Theorem nbuhgr2vtx1edgblem 39419

Description: Lemma for nbuhgr2vtx1edgb 39420. This reverse direction of nbgr2vtx1edg 39418 only holds for classes whose edges are subsets of the set of vertices (hypergraphs!) (Contributed by AV, 2-Nov-2020.)

Hypotheses

Ref	Expression
nbgr2vtx1edg.v	|- V = (Vtx ` G )
nbgr2vtx1edg.e	|- E = (Edg ` G )

Assertion

Ref	Expression
nbuhgr2vtx1edgblem	|- ( ( G e. UHGraph /\ V = { a , b } /\ a e. ( G NeighbVtx b ) ) -> { a , b } e. E )

Distinct variable groups:   E, a, b   G, a, b   V, a, b

Proof of Theorem nbuhgr2vtx1edgblem

Dummy variable e is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nbgr2vtx1edg.v	. . . . 5 |- V = (Vtx ` G )
2		nbgr2vtx1edg.e	. . . . 5 |- E = (Edg ` G )
3	1, 2	nbgrel 39410	. . . 4 |- ( G e. UHGraph -> ( a e. ( G NeighbVtx b ) <-> ( ( a e. V /\ b e. V ) /\ a =/= b /\ E. e e. E { b , a } C_ e ) ) )
4	3	adantr 467	. . 3 |- ( ( G e. UHGraph /\ V = { a , b } ) -> ( a e. ( G NeighbVtx b ) <-> ( ( a e. V /\ b e. V ) /\ a =/= b /\ E. e e. E { b , a } C_ e ) ) )
5	2	eleq2i 2521	. . . . . . . . . 10 |- ( e e. E <-> e e. (Edg ` G ) )
6		edguhgr 39221	. . . . . . . . . 10 |- ( ( G e. UHGraph /\ e e. (Edg ` G ) ) -> e e. ~P (Vtx ` G ) )
7	5, 6	sylan2b 478	. . . . . . . . 9 |- ( ( G e. UHGraph /\ e e. E ) -> e e. ~P (Vtx ` G ) )
8	1	eqeq1i 2456	. . . . . . . . . . . . 13 |- ( V = { a , b } <-> (Vtx ` G ) = { a , b } )
9		pweq 3954	. . . . . . . . . . . . . . 15 |- ( (Vtx ` G ) = { a , b } -> ~P (Vtx ` G ) = ~P { a , b } )
10	9	eleq2d 2514	. . . . . . . . . . . . . 14 |- ( (Vtx ` G ) = { a , b } -> ( e e. ~P (Vtx ` G ) <-> e e. ~P { a , b } ) )
11		selpw 3958	. . . . . . . . . . . . . 14 |- ( e e. ~P { a , b } <-> e C_ { a , b } )
12	10, 11	syl6bb 265	. . . . . . . . . . . . 13 |- ( (Vtx ` G ) = { a , b } -> ( e e. ~P (Vtx ` G ) <-> e C_ { a , b } ) )
13	8, 12	sylbi 199	. . . . . . . . . . . 12 |- ( V = { a , b } -> ( e e. ~P (Vtx ` G ) <-> e C_ { a , b } ) )
14	13	adantl 468	. . . . . . . . . . 11 |- ( ( ( G e. UHGraph /\ e e. E ) /\ V = { a , b } ) -> ( e e. ~P (Vtx ` G ) <-> e C_ { a , b } ) )
15		prcom 4050	. . . . . . . . . . . . . . . 16 |- { b , a } = { a , b }
16	15	sseq1i 3456	. . . . . . . . . . . . . . 15 |- ( { b , a } C_ e <-> { a , b } C_ e )
17		eqss 3447	. . . . . . . . . . . . . . . . 17 |- ( { a , b } = e <-> ( { a , b } C_ e /\ e C_ { a , b } ) )
18		eleq1a 2524	. . . . . . . . . . . . . . . . . . 19 |- ( e e. E -> ( { a , b } = e -> { a , b } e. E ) )
19	18	a1i 11	. . . . . . . . . . . . . . . . . 18 |- ( ( ( a e. V /\ b e. V ) /\ a =/= b ) -> ( e e. E -> ( { a , b } = e -> { a , b } e. E ) ) )
20	19	com13 83	. . . . . . . . . . . . . . . . 17 |- ( { a , b } = e -> ( e e. E -> ( ( ( a e. V /\ b e. V ) /\ a =/= b ) -> { a , b } e. E ) ) )
21	17, 20	sylbir 217	. . . . . . . . . . . . . . . 16 |- ( ( { a , b } C_ e /\ e C_ { a , b } ) -> ( e e. E -> ( ( ( a e. V /\ b e. V ) /\ a =/= b ) -> { a , b } e. E ) ) )
22	21	ex 436	. . . . . . . . . . . . . . 15 |- ( { a , b } C_ e -> ( e C_ { a , b } -> ( e e. E -> ( ( ( a e. V /\ b e. V ) /\ a =/= b ) -> { a , b } e. E ) ) ) )
23	16, 22	sylbi 199	. . . . . . . . . . . . . 14 |- ( { b , a } C_ e -> ( e C_ { a , b } -> ( e e. E -> ( ( ( a e. V /\ b e. V ) /\ a =/= b ) -> { a , b } e. E ) ) ) )
24	23	com13 83	. . . . . . . . . . . . 13 |- ( e e. E -> ( e C_ { a , b } -> ( { b , a } C_ e -> ( ( ( a e. V /\ b e. V ) /\ a =/= b ) -> { a , b } e. E ) ) ) )
25	24	adantl 468	. . . . . . . . . . . 12 |- ( ( G e. UHGraph /\ e e. E ) -> ( e C_ { a , b } -> ( { b , a } C_ e -> ( ( ( a e. V /\ b e. V ) /\ a =/= b ) -> { a , b } e. E ) ) ) )
26	25	adantr 467	. . . . . . . . . . 11 |- ( ( ( G e. UHGraph /\ e e. E ) /\ V = { a , b } ) -> ( e C_ { a , b } -> ( { b , a } C_ e -> ( ( ( a e. V /\ b e. V ) /\ a =/= b ) -> { a , b } e. E ) ) ) )
27	14, 26	sylbid 219	. . . . . . . . . 10 |- ( ( ( G e. UHGraph /\ e e. E ) /\ V = { a , b } ) -> ( e e. ~P (Vtx ` G ) -> ( { b , a } C_ e -> ( ( ( a e. V /\ b e. V ) /\ a =/= b ) -> { a , b } e. E ) ) ) )
28	27	ex 436	. . . . . . . . 9 |- ( ( G e. UHGraph /\ e e. E ) -> ( V = { a , b } -> ( e e. ~P (Vtx ` G ) -> ( { b , a } C_ e -> ( ( ( a e. V /\ b e. V ) /\ a =/= b ) -> { a , b } e. E ) ) ) ) )
29	7, 28	mpid 42	. . . . . . . 8 |- ( ( G e. UHGraph /\ e e. E ) -> ( V = { a , b } -> ( { b , a } C_ e -> ( ( ( a e. V /\ b e. V ) /\ a =/= b ) -> { a , b } e. E ) ) ) )
30	29	impancom 442	. . . . . . 7 |- ( ( G e. UHGraph /\ V = { a , b } ) -> ( e e. E -> ( { b , a } C_ e -> ( ( ( a e. V /\ b e. V ) /\ a =/= b ) -> { a , b } e. E ) ) ) )
31	30	com14 91	. . . . . 6 |- ( ( ( a e. V /\ b e. V ) /\ a =/= b ) -> ( e e. E -> ( { b , a } C_ e -> ( ( G e. UHGraph /\ V = { a , b } ) -> { a , b } e. E ) ) ) )
32	31	rexlimdv 2877	. . . . 5 |- ( ( ( a e. V /\ b e. V ) /\ a =/= b ) -> ( E. e e. E { b , a } C_ e -> ( ( G e. UHGraph /\ V = { a , b } ) -> { a , b } e. E ) ) )
33	32	3impia 1205	. . . 4 |- ( ( ( a e. V /\ b e. V ) /\ a =/= b /\ E. e e. E { b , a } C_ e ) -> ( ( G e. UHGraph /\ V = { a , b } ) -> { a , b } e. E ) )
34	33	com12 32	. . 3 |- ( ( G e. UHGraph /\ V = { a , b } ) -> ( ( ( a e. V /\ b e. V ) /\ a =/= b /\ E. e e. E { b , a } C_ e ) -> { a , b } e. E ) )
35	4, 34	sylbid 219	. 2 |- ( ( G e. UHGraph /\ V = { a , b } ) -> ( a e. ( G NeighbVtx b ) -> { a , b } e. E ) )
36	35	3impia 1205	1 |- ( ( G e. UHGraph /\ V = { a , b } /\ a e. ( G NeighbVtx b ) ) -> { a , b } e. E )

Syntax hints:   -> wi 4   <-> wb 188   /\ wa 371   /\ w3a 985   = wceq 1444   e. wcel 1887   =/= wne 2622   E.wrex 2738   C_ wss 3404   ~Pcpw 3951   {cpr 3970   ` cfv 5582  (class class class)co 6290  Vtxcvtx 39101   UHGraph cuhgr 39147  Edgcedga 39210   NeighbVtx cnbgr 39397

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-fal 1450  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-fv 5590  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-1st 6793  df-2nd 6794  df-uhgr 39149  df-edga 39211  df-nbgr 39401

This theorem is referenced by:  nbuhgr2vtx1edgb  39420