Theorem nbuhgr2vtx1edgblem 26247

Description: Lemma for nbuhgr2vtx1edgb 26248. This reverse direction of nbgr2vtx1edg 26246 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 26238	. . . 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 481	. . 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 2693	. . . . . . . . . 10 |- ( e e. E <-> e e. (Edg ` G ) )
6		edguhgr 26024	. . . . . . . . . 10 |- ( ( G e. UHGraph /\ e e. (Edg ` G ) ) -> e e. ~P (Vtx ` G ) )
7	5, 6	sylan2b 492	. . . . . . . . 9 |- ( ( G e. UHGraph /\ e e. E ) -> e e. ~P (Vtx ` G ) )
8	1	eqeq1i 2627	. . . . . . . . . . . . 13 |- ( V = { a , b } <-> (Vtx ` G ) = { a , b } )
9		pweq 4161	. . . . . . . . . . . . . . 15 |- ( (Vtx ` G ) = { a , b } -> ~P (Vtx ` G ) = ~P { a , b } )
10	9	eleq2d 2687	. . . . . . . . . . . . . 14 |- ( (Vtx ` G ) = { a , b } -> ( e e. ~P (Vtx ` G ) <-> e e. ~P { a , b } ) )
11		selpw 4165	. . . . . . . . . . . . . 14 |- ( e e. ~P { a , b } <-> e C_ { a , b } )
12	10, 11	syl6bb 276	. . . . . . . . . . . . 13 |- ( (Vtx ` G ) = { a , b } -> ( e e. ~P (Vtx ` G ) <-> e C_ { a , b } ) )
13	8, 12	sylbi 207	. . . . . . . . . . . 12 |- ( V = { a , b } -> ( e e. ~P (Vtx ` G ) <-> e C_ { a , b } ) )
14	13	adantl 482	. . . . . . . . . . 11 |- ( ( ( G e. UHGraph /\ e e. E ) /\ V = { a , b } ) -> ( e e. ~P (Vtx ` G ) <-> e C_ { a , b } ) )
15		prcom 4267	. . . . . . . . . . . . . . . 16 |- { b , a } = { a , b }
16	15	sseq1i 3629	. . . . . . . . . . . . . . 15 |- ( { b , a } C_ e <-> { a , b } C_ e )
17		eqss 3618	. . . . . . . . . . . . . . . . 17 |- ( { a , b } = e <-> ( { a , b } C_ e /\ e C_ { a , b } ) )
18		eleq1a 2696	. . . . . . . . . . . . . . . . . . 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 88	. . . . . . . . . . . . . . . . 17 |- ( { a , b } = e -> ( e e. E -> ( ( ( a e. V /\ b e. V ) /\ a =/= b ) -> { a , b } e. E ) ) )
21	17, 20	sylbir 225	. . . . . . . . . . . . . . . 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 450	. . . . . . . . . . . . . . 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 207	. . . . . . . . . . . . . 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 88	. . . . . . . . . . . . 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 482	. . . . . . . . . . . 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 481	. . . . . . . . . . 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 230	. . . . . . . . . 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 450	. . . . . . . . 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 44	. . . . . . . 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 456	. . . . . . 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 96	. . . . . 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 3030	. . . . 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 1261	. . . 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 230	. 2 |- ( ( G e. UHGraph /\ V = { a , b } ) -> ( a e. ( G NeighbVtx b ) -> { a , b } e. E ) )
36	35	3impia 1261	1 |- ( ( G e. UHGraph /\ V = { a , b } /\ a e. ( G NeighbVtx b ) ) -> { a , b } e. E )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   C_ wss 3574   ~Pcpw 4158   {cpr 4179   ` cfv 5888  (class class class)co 6650  Vtxcvtx 25874  Edgcedg 25939   UHGraph cuhgr 25951   NeighbVtx cnbgr 26224

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-8 1992  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-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-fal 1489  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-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  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-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-edg 25940  df-uhgr 25953  df-nbgr 26228

This theorem is referenced by:  nbuhgr2vtx1edgb  26248