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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.
StepHypRef Expression
1 nbgr2vtx1edg.v . . . . 5  |-  V  =  (Vtx `  G )
2 nbgr2vtx1edg.e . . . . 5  |-  E  =  (Edg `  G )
31, 2nbgrel 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
) ) )
43adantr 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 )
) )
52eleq2i 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 )
)
75, 6sylan2b 478 . . . . . . . . 9  |-  ( ( G  e. UHGraph  /\  e  e.  E )  ->  e  e.  ~P (Vtx `  G
) )
81eqeq1i 2456 . . . . . . . . . . . . 13  |-  ( V  =  { a ,  b }  <->  (Vtx `  G
)  =  { a ,  b } )
9 pweq 3954 . . . . . . . . . . . . . . 15  |-  ( (Vtx
`  G )  =  { a ,  b }  ->  ~P (Vtx `  G )  =  ~P { a ,  b } )
109eleq2d 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 } )
1210, 11syl6bb 265 . . . . . . . . . . . . 13  |-  ( (Vtx
`  G )  =  { a ,  b }  ->  ( e  e.  ~P (Vtx `  G
)  <->  e  C_  { a ,  b } ) )
138, 12sylbi 199 . . . . . . . . . . . 12  |-  ( V  =  { a ,  b }  ->  (
e  e.  ~P (Vtx `  G )  <->  e  C_  { a ,  b } ) )
1413adantl 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 }
1615sseq1i 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 ) )
1918a1i 11 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( a  e.  V  /\  b  e.  V
)  /\  a  =/=  b )  ->  (
e  e.  E  -> 
( { a ,  b }  =  e  ->  { a ,  b }  e.  E
) ) )
2019com13 83 . . . . . . . . . . . . . . . . 17  |-  ( { a ,  b }  =  e  ->  (
e  e.  E  -> 
( ( ( a  e.  V  /\  b  e.  V )  /\  a  =/=  b )  ->  { a ,  b }  e.  E ) ) )
2117, 20sylbir 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 ) ) )
2221ex 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 ) ) ) )
2316, 22sylbi 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 ) ) ) )
2423com13 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 ) ) ) )
2524adantl 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 ) ) ) )
2625adantr 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 ) ) ) )
2714, 26sylbid 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 ) ) ) )
2827ex 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 ) ) ) ) )
297, 28mpid 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 ) ) ) )
3029impancom 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 ) ) ) )
3130com14 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 ) ) ) )
3231rexlimdv 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 ) ) )
33323impia 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 ) )
3433com12 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 ) )
354, 34sylbid 219 . 2  |-  ( ( G  e. UHGraph  /\  V  =  { a ,  b } )  ->  (
a  e.  ( G NeighbVtx  b )  ->  { a ,  b }  e.  E ) )
36353impia 1205 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 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
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