Theorem frisusgranb 30760

Description: In a friendship graph, the neighborhoods of two different vertices have exactly one vertex in common. (Contributed by Alexander van der Vekens, 19-Dec-2017.)

Assertion

Ref	Expression
frisusgranb	|- ( V FriendGrph E -> A. k e. V A. l e. ( V \ { k } ) E. x e. V ( ( <. V , E >. Neighbors k ) i^i ( <. V , E >. Neighbors l ) ) = { x } )

Distinct variable groups:   k, V, l, x   k, E, l, x

Proof of Theorem frisusgranb

Dummy variable n is distinct from all other variables.

Step	Hyp	Ref	Expression
1		frisusgrapr 30754	. 2 |- ( V FriendGrph E -> ( V USGrph E /\ A. k e. V A. l e. ( V \ { k } ) E! n e. V { { n , k } , { n , l } } C_ ran E ) )
2		ssrab2 3548	. . . . . . . . . . . 12 |- { n e. V | { { n , k } , { n , l } } C_ ran E } C_ V
3		sseq1 3488	. . . . . . . . . . . 12 |- ( { n e. V | { { n , k } , { n , l } } C_ ran E } = { x } -> ( { n e. V | { { n , k } , { n , l } } C_ ran E } C_ V <-> { x } C_ V ) )
4	2, 3	mpbii 211	. . . . . . . . . . 11 |- ( { n e. V | { { n , k } , { n , l } } C_ ran E } = { x } -> { x } C_ V )
5		vex 3081	. . . . . . . . . . . 12 |- x e. _V
6	5	snss 4110	. . . . . . . . . . 11 |- ( x e. V <-> { x } C_ V )
7	4, 6	sylibr 212	. . . . . . . . . 10 |- ( { n e. V | { { n , k } , { n , l } } C_ ran E } = { x } -> x e. V )
8	7	adantl 466	. . . . . . . . 9 |- ( ( ( ( V USGrph E /\ k e. V ) /\ l e. ( V \ { k } ) ) /\ { n e. V | { { n , k } , { n , l } } C_ ran E } = { x } ) -> x e. V )
9		nbusgra 23518	. . . . . . . . . . . . 13 |- ( V USGrph E -> ( <. V , E >. Neighbors k ) = { n e. V | { k , n } e. ran E } )
10		nbusgra 23518	. . . . . . . . . . . . 13 |- ( V USGrph E -> ( <. V , E >. Neighbors l ) = { n e. V | { l , n } e. ran E } )
11	9, 10	ineq12d 3664	. . . . . . . . . . . 12 |- ( V USGrph E -> ( ( <. V , E >. Neighbors k ) i^i ( <. V , E >. Neighbors l ) ) = ( { n e. V | { k , n } e. ran E } i^i { n e. V | { l , n } e. ran E } ) )
12	11	ad3antrrr 729	. . . . . . . . . . 11 |- ( ( ( ( V USGrph E /\ k e. V ) /\ l e. ( V \ { k } ) ) /\ { n e. V | { { n , k } , { n , l } } C_ ran E } = { x } ) -> ( ( <. V , E >. Neighbors k ) i^i ( <. V , E >. Neighbors l ) ) = ( { n e. V | { k , n } e. ran E } i^i { n e. V | { l , n } e. ran E } ) )
13		inrab 3733	. . . . . . . . . . 11 |- ( { n e. V | { k , n } e. ran E } i^i { n e. V | { l , n } e. ran E } ) = { n e. V | ( { k , n } e. ran E /\ { l , n } e. ran E ) }
14	12, 13	syl6eq 2511	. . . . . . . . . 10 |- ( ( ( ( V USGrph E /\ k e. V ) /\ l e. ( V \ { k } ) ) /\ { n e. V | { { n , k } , { n , l } } C_ ran E } = { x } ) -> ( ( <. V , E >. Neighbors k ) i^i ( <. V , E >. Neighbors l ) ) = { n e. V | ( { k , n } e. ran E /\ { l , n } e. ran E ) } )
15		prcom 4064	. . . . . . . . . . . . . . . 16 |- { k , n } = { n , k }
16	15	eleq1i 2531	. . . . . . . . . . . . . . 15 |- ( { k , n } e. ran E <-> { n , k } e. ran E )
17		prcom 4064	. . . . . . . . . . . . . . . 16 |- { l , n } = { n , l }
18	17	eleq1i 2531	. . . . . . . . . . . . . . 15 |- ( { l , n } e. ran E <-> { n , l } e. ran E )
19	16, 18	anbi12i 697	. . . . . . . . . . . . . 14 |- ( ( { k , n } e. ran E /\ { l , n } e. ran E ) <-> ( { n , k } e. ran E /\ { n , l } e. ran E ) )
20		zfpair2 4643	. . . . . . . . . . . . . . 15 |- { n , k } e. _V
21		zfpair2 4643	. . . . . . . . . . . . . . 15 |- { n , l } e. _V
22	20, 21	prss 4138	. . . . . . . . . . . . . 14 |- ( ( { n , k } e. ran E /\ { n , l } e. ran E ) <-> { { n , k } , { n , l } } C_ ran E )
23	19, 22	bitri 249	. . . . . . . . . . . . 13 |- ( ( { k , n } e. ran E /\ { l , n } e. ran E ) <-> { { n , k } , { n , l } } C_ ran E )
24	23	a1i 11	. . . . . . . . . . . 12 |- ( ( ( ( V USGrph E /\ k e. V ) /\ l e. ( V \ { k } ) ) /\ n e. V ) -> ( ( { k , n } e. ran E /\ { l , n } e. ran E ) <-> { { n , k } , { n , l } } C_ ran E ) )
25	24	rabbidva 3069	. . . . . . . . . . 11 |- ( ( ( V USGrph E /\ k e. V ) /\ l e. ( V \ { k } ) ) -> { n e. V | ( { k , n } e. ran E /\ { l , n } e. ran E ) } = { n e. V | { { n , k } , { n , l } } C_ ran E } )
26	25	adantr 465	. . . . . . . . . 10 |- ( ( ( ( V USGrph E /\ k e. V ) /\ l e. ( V \ { k } ) ) /\ { n e. V | { { n , k } , { n , l } } C_ ran E } = { x } ) -> { n e. V | ( { k , n } e. ran E /\ { l , n } e. ran E ) } = { n e. V | { { n , k } , { n , l } } C_ ran E } )
27		simpr 461	. . . . . . . . . 10 |- ( ( ( ( V USGrph E /\ k e. V ) /\ l e. ( V \ { k } ) ) /\ { n e. V | { { n , k } , { n , l } } C_ ran E } = { x } ) -> { n e. V | { { n , k } , { n , l } } C_ ran E } = { x } )
28	14, 26, 27	3eqtrd 2499	. . . . . . . . 9 |- ( ( ( ( V USGrph E /\ k e. V ) /\ l e. ( V \ { k } ) ) /\ { n e. V | { { n , k } , { n , l } } C_ ran E } = { x } ) -> ( ( <. V , E >. Neighbors k ) i^i ( <. V , E >. Neighbors l ) ) = { x } )
29	8, 28	jca 532	. . . . . . . 8 |- ( ( ( ( V USGrph E /\ k e. V ) /\ l e. ( V \ { k } ) ) /\ { n e. V | { { n , k } , { n , l } } C_ ran E } = { x } ) -> ( x e. V /\ ( ( <. V , E >. Neighbors k ) i^i ( <. V , E >. Neighbors l ) ) = { x } ) )
30	29	ex 434	. . . . . . 7 |- ( ( ( V USGrph E /\ k e. V ) /\ l e. ( V \ { k } ) ) -> ( { n e. V | { { n , k } , { n , l } } C_ ran E } = { x } -> ( x e. V /\ ( ( <. V , E >. Neighbors k ) i^i ( <. V , E >. Neighbors l ) ) = { x } ) ) )
31	30	eximdv 1677	. . . . . 6 |- ( ( ( V USGrph E /\ k e. V ) /\ l e. ( V \ { k } ) ) -> ( E. x { n e. V | { { n , k } , { n , l } } C_ ran E } = { x } -> E. x ( x e. V /\ ( ( <. V , E >. Neighbors k ) i^i ( <. V , E >. Neighbors l ) ) = { x } ) ) )
32		reusn 4059	. . . . . 6 |- ( E! n e. V { { n , k } , { n , l } } C_ ran E <-> E. x { n e. V | { { n , k } , { n , l } } C_ ran E } = { x } )
33		df-rex 2805	. . . . . 6 |- ( E. x e. V ( ( <. V , E >. Neighbors k ) i^i ( <. V , E >. Neighbors l ) ) = { x } <-> E. x ( x e. V /\ ( ( <. V , E >. Neighbors k ) i^i ( <. V , E >. Neighbors l ) ) = { x } ) )
34	31, 32, 33	3imtr4g 270	. . . . 5 |- ( ( ( V USGrph E /\ k e. V ) /\ l e. ( V \ { k } ) ) -> ( E! n e. V { { n , k } , { n , l } } C_ ran E -> E. x e. V ( ( <. V , E >. Neighbors k ) i^i ( <. V , E >. Neighbors l ) ) = { x } ) )
35	34	ralimdva 2832	. . . 4 |- ( ( V USGrph E /\ k e. V ) -> ( A. l e. ( V \ { k } ) E! n e. V { { n , k } , { n , l } } C_ ran E -> A. l e. ( V \ { k } ) E. x e. V ( ( <. V , E >. Neighbors k ) i^i ( <. V , E >. Neighbors l ) ) = { x } ) )
36	35	ralimdva 2832	. . 3 |- ( V USGrph E -> ( A. k e. V A. l e. ( V \ { k } ) E! n e. V { { n , k } , { n , l } } C_ ran E -> A. k e. V A. l e. ( V \ { k } ) E. x e. V ( ( <. V , E >. Neighbors k ) i^i ( <. V , E >. Neighbors l ) ) = { x } ) )
37	36	imp 429	. 2 |- ( ( V USGrph E /\ A. k e. V A. l e. ( V \ { k } ) E! n e. V { { n , k } , { n , l } } C_ ran E ) -> A. k e. V A. l e. ( V \ { k } ) E. x e. V ( ( <. V , E >. Neighbors k ) i^i ( <. V , E >. Neighbors l ) ) = { x } )
38	1, 37	syl 16	1 |- ( V FriendGrph E -> A. k e. V A. l e. ( V \ { k } ) E. x e. V ( ( <. V , E >. Neighbors k ) i^i ( <. V , E >. Neighbors l ) ) = { x } )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1370   E.wex 1587   e. wcel 1758   A.wral 2799   E.wrex 2800   E!wreu 2801   {crab 2803   \ cdif 3436   i^i cin 3438   C_ wss 3439   {csn 3988   {cpr 3990   <.cop 3994   class class class wbr 4403   ran crn 4952  (class class class)co 6203   USGrph cusg 23443   Neighbors cnbgra 23508   FriendGrph cfrgra 30751

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9453  ax-resscn 9454  ax-1cn 9455  ax-icn 9456  ax-addcl 9457  ax-addrcl 9458  ax-mulcl 9459  ax-mulrcl 9460  ax-mulcom 9461  ax-addass 9462  ax-mulass 9463  ax-distr 9464  ax-i2m1 9465  ax-1ne0 9466  ax-1rid 9467  ax-rnegex 9468  ax-rrecex 9469  ax-cnre 9470  ax-pre-lttri 9471  ax-pre-lttrn 9472  ax-pre-ltadd 9473  ax-pre-mulgt0 9474

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-recs 6945  df-rdg 6979  df-1o 7033  df-er 7214  df-en 7424  df-dom 7425  df-sdom 7426  df-fin 7427  df-card 8224  df-pnf 9535  df-mnf 9536  df-xr 9537  df-ltxr 9538  df-le 9539  df-sub 9712  df-neg 9713  df-nn 10438  df-2 10495  df-n0 10695  df-z 10762  df-uz 10977  df-fz 11559  df-hash 12225  df-usgra 23445  df-nbgra 23511  df-frgra 30752

This theorem is referenced by:  frgrancvvdeqlem4  30797