Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  usgrnbcnvfv Structured version   Visualization version   Unicode version

Theorem usgrnbcnvfv 39603
Description: Applying the edge function on the converse edge function applied on a pair of a vertex and one of its neighbors is this pair in a simple graph. (Contributed by Alexander van der Vekens, 18-Dec-2017.) (Revised by AV, 27-Oct-2020.)
Hypothesis
Ref Expression
usgrnbcnvfv.i  |-  I  =  (iEdg `  G )
Assertion
Ref Expression
usgrnbcnvfv  |-  ( ( G  e. USGraph  /\  N  e.  ( G NeighbVtx  K )
)  ->  ( I `  ( `' I `  { K ,  N }
) )  =  { K ,  N }
)

Proof of Theorem usgrnbcnvfv
StepHypRef Expression
1 usgrnbcnvfv.i . . . 4  |-  I  =  (iEdg `  G )
21usgrf1o 39419 . . 3  |-  ( G  e. USGraph  ->  I : dom  I
-1-1-onto-> ran  I )
32adantr 472 . 2  |-  ( ( G  e. USGraph  /\  N  e.  ( G NeighbVtx  K )
)  ->  I : dom  I -1-1-onto-> ran  I )
4 prcom 4041 . . 3  |-  { N ,  K }  =  { K ,  N }
5 eqid 2471 . . . . . 6  |-  (Edg `  G )  =  (Edg
`  G )
65nbusgreledg 39585 . . . . 5  |-  ( G  e. USGraph  ->  ( N  e.  ( G NeighbVtx  K )  <->  { N ,  K }  e.  (Edg `  G )
) )
7 edgaval 39373 . . . . . . 7  |-  ( G  e. USGraph  ->  (Edg `  G
)  =  ran  (iEdg `  G ) )
81eqcomi 2480 . . . . . . . 8  |-  (iEdg `  G )  =  I
98rneqi 5067 . . . . . . 7  |-  ran  (iEdg `  G )  =  ran  I
107, 9syl6eq 2521 . . . . . 6  |-  ( G  e. USGraph  ->  (Edg `  G
)  =  ran  I
)
1110eleq2d 2534 . . . . 5  |-  ( G  e. USGraph  ->  ( { N ,  K }  e.  (Edg
`  G )  <->  { N ,  K }  e.  ran  I ) )
126, 11bitrd 261 . . . 4  |-  ( G  e. USGraph  ->  ( N  e.  ( G NeighbVtx  K )  <->  { N ,  K }  e.  ran  I ) )
1312biimpa 492 . . 3  |-  ( ( G  e. USGraph  /\  N  e.  ( G NeighbVtx  K )
)  ->  { N ,  K }  e.  ran  I )
144, 13syl5eqelr 2554 . 2  |-  ( ( G  e. USGraph  /\  N  e.  ( G NeighbVtx  K )
)  ->  { K ,  N }  e.  ran  I )
15 f1ocnvfv2 6194 . 2  |-  ( ( I : dom  I -1-1-onto-> ran  I  /\  { K ,  N }  e.  ran  I )  ->  (
I `  ( `' I `  { K ,  N } ) )  =  { K ,  N } )
163, 14, 15syl2anc 673 1  |-  ( ( G  e. USGraph  /\  N  e.  ( G NeighbVtx  K )
)  ->  ( I `  ( `' I `  { K ,  N }
) )  =  { K ,  N }
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 376    = wceq 1452    e. wcel 1904   {cpr 3961   `'ccnv 4838   dom cdm 4839   ran crn 4840   -1-1-onto->wf1o 5588   ` cfv 5589  (class class class)co 6308  iEdgciedg 39252  Edgcedga 39371   USGraph cusgr 39397   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-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  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-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  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-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  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-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-2o 7201  df-oadd 7204  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-card 8391  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-2 10690  df-n0 10894  df-z 10962  df-uz 11183  df-fz 11811  df-hash 12554  df-upgr 39328  df-umgr 39329  df-edga 39372  df-usgr 39399  df-nbgr 39565
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator