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Theorem snstriedgval 39138
Description: The set of indexed edges of a graph without edges represented as an extensible structure with vertices as base set and no indexed edges. See iedgvalsnop 39142 for the (degenerated) case where  V  =  (
Base `  ndx ). (Contributed by AV, 24-Sep-2020.)
Hypotheses
Ref Expression
snstrvtxval.v  |-  V  e.  X
snstrvtxval.g  |-  G  =  { <. ( Base `  ndx ) ,  V >. }
Assertion
Ref Expression
snstriedgval  |-  ( V  =/=  ( Base `  ndx )  ->  (iEdg `  G
)  =  (/) )

Proof of Theorem snstriedgval
StepHypRef Expression
1 snstrvtxval.g . . . 4  |-  G  =  { <. ( Base `  ndx ) ,  V >. }
2 snex 4641 . . . 4  |-  { <. (
Base `  ndx ) ,  V >. }  e.  _V
31, 2eqeltri 2525 . . 3  |-  G  e. 
_V
4 iedgval 39106 . . 3  |-  ( G  e.  _V  ->  (iEdg `  G )  =  if ( G  e.  ( _V  X.  _V ) ,  ( 2nd `  G
) ,  (.ef `  G ) ) )
53, 4mp1i 13 . 2  |-  ( V  =/=  ( Base `  ndx )  ->  (iEdg `  G
)  =  if ( G  e.  ( _V 
X.  _V ) ,  ( 2nd `  G ) ,  (.ef `  G
) ) )
6 necom 2677 . . . 4  |-  ( V  =/=  ( Base `  ndx ) 
<->  ( Base `  ndx )  =/=  V )
7 fvex 5875 . . . . 5  |-  ( Base `  ndx )  e.  _V
8 snstrvtxval.v . . . . 5  |-  V  e.  X
97, 8, 1funsndifnop 39022 . . . 4  |-  ( (
Base `  ndx )  =/= 
V  ->  -.  G  e.  ( _V  X.  _V ) )
106, 9sylbi 199 . . 3  |-  ( V  =/=  ( Base `  ndx )  ->  -.  G  e.  ( _V  X.  _V )
)
1110iffalsed 3892 . 2  |-  ( V  =/=  ( Base `  ndx )  ->  if ( G  e.  ( _V  X.  _V ) ,  ( 2nd `  G ) ,  (.ef
`  G ) )  =  (.ef `  G
) )
122a1i 11 . . . . 5  |-  ( G  =  { <. ( Base `  ndx ) ,  V >. }  ->  { <. (
Base `  ndx ) ,  V >. }  e.  _V )
131, 12syl5eqel 2533 . . . 4  |-  ( G  =  { <. ( Base `  ndx ) ,  V >. }  ->  G  e.  _V )
14 edgfndxid 39098 . . . 4  |-  ( G  e.  _V  ->  (.ef `  G )  =  ( G `  (.ef `  ndx ) ) )
151, 13, 14mp2b 10 . . 3  |-  (.ef `  G )  =  ( G `  (.ef `  ndx ) )
16 slotsbaseefdif 39100 . . . . . . . 8  |-  ( Base `  ndx )  =/=  (.ef ` 
ndx )
1716nesymi 2681 . . . . . . 7  |-  -.  (.ef ` 
ndx )  =  (
Base `  ndx )
1817a1i 11 . . . . . 6  |-  ( V  =/=  ( Base `  ndx )  ->  -.  (.ef `  ndx )  =  ( Base ` 
ndx ) )
19 fvex 5875 . . . . . . 7  |-  (.ef `  ndx )  e.  _V
2019elsnc 3992 . . . . . 6  |-  ( (.ef
`  ndx )  e.  {
( Base `  ndx ) }  <-> 
(.ef `  ndx )  =  ( Base `  ndx ) )
2118, 20sylnibr 307 . . . . 5  |-  ( V  =/=  ( Base `  ndx )  ->  -.  (.ef `  ndx )  e.  { ( Base `  ndx ) } )
221dmeqi 5036 . . . . . 6  |-  dom  G  =  dom  { <. ( Base `  ndx ) ,  V >. }
23 dmsnopg 5307 . . . . . . 7  |-  ( V  e.  X  ->  dom  {
<. ( Base `  ndx ) ,  V >. }  =  { ( Base `  ndx ) } )
248, 23mp1i 13 . . . . . 6  |-  ( V  =/=  ( Base `  ndx )  ->  dom  { <. ( Base `  ndx ) ,  V >. }  =  {
( Base `  ndx ) } )
2522, 24syl5eq 2497 . . . . 5  |-  ( V  =/=  ( Base `  ndx )  ->  dom  G  =  { ( Base `  ndx ) } )
2621, 25neleqtrrd 2551 . . . 4  |-  ( V  =/=  ( Base `  ndx )  ->  -.  (.ef `  ndx )  e.  dom  G )
27 ndmfv 5889 . . . 4  |-  ( -.  (.ef `  ndx )  e. 
dom  G  ->  ( G `
 (.ef `  ndx ) )  =  (/) )
2826, 27syl 17 . . 3  |-  ( V  =/=  ( Base `  ndx )  ->  ( G `  (.ef `  ndx ) )  =  (/) )
2915, 28syl5eq 2497 . 2  |-  ( V  =/=  ( Base `  ndx )  ->  (.ef `  G
)  =  (/) )
305, 11, 293eqtrd 2489 1  |-  ( V  =/=  ( Base `  ndx )  ->  (iEdg `  G
)  =  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1444    e. wcel 1887    =/= wne 2622   _Vcvv 3045   (/)c0 3731   ifcif 3881   {csn 3968   <.cop 3974    X. cxp 4832   dom cdm 4834   ` cfv 5582   2ndc2nd 6792   ndxcnx 15118   Basecbs 15121  .efcedgf 39095  iEdgciedg 39102
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  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  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-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  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-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  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-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-er 7363  df-en 7570  df-dom 7571  df-sdom 7572  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-ndx 15124  df-slot 15125  df-base 15126  df-edgf 39096  df-iedg 39104
This theorem is referenced by: (None)
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