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Theorem rrgmex 13741
Description: A structure whose set of left-regular elements is inhabited is a set. (Contributed by Jim Kingdon, 12-Aug-2025.)
Hypothesis
Ref Expression
rrgmex.e  |-  E  =  (RLReg `  R )
Assertion
Ref Expression
rrgmex  |-  ( A  e.  E  ->  R  e.  _V )

Proof of Theorem rrgmex
Dummy variables  x  r  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mptrel 4784 . . . 4  |-  Rel  (
r  e.  _V  |->  { x  e.  ( Base `  r )  |  A. y  e.  ( Base `  r ) ( ( x ( .r `  r ) y )  =  ( 0g `  r )  ->  y  =  ( 0g `  r ) ) } )
2 df-rlreg 13738 . . . . 5  |- RLReg  =  ( r  e.  _V  |->  { x  e.  ( Base `  r )  |  A. y  e.  ( Base `  r ) ( ( x ( .r `  r ) y )  =  ( 0g `  r )  ->  y  =  ( 0g `  r ) ) } )
32releqi 4738 . . . 4  |-  ( Rel RLReg  <->  Rel  ( r  e.  _V  |->  { x  e.  ( Base `  r )  | 
A. y  e.  (
Base `  r )
( ( x ( .r `  r ) y )  =  ( 0g `  r )  ->  y  =  ( 0g `  r ) ) } ) )
41, 3mpbir 146 . . 3  |-  Rel RLReg
5 rrgmex.e . . . . 5  |-  E  =  (RLReg `  R )
65eleq2i 2260 . . . 4  |-  ( A  e.  E  <->  A  e.  (RLReg `  R ) )
76biimpi 120 . . 3  |-  ( A  e.  E  ->  A  e.  (RLReg `  R )
)
8 relelfvdm 5578 . . 3  |-  ( ( Rel RLReg  /\  A  e.  (RLReg `  R ) )  ->  R  e.  dom RLReg )
94, 7, 8sylancr 414 . 2  |-  ( A  e.  E  ->  R  e.  dom RLReg )
109elexd 2773 1  |-  ( A  e.  E  ->  R  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1364    e. wcel 2164   A.wral 2472   {crab 2476   _Vcvv 2760    |-> cmpt 4090   dom cdm 4655   Rel wrel 4660   ` cfv 5246  (class class class)co 5910   Basecbs 12608   .rcmulr 12686   0gc0g 12857  RLRegcrlreg 13735
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-mpt 4092  df-xp 4661  df-rel 4662  df-dm 4665  df-iota 5207  df-fv 5254  df-rlreg 13738
This theorem is referenced by:  rrgval  13742
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