Theorem frisusgranb 25774

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 25768	. 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 3526	. . . . . . . . . . . 12 |- { n e. V | { { n , k } , { n , l } } C_ ran E } C_ V
3		sseq1 3465	. . . . . . . . . . . 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 216	. . . . . . . . . . 11 |- ( { n e. V | { { n , k } , { n , l } } C_ ran E } = { x } -> { x } C_ V )
5		vex 3060	. . . . . . . . . . . 12 |- x e. _V
6	5	snss 4109	. . . . . . . . . . 11 |- ( x e. V <-> { x } C_ V )
7	4, 6	sylibr 217	. . . . . . . . . 10 |- ( { n e. V | { { n , k } , { n , l } } C_ ran E } = { x } -> x e. V )
8	7	adantl 472	. . . . . . . . 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 25205	. . . . . . . . . . . . 13 |- ( V USGrph E -> ( <. V , E >. Neighbors k ) = { n e. V | { k , n } e. ran E } )
10		nbusgra 25205	. . . . . . . . . . . . 13 |- ( V USGrph E -> ( <. V , E >. Neighbors l ) = { n e. V | { l , n } e. ran E } )
11	9, 10	ineq12d 3647	. . . . . . . . . . . 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 741	. . . . . . . . . . 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 3727	. . . . . . . . . . 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 2512	. . . . . . . . . 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 4063	. . . . . . . . . . . . . . . 16 |- { k , n } = { n , k }
16	15	eleq1i 2531	. . . . . . . . . . . . . . 15 |- ( { k , n } e. ran E <-> { n , k } e. ran E )
17		prcom 4063	. . . . . . . . . . . . . . . 16 |- { l , n } = { n , l }
18	17	eleq1i 2531	. . . . . . . . . . . . . . 15 |- ( { l , n } e. ran E <-> { n , l } e. ran E )
19	16, 18	anbi12i 708	. . . . . . . . . . . . . 14 |- ( ( { k , n } e. ran E /\ { l , n } e. ran E ) <-> ( { n , k } e. ran E /\ { n , l } e. ran E ) )
20		zfpair2 4654	. . . . . . . . . . . . . . 15 |- { n , k } e. _V
21		zfpair2 4654	. . . . . . . . . . . . . . 15 |- { n , l } e. _V
22	20, 21	prss 4139	. . . . . . . . . . . . . 14 |- ( ( { n , k } e. ran E /\ { n , l } e. ran E ) <-> { { n , k } , { n , l } } C_ ran E )
23	19, 22	bitri 257	. . . . . . . . . . . . 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 3047	. . . . . . . . . . 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 471	. . . . . . . . . 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 467	. . . . . . . . . 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 2500	. . . . . . . . 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 539	. . . . . . . 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 440	. . . . . . 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 1775	. . . . . 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 4058	. . . . . 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 2755	. . . . . 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 278	. . . . 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 2808	. . . 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 2808	. . 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 435	. 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 17	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 189   /\ wa 375   = wceq 1455   E.wex 1674   e. wcel 1898   A.wral 2749   E.wrex 2750   E!wreu 2751   {crab 2753   \ cdif 3413   i^i cin 3415   C_ wss 3416   {csn 3980   {cpr 3982   <.cop 3986   class class class wbr 4416   ran crn 4854  (class class class)co 6315   USGrph cusg 25106   Neighbors cnbgra 25194   FriendGrph cfrgra 25765

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4539  ax-nul 4548  ax-pow 4595  ax-pr 4653  ax-un 6610  ax-cnex 9621  ax-resscn 9622  ax-1cn 9623  ax-icn 9624  ax-addcl 9625  ax-addrcl 9626  ax-mulcl 9627  ax-mulrcl 9628  ax-mulcom 9629  ax-addass 9630  ax-mulass 9631  ax-distr 9632  ax-i2m1 9633  ax-1ne0 9634  ax-1rid 9635  ax-rnegex 9636  ax-rrecex 9637  ax-cnre 9638  ax-pre-lttri 9639  ax-pre-lttrn 9640  ax-pre-ltadd 9641  ax-pre-mulgt0 9642

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-nel 2636  df-ral 2754  df-rex 2755  df-reu 2756  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4213  df-int 4249  df-iun 4294  df-br 4417  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6277  df-ov 6318  df-oprab 6319  df-mpt2 6320  df-om 6720  df-1st 6820  df-2nd 6821  df-wrecs 7054  df-recs 7116  df-rdg 7154  df-1o 7208  df-er 7389  df-en 7596  df-dom 7597  df-sdom 7598  df-fin 7599  df-card 8399  df-pnf 9703  df-mnf 9704  df-xr 9705  df-ltxr 9706  df-le 9707  df-sub 9888  df-neg 9889  df-nn 10638  df-2 10696  df-n0 10899  df-z 10967  df-uz 11189  df-fz 11814  df-hash 12548  df-usgra 25109  df-nbgra 25197  df-frgra 25766

This theorem is referenced by:  frgrancvvdeqlem4  25810