Theorem 2wspmdisj 27201

Description: The sets of paths of length 2 with a given vertex in the middle are distinct for different vertices in the middle. (Contributed by Alexander van der Vekens, 11-Mar-2018.) (Revised by AV, 18-May-2021.) (Proof shortened by AV, 10-Jan-2022.)

Hypotheses

Ref	Expression
frgrhash2wsp.v	|- V = (Vtx ` G )
fusgreg2wsp.m	|- M = ( a e. V |-> { w e. ( 2 WSPathsN G ) | ( w ` 1 ) = a } )

Assertion

Ref	Expression
2wspmdisj	|- Disj x e. V ( M ` x )

Distinct variable groups:   G, a   V, a   w, G, a, x   x, V, a, w   x, M   w, V

Allowed substitution hints:   M( w, a)

Proof of Theorem 2wspmdisj

Dummy variables y t are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		orc 400	. . . . 5 |- ( x = y -> ( x = y \/ ( ( M ` x ) i^i ( M ` y ) ) = (/) ) )
2	1	a1d 25	. . . 4 |- ( x = y -> ( ( x e. V /\ y e. V ) -> ( x = y \/ ( ( M ` x ) i^i ( M ` y ) ) = (/) ) ) )
3		frgrhash2wsp.v	. . . . . . . . . . . . . 14 |- V = (Vtx ` G )
4		fusgreg2wsp.m	. . . . . . . . . . . . . 14 |- M = ( a e. V |-> { w e. ( 2 WSPathsN G ) | ( w ` 1 ) = a } )
5	3, 4	fusgreg2wsplem 27197	. . . . . . . . . . . . 13 |- ( y e. V -> ( t e. ( M ` y ) <-> ( t e. ( 2 WSPathsN G ) /\ ( t ` 1 ) = y ) ) )
6	5	adantl 482	. . . . . . . . . . . 12 |- ( ( x e. V /\ y e. V ) -> ( t e. ( M ` y ) <-> ( t e. ( 2 WSPathsN G ) /\ ( t ` 1 ) = y ) ) )
7	6	adantr 481	. . . . . . . . . . 11 |- ( ( ( x e. V /\ y e. V ) /\ t e. ( M ` x ) ) -> ( t e. ( M ` y ) <-> ( t e. ( 2 WSPathsN G ) /\ ( t ` 1 ) = y ) ) )
8	3, 4	fusgreg2wsplem 27197	. . . . . . . . . . . . . 14 |- ( x e. V -> ( t e. ( M ` x ) <-> ( t e. ( 2 WSPathsN G ) /\ ( t ` 1 ) = x ) ) )
9		eqtr2 2642	. . . . . . . . . . . . . . . . . 18 |- ( ( ( t ` 1 ) = x /\ ( t ` 1 ) = y ) -> x = y )
10	9	expcom 451	. . . . . . . . . . . . . . . . 17 |- ( ( t ` 1 ) = y -> ( ( t ` 1 ) = x -> x = y ) )
11	10	adantl 482	. . . . . . . . . . . . . . . 16 |- ( ( t e. ( 2 WSPathsN G ) /\ ( t ` 1 ) = y ) -> ( ( t ` 1 ) = x -> x = y ) )
12	11	com12 32	. . . . . . . . . . . . . . 15 |- ( ( t ` 1 ) = x -> ( ( t e. ( 2 WSPathsN G ) /\ ( t ` 1 ) = y ) -> x = y ) )
13	12	adantl 482	. . . . . . . . . . . . . 14 |- ( ( t e. ( 2 WSPathsN G ) /\ ( t ` 1 ) = x ) -> ( ( t e. ( 2 WSPathsN G ) /\ ( t ` 1 ) = y ) -> x = y ) )
14	8, 13	syl6bi 243	. . . . . . . . . . . . 13 |- ( x e. V -> ( t e. ( M ` x ) -> ( ( t e. ( 2 WSPathsN G ) /\ ( t ` 1 ) = y ) -> x = y ) ) )
15	14	adantr 481	. . . . . . . . . . . 12 |- ( ( x e. V /\ y e. V ) -> ( t e. ( M ` x ) -> ( ( t e. ( 2 WSPathsN G ) /\ ( t ` 1 ) = y ) -> x = y ) ) )
16	15	imp 445	. . . . . . . . . . 11 |- ( ( ( x e. V /\ y e. V ) /\ t e. ( M ` x ) ) -> ( ( t e. ( 2 WSPathsN G ) /\ ( t ` 1 ) = y ) -> x = y ) )
17	7, 16	sylbid 230	. . . . . . . . . 10 |- ( ( ( x e. V /\ y e. V ) /\ t e. ( M ` x ) ) -> ( t e. ( M ` y ) -> x = y ) )
18	17	con3d 148	. . . . . . . . 9 |- ( ( ( x e. V /\ y e. V ) /\ t e. ( M ` x ) ) -> ( -. x = y -> -. t e. ( M ` y ) ) )
19	18	impancom 456	. . . . . . . 8 |- ( ( ( x e. V /\ y e. V ) /\ -. x = y ) -> ( t e. ( M ` x ) -> -. t e. ( M ` y ) ) )
20	19	ralrimiv 2965	. . . . . . 7 |- ( ( ( x e. V /\ y e. V ) /\ -. x = y ) -> A. t e. ( M ` x ) -. t e. ( M ` y ) )
21		disj 4017	. . . . . . 7 |- ( ( ( M ` x ) i^i ( M ` y ) ) = (/) <-> A. t e. ( M ` x ) -. t e. ( M ` y ) )
22	20, 21	sylibr 224	. . . . . 6 |- ( ( ( x e. V /\ y e. V ) /\ -. x = y ) -> ( ( M ` x ) i^i ( M ` y ) ) = (/) )
23	22	olcd 408	. . . . 5 |- ( ( ( x e. V /\ y e. V ) /\ -. x = y ) -> ( x = y \/ ( ( M ` x ) i^i ( M ` y ) ) = (/) ) )
24	23	expcom 451	. . . 4 |- ( -. x = y -> ( ( x e. V /\ y e. V ) -> ( x = y \/ ( ( M ` x ) i^i ( M ` y ) ) = (/) ) ) )
25	2, 24	pm2.61i 176	. . 3 |- ( ( x e. V /\ y e. V ) -> ( x = y \/ ( ( M ` x ) i^i ( M ` y ) ) = (/) ) )
26	25	rgen2a 2977	. 2 |- A. x e. V A. y e. V ( x = y \/ ( ( M ` x ) i^i ( M ` y ) ) = (/) )
27		fveq2 6191	. . 3 |- ( x = y -> ( M ` x ) = ( M ` y ) )
28	27	disjor 4634	. 2 |- (Disj x e. V ( M ` x ) <-> A. x e. V A. y e. V ( x = y \/ ( ( M ` x ) i^i ( M ` y ) ) = (/) ) )
29	26, 28	mpbir 221	1 |- Disj x e. V ( M ` x )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   i^i cin 3573   (/)c0 3915  Disj wdisj 4620   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   1c1 9937   2c2 11070  Vtxcvtx 25874   WSPathsN cwwspthsn 26720

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-ral 2917  df-rex 2918  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-disj 4621  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653

This theorem is referenced by:  fusgreghash2wsp  27202