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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.
StepHypRef Expression
1 nbgr2vtx1edg.v . . . . 5  |-  V  =  (Vtx `  G )
2 nbgr2vtx1edg.e . . . . 5  |-  E  =  (Edg `  G )
31, 2nbgrel 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
) ) )
43adantr 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 )
) )
52eleq2i 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 )
)
75, 6sylan2b 483 . . . . . . . . 9  |-  ( ( G  e. UHGraph  /\  e  e.  E )  ->  e  e.  ~P (Vtx `  G
) )
81eqeq1i 2476 . . . . . . . . . . . . 13  |-  ( V  =  { a ,  b }  <->  (Vtx `  G
)  =  { a ,  b } )
9 pweq 3945 . . . . . . . . . . . . . . 15  |-  ( (Vtx
`  G )  =  { a ,  b }  ->  ~P (Vtx `  G )  =  ~P { a ,  b } )
109eleq2d 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 } )
1210, 11syl6bb 269 . . . . . . . . . . . . 13  |-  ( (Vtx
`  G )  =  { a ,  b }  ->  ( e  e.  ~P (Vtx `  G
)  <->  e  C_  { a ,  b } ) )
138, 12sylbi 200 . . . . . . . . . . . 12  |-  ( V  =  { a ,  b }  ->  (
e  e.  ~P (Vtx `  G )  <->  e  C_  { a ,  b } ) )
1413adantl 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 }
1615sseq1i 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 ) )
1918a1i 11 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( a  e.  V  /\  b  e.  V
)  /\  a  =/=  b )  ->  (
e  e.  E  -> 
( { a ,  b }  =  e  ->  { a ,  b }  e.  E
) ) )
2019com13 82 . . . . . . . . . . . . . . . . 17  |-  ( { a ,  b }  =  e  ->  (
e  e.  E  -> 
( ( ( a  e.  V  /\  b  e.  V )  /\  a  =/=  b )  ->  { a ,  b }  e.  E ) ) )
2117, 20sylbir 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 ) ) )
2221ex 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 ) ) ) )
2316, 22sylbi 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 ) ) ) )
2423com13 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 ) ) ) )
2524adantl 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 ) ) ) )
2625adantr 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 ) ) ) )
2714, 26sylbid 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 ) ) ) )
2827ex 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 ) ) ) ) )
297, 28mpid 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 ) ) ) )
3029impancom 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 ) ) ) )
3130com14 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 ) ) ) )
3231rexlimdv 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 ) ) )
33323impia 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 ) )
3433com12 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 ) )
354, 34sylbid 223 . 2  |-  ( ( G  e. UHGraph  /\  V  =  { a ,  b } )  ->  (
a  e.  ( G NeighbVtx  b )  ->  { a ,  b }  e.  E ) )
36353impia 1228 1  |-  ( ( G  e. UHGraph  /\  V  =  { a ,  b }  /\  a  e.  ( G NeighbVtx  b )
)  ->  { a ,  b }  e.  E )
Colors of variables: wff setvar class
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
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