Theorem frisusgranb 25124

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 25118	. 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 3581	. . . . . . . . . . . 12 |- { n e. V | { { n , k } , { n , l } } C_ ran E } C_ V
3		sseq1 3520	. . . . . . . . . . . 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 3112	. . . . . . . . . . . 12 |- x e. _V
6	5	snss 4156	. . . . . . . . . . 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 24555	. . . . . . . . . . . . 13 |- ( V USGrph E -> ( <. V , E >. Neighbors k ) = { n e. V | { k , n } e. ran E } )
10		nbusgra 24555	. . . . . . . . . . . . 13 |- ( V USGrph E -> ( <. V , E >. Neighbors l ) = { n e. V | { l , n } e. ran E } )
11	9, 10	ineq12d 3697	. . . . . . . . . . . 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 3777	. . . . . . . . . . 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 2514	. . . . . . . . . 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 4110	. . . . . . . . . . . . . . . 16 |- { k , n } = { n , k }
16	15	eleq1i 2534	. . . . . . . . . . . . . . 15 |- ( { k , n } e. ran E <-> { n , k } e. ran E )
17		prcom 4110	. . . . . . . . . . . . . . . 16 |- { l , n } = { n , l }
18	17	eleq1i 2534	. . . . . . . . . . . . . . 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 4696	. . . . . . . . . . . . . . 15 |- { n , k } e. _V
21		zfpair2 4696	. . . . . . . . . . . . . . 15 |- { n , l } e. _V
22	20, 21	prss 4186	. . . . . . . . . . . . . 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 3100	. . . . . . . . . . 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 2502	. . . . . . . . 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 1711	. . . . . 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 4105	. . . . . 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 2813	. . . . . 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 2865	. . . 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 2865	. . 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 1395   E.wex 1613   e. wcel 1819   A.wral 2807   E.wrex 2808   E!wreu 2809   {crab 2811   \ cdif 3468   i^i cin 3470   C_ wss 3471   {csn 4032   {cpr 4034   <.cop 4038   class class class wbr 4456   ran crn 5009  (class class class)co 6296   USGrph cusg 24457   Neighbors cnbgra 24544   FriendGrph cfrgra 25115

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-card 8337  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-n0 10817  df-z 10886  df-uz 11107  df-fz 11698  df-hash 12409  df-usgra 24460  df-nbgra 24547  df-frgra 25116

This theorem is referenced by:  frgrancvvdeqlem4  25160