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Theorem usgredgedga 39293
Description: In a simple graph there is a 1-1 onto mapping between the indexed edges containing a fixed vertex and the set of edges containing this vertex. (Contributed by by AV, 18-Oct-2020.) (Proof shortened by AV, 11-Dec-2020.)
Hypotheses
Ref Expression
ushgredgedga.e  |-  E  =  (Edg `  G )
ushgredgedga.i  |-  I  =  (iEdg `  G )
ushgredgedga.v  |-  V  =  (Vtx `  G )
ushgredgedga.a  |-  A  =  { i  e.  dom  I  |  N  e.  ( I `  i
) }
ushgredgedga.b  |-  B  =  { e  e.  E  |  N  e.  e }
ushgredgedga.f  |-  F  =  ( x  e.  A  |->  ( I `  x
) )
Assertion
Ref Expression
usgredgedga  |-  ( ( G  e. USGraph  /\  N  e.  V )  ->  F : A -1-1-onto-> B )
Distinct variable groups:    B, e    e, E, i    e, G, i, x    e, I, i, x    e, N, i, x    e, V, i, x
Allowed substitution hints:    A( x, e, i)    B( x, i)    E( x)    F( x, e, i)

Proof of Theorem usgredgedga
StepHypRef Expression
1 usgruspgr 39251 . . 3  |-  ( G  e. USGraph  ->  G  e. USPGraph  )
2 uspgrushgr 39248 . . 3  |-  ( G  e. USPGraph  ->  G  e. USHGraph  )
31, 2syl 17 . 2  |-  ( G  e. USGraph  ->  G  e. USHGraph  )
4 ushgredgedga.e . . 3  |-  E  =  (Edg `  G )
5 ushgredgedga.i . . 3  |-  I  =  (iEdg `  G )
6 ushgredgedga.v . . 3  |-  V  =  (Vtx `  G )
7 ushgredgedga.a . . 3  |-  A  =  { i  e.  dom  I  |  N  e.  ( I `  i
) }
8 ushgredgedga.b . . 3  |-  B  =  { e  e.  E  |  N  e.  e }
9 ushgredgedga.f . . 3  |-  F  =  ( x  e.  A  |->  ( I `  x
) )
104, 5, 6, 7, 8, 9ushgredgedga 39292 . 2  |-  ( ( G  e. USHGraph  /\  N  e.  V )  ->  F : A -1-1-onto-> B )
113, 10sylan 474 1  |-  ( ( G  e. USGraph  /\  N  e.  V )  ->  F : A -1-1-onto-> B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371    = wceq 1443    e. wcel 1886   {crab 2740    |-> cmpt 4460   dom cdm 4833   -1-1-onto->wf1o 5580   ` cfv 5581  Vtxcvtx 39087  iEdgciedg 39088   USHGraph cushgr 39134  Edgcedga 39196   USPGraph cuspgr 39221   USGraph cusgr 39222
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-resscn 9593  ax-1cn 9594  ax-icn 9595  ax-addcl 9596  ax-addrcl 9597  ax-mulcl 9598  ax-mulrcl 9599  ax-i2m1 9604  ax-1ne0 9605  ax-rrecex 9608  ax-cnre 9609  ax-pre-lttri 9610  ax-pre-lttrn 9611
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-nel 2624  df-ral 2741  df-rex 2742  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-op 3974  df-uni 4198  df-br 4402  df-opab 4461  df-mpt 4462  df-id 4748  df-po 4754  df-so 4755  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-ov 6291  df-er 7360  df-en 7567  df-dom 7568  df-sdom 7569  df-pnf 9674  df-mnf 9675  df-xr 9676  df-ltxr 9677  df-le 9678  df-2 10665  df-uhgr 39135  df-ushgr 39136  df-edga 39197  df-uspgr 39223  df-usgr 39224
This theorem is referenced by:  usgredgaleordALT  39297
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