Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  structiedg0val Structured version   Visualization version   Unicode version

Theorem structiedg0val 39277
Description: The set of indexed edges of an extensible structure with a base set and another slot not being the slot for edge functions is empty. (Contributed by AV, 23-Sep-2020.)
Hypotheses
Ref Expression
structvtxvallem.s  |-  S  e.  NN
structvtxvallem.b  |-  ( Base `  ndx )  <  S
structvtxvallem.g  |-  G  =  { <. ( Base `  ndx ) ,  V >. , 
<. S ,  E >. }
Assertion
Ref Expression
structiedg0val  |-  ( ( V  e.  X  /\  E  e.  Y  /\  S  =/=  (.ef `  ndx ) )  ->  (iEdg `  G )  =  (/) )

Proof of Theorem structiedg0val
StepHypRef Expression
1 structvtxvallem.s . . . . . 6  |-  S  e.  NN
2 structvtxvallem.b . . . . . 6  |-  ( Base `  ndx )  <  S
3 structvtxvallem.g . . . . . 6  |-  G  =  { <. ( Base `  ndx ) ,  V >. , 
<. S ,  E >. }
41, 2, 3structvtxvallem 39275 . . . . 5  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( G  e.  _V  /\ 
Fun  G  /\  { (
Base `  ndx ) ,  S }  C_  dom  G ) )
543adant3 1050 . . . 4  |-  ( ( V  e.  X  /\  E  e.  Y  /\  S  =/=  (.ef `  ndx ) )  ->  ( G  e.  _V  /\  Fun  G  /\  { ( Base `  ndx ) ,  S }  C_  dom  G ) )
6 3simpa 1027 . . . . . 6  |-  ( ( G  e.  _V  /\  Fun  G  /\  { (
Base `  ndx ) ,  S }  C_  dom  G )  ->  ( G  e.  _V  /\  Fun  G
) )
76adantl 473 . . . . 5  |-  ( ( ( V  e.  X  /\  E  e.  Y  /\  S  =/=  (.ef ` 
ndx ) )  /\  ( G  e.  _V  /\ 
Fun  G  /\  { (
Base `  ndx ) ,  S }  C_  dom  G ) )  ->  ( G  e.  _V  /\  Fun  G ) )
8 basendxnn 15252 . . . . . . . 8  |-  ( Base `  ndx )  e.  NN
98nnrei 10640 . . . . . . 7  |-  ( Base `  ndx )  e.  RR
109, 2ltneii 9765 . . . . . 6  |-  ( Base `  ndx )  =/=  S
1110a1i 11 . . . . 5  |-  ( ( ( V  e.  X  /\  E  e.  Y  /\  S  =/=  (.ef ` 
ndx ) )  /\  ( G  e.  _V  /\ 
Fun  G  /\  { (
Base `  ndx ) ,  S }  C_  dom  G ) )  ->  ( Base `  ndx )  =/= 
S )
12 simpr3 1038 . . . . 5  |-  ( ( ( V  e.  X  /\  E  e.  Y  /\  S  =/=  (.ef ` 
ndx ) )  /\  ( G  e.  _V  /\ 
Fun  G  /\  { (
Base `  ndx ) ,  S }  C_  dom  G ) )  ->  { (
Base `  ndx ) ,  S }  C_  dom  G )
137, 11, 123jca 1210 . . . 4  |-  ( ( ( V  e.  X  /\  E  e.  Y  /\  S  =/=  (.ef ` 
ndx ) )  /\  ( G  e.  _V  /\ 
Fun  G  /\  { (
Base `  ndx ) ,  S }  C_  dom  G ) )  ->  (
( G  e.  _V  /\ 
Fun  G )  /\  ( Base `  ndx )  =/= 
S  /\  { ( Base `  ndx ) ,  S }  C_  dom  G ) )
145, 13mpdan 681 . . 3  |-  ( ( V  e.  X  /\  E  e.  Y  /\  S  =/=  (.ef `  ndx ) )  ->  (
( G  e.  _V  /\ 
Fun  G )  /\  ( Base `  ndx )  =/= 
S  /\  { ( Base `  ndx ) ,  S }  C_  dom  G ) )
15 fvex 5889 . . . 4  |-  ( Base `  ndx )  e.  _V
161elexi 3041 . . . 4  |-  S  e. 
_V
1715, 16funiedgdm2val 39267 . . 3  |-  ( ( ( G  e.  _V  /\ 
Fun  G )  /\  ( Base `  ndx )  =/= 
S  /\  { ( Base `  ndx ) ,  S }  C_  dom  G )  ->  (iEdg `  G
)  =  (.ef `  G ) )
1814, 17syl 17 . 2  |-  ( ( V  e.  X  /\  E  e.  Y  /\  S  =/=  (.ef `  ndx ) )  ->  (iEdg `  G )  =  (.ef
`  G ) )
19 prex 4642 . . . . . 6  |-  { <. (
Base `  ndx ) ,  V >. ,  <. S ,  E >. }  e.  _V
2019a1i 11 . . . . 5  |-  ( G  =  { <. ( Base `  ndx ) ,  V >. ,  <. S ,  E >. }  ->  { <. (
Base `  ndx ) ,  V >. ,  <. S ,  E >. }  e.  _V )
213, 20syl5eqel 2553 . . . 4  |-  ( G  =  { <. ( Base `  ndx ) ,  V >. ,  <. S ,  E >. }  ->  G  e.  _V )
22 edgfndxid 39248 . . . 4  |-  ( G  e.  _V  ->  (.ef `  G )  =  ( G `  (.ef `  ndx ) ) )
233, 21, 22mp2b 10 . . 3  |-  (.ef `  G )  =  ( G `  (.ef `  ndx ) )
24 slotsbaseefdif 39250 . . . . . . . . 9  |-  ( Base `  ndx )  =/=  (.ef ` 
ndx )
2524nesymi 2700 . . . . . . . 8  |-  -.  (.ef ` 
ndx )  =  (
Base `  ndx )
2625a1i 11 . . . . . . 7  |-  ( ( V  e.  X  /\  E  e.  Y  /\  S  =/=  (.ef `  ndx ) )  ->  -.  (.ef `  ndx )  =  ( Base `  ndx ) )
27 neneq 2649 . . . . . . . . 9  |-  ( S  =/=  (.ef `  ndx )  ->  -.  S  =  (.ef `  ndx ) )
28 eqcom 2478 . . . . . . . . 9  |-  ( (.ef
`  ndx )  =  S  <-> 
S  =  (.ef `  ndx ) )
2927, 28sylnibr 312 . . . . . . . 8  |-  ( S  =/=  (.ef `  ndx )  ->  -.  (.ef `  ndx )  =  S )
30293ad2ant3 1053 . . . . . . 7  |-  ( ( V  e.  X  /\  E  e.  Y  /\  S  =/=  (.ef `  ndx ) )  ->  -.  (.ef `  ndx )  =  S )
31 ioran 498 . . . . . . 7  |-  ( -.  ( (.ef `  ndx )  =  ( Base ` 
ndx )  \/  (.ef ` 
ndx )  =  S )  <->  ( -.  (.ef ` 
ndx )  =  (
Base `  ndx )  /\  -.  (.ef `  ndx )  =  S ) )
3226, 30, 31sylanbrc 677 . . . . . 6  |-  ( ( V  e.  X  /\  E  e.  Y  /\  S  =/=  (.ef `  ndx ) )  ->  -.  ( (.ef `  ndx )  =  ( Base `  ndx )  \/  (.ef `  ndx )  =  S )
)
33 fvex 5889 . . . . . . 7  |-  (.ef `  ndx )  e.  _V
3433elpr 3977 . . . . . 6  |-  ( (.ef
`  ndx )  e.  {
( Base `  ndx ) ,  S }  <->  ( (.ef ` 
ndx )  =  (
Base `  ndx )  \/  (.ef `  ndx )  =  S ) )
3532, 34sylnibr 312 . . . . 5  |-  ( ( V  e.  X  /\  E  e.  Y  /\  S  =/=  (.ef `  ndx ) )  ->  -.  (.ef `  ndx )  e. 
{ ( Base `  ndx ) ,  S }
)
363dmeqi 5041 . . . . . 6  |-  dom  G  =  dom  { <. ( Base `  ndx ) ,  V >. ,  <. S ,  E >. }
37 dmpropg 5316 . . . . . . 7  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  dom  { <. ( Base `  ndx ) ,  V >. ,  <. S ,  E >. }  =  {
( Base `  ndx ) ,  S } )
38373adant3 1050 . . . . . 6  |-  ( ( V  e.  X  /\  E  e.  Y  /\  S  =/=  (.ef `  ndx ) )  ->  dom  {
<. ( Base `  ndx ) ,  V >. , 
<. S ,  E >. }  =  { ( Base `  ndx ) ,  S } )
3936, 38syl5eq 2517 . . . . 5  |-  ( ( V  e.  X  /\  E  e.  Y  /\  S  =/=  (.ef `  ndx ) )  ->  dom  G  =  { ( Base `  ndx ) ,  S } )
4035, 39neleqtrrd 2571 . . . 4  |-  ( ( V  e.  X  /\  E  e.  Y  /\  S  =/=  (.ef `  ndx ) )  ->  -.  (.ef `  ndx )  e. 
dom  G )
41 ndmfv 5903 . . . 4  |-  ( -.  (.ef `  ndx )  e. 
dom  G  ->  ( G `
 (.ef `  ndx ) )  =  (/) )
4240, 41syl 17 . . 3  |-  ( ( V  e.  X  /\  E  e.  Y  /\  S  =/=  (.ef `  ndx ) )  ->  ( G `  (.ef `  ndx ) )  =  (/) )
4323, 42syl5eq 2517 . 2  |-  ( ( V  e.  X  /\  E  e.  Y  /\  S  =/=  (.ef `  ndx ) )  ->  (.ef `  G )  =  (/) )
4418, 43eqtrd 2505 1  |-  ( ( V  e.  X  /\  E  e.  Y  /\  S  =/=  (.ef `  ndx ) )  ->  (iEdg `  G )  =  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 375    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904    =/= wne 2641   _Vcvv 3031    C_ wss 3390   (/)c0 3722   {cpr 3961   <.cop 3965   class class class wbr 4395   dom cdm 4839   Fun wfun 5583   ` cfv 5589    < clt 9693   NNcn 10631   ndxcnx 15196   Basecbs 15199  .efcedgf 39245  iEdgciedg 39252
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-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-3 10691  df-4 10692  df-5 10693  df-6 10694  df-7 10695  df-8 10696  df-9 10697  df-10 10698  df-n0 10894  df-z 10962  df-dec 11075  df-uz 11183  df-fz 11811  df-hash 12554  df-ndx 15202  df-slot 15203  df-base 15204  df-edgf 39246  df-iedg 39254
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator