn ` G ) B ) P -> f ( A (WalksOn ` G ) B ) P )
12		trlsonwlkon 26606	. . . . . . . . . . 11 |- ( h ( C (TrailsOn ` G ) D ) P -> h ( C (WalksOn ` G ) D ) P )
13	11, 12	anim12i 590	. . . . . . . . . 10 |- ( ( f ( A (TrailsOn ` G ) B ) P /\ h ( C (TrailsOn ` G ) D ) P ) -> ( f ( A (WalksOn ` G ) B ) P /\ h ( C (WalksOn ` G ) D ) P ) )
14	9, 10, 13	syl2an 494	. . . . . . . . 9 |- ( ( f ( A (PathsOn ` G ) B ) P /\ h ( C (PathsOn ` G ) D ) P ) -> ( f ( A (WalksOn ` G ) B ) P /\ h ( C (WalksOn ` G ) D ) P ) )
15		wlksoneq1eq2 26560	. . . . . . . . 9 |- ( ( f ( A (WalksOn ` G ) B ) P /\ h ( C (WalksOn ` G ) D ) P ) -> ( A = C /\ B = D ) )
16	8, 14, 15	3syl 18	. . . . . . . 8 |- ( ( f ( A (SPathsOn ` G ) B ) P /\ h ( C (SPathsOn ` G ) D ) P ) -> ( A = C /\ B = D ) )
17	16	expcom 451	. . . . . . 7 |- ( h ( C (SPathsOn ` G ) D ) P -> ( f ( A (SPathsOn ` G ) B ) P -> ( A = C /\ B = D ) ) )
18	17	exlimiv 1858	. . . . . 6 |- ( E. h h ( C (SPathsOn ` G ) D ) P -> ( f ( A (SPathsOn ` G ) B ) P -> ( A = C /\ B = D ) ) )
19	18	com12 32	. . . . 5 |- ( f ( A (SPathsOn ` G ) B ) P -> ( E. h h ( C (SPathsOn ` G ) D ) P -> ( A = C /\ B = D ) ) )
20	19	exlimiv 1858	. . . 4 |- ( E. f f ( A (SPathsOn ` G ) B ) P -> ( E. h h ( C (SPathsOn ` G ) D ) P -> ( A = C /\ B = D ) ) )
21	20	imp 445	. . 3 |- ( ( E. f f ( A (SPathsOn ` G ) B ) P /\ E. h h ( C (SPathsOn ` G ) D ) P ) -> ( A = C /\ B = D ) )
22	4, 5, 21	syl2an 494	. 2 |- ( ( ( ( N e. NN0 /\ G e. _V ) /\ ( A e. (Vtx ` G ) /\ B e. (Vtx ` G ) ) /\ ( P e. ( A ( N WWalksNOn G ) B ) /\ E. f f ( A (SPathsOn ` G ) B ) P ) ) /\ ( ( N e. NN0 /\ G e. _V ) /\ ( C e. (Vtx ` G ) /\ D e. (Vtx ` G ) ) /\ ( P e. ( C ( N WWalksNOn G ) D ) /\ E. h h ( C (SPathsOn ` G ) D ) P ) ) ) -> ( A = C /\ B = D ) )
23	2, 3, 22	syl2an 494	1 |- ( ( P e. ( A ( N WSPathsNOn G ) B ) /\ P e. ( C ( N WSPathsNOn G ) D ) ) -> ( A = C /\ B = D ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   _Vcvv 3200   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   NN0cn0 11292  Vtxcvtx 25874  WalksOncwlkson 26493  TrailsOnctrlson 26588  PathsOncpthson 26610  SPathsOncspthson 26611   WWalksNOn cwwlksnon 26719   WSPathsNOn cwwspthsnon 26721

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-fal 1489  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-oadd 7564  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  df-wlkson 26496  df-trls 26589  df-trlson 26590  df-pths 26612  df-spths 26613  df-pthson 26614  df-spthson 26615  df-wwlksnon 26724  df-wspthsnon 26726

This theorem is referenced by:  2wspdisj  26855  2wspiundisj  26856