Theorem nbuhgr2vtx1edgblem 39583

Description: Lemma for nbuhgr2vtx1edgb 39584. This reverse direction of nbgr2vtx1edg 39582 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 39574	. . . 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 472	. . 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 2541	. . . . . . . . . 10 |- ( e e. E <-> e e. (Edg ` G ) )
6		edguhgr 39382	. . . . . . . . . 10 |- ( ( G e. UHGraph /\ e e. (Edg ` G ) ) -> e e. ~P (Vtx ` G ) )
7	5, 6	sylan2b 483	. . . . . . . . 9 |- ( ( G e. UHGraph /\ e e. E ) -> e e. ~P (Vtx ` G ) )
8	1	eqeq1i 2476	. . . . . . . . . . . . 13 |- ( V = { a , b } <-> (Vtx ` G ) = { a , b } )
9		pweq 3945	. . . . . . . . . . . . . . 15 |- ( (Vtx ` G ) = { a , b } -> ~P (Vtx ` G ) = ~P { a , b } )
10	9	eleq2d 2534	. . . . . . . . . . . . . 14 |- ( (Vtx ` G ) = { a , b } -> ( e e. ~P (Vtx ` G ) <-> e e. ~P { a , b } ) )
11		selpw 3949	. . . . . . . . . . . . . 14 |- ( e e. ~P { a , b } <-> e C_ { a , b } )
12	10, 11	syl6bb 269	. . . . . . . . . . . . 13 |- ( (Vtx ` G ) = { a , b } -> ( e e. ~P (Vtx ` G ) <-> e C_ { a , b } ) )
13	8, 12	sylbi 200	. . . . . . . . . . . 12 |- ( V = { a , b } -> ( e e. ~P (Vtx ` G ) <-> e C_ { a , b } ) )
14	13	adantl 473	. . . . . . . . . . 11 |- ( ( ( G e. UHGraph /\ e e. E ) /\ V = { a , b } ) -> ( e e. ~P (Vtx ` G ) <-> e C_ { a , b } ) )
15		prcom 4041	. . . . . . . . . . . . . . . 16 |- { b , a } = { a , b }
16	15	sseq1i 3442	. . . . . . . . . . . . . . 15 |- ( { b , a } C_ e <-> { a , b } C_ e )
17		eqss 3433	. . . . . . . . . . . . . . . . 17 |- ( { a , b } = e <-> ( { a , b } C_ e /\ e C_ { a , b } ) )
18		eleq1a 2544	. . . . . . . . . . . . . . . . . . 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 82	. . . . . . . . . . . . . . . . 17 |- ( { a , b } = e -> ( e e. E -> ( ( ( a e. V /\ b e. V ) /\ a =/= b ) -> { a , b } e. E ) ) )
21	17, 20	sylbir 218	. . . . . . . . . . . . . . . 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 441	. . . . . . . . . . . . . . 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 200	. . . . . . . . . . . . . 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 82	. . . . . . . . . . . . 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 473	. . . . . . . . . . . 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 472	. . . . . . . . . . 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 223	. . . . . . . . . 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 441	. . . . . . . . 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 41	. . . . . . . 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 447	. . . . . . 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 90	. . . . . 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 2870	. . . . 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 1228	. . . 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 31	. . 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 223	. 2 |- ( ( G e. UHGraph /\ V = { a , b } ) -> ( a e. ( G NeighbVtx b ) -> { a , b } e. E ) )
36	35	3impia 1228	1 |- ( ( G e. UHGraph /\ V = { a , b } /\ a e. ( G NeighbVtx b ) ) -> { a , b } e. E )

Syntax hints:   -> wi 4   <-> wb 189   /\ wa 376   /\ w3a 1007   = wceq 1452   e. wcel 1904   =/= wne 2641   E.wrex 2757   C_ wss 3390   ~Pcpw 3942   {cpr 3961   ` cfv 5589  (class class class)co 6308  Vtxcvtx 39251   UHGraph cuhgr 39300  Edgcedga 39371   NeighbVtx cnbgr 39561

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-fv 5597  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-1st 6812  df-2nd 6813  df-uhgr 39302  df-edga 39372  df-nbgr 39565

This theorem is referenced by:  nbuhgr2vtx1edgb  39584