Theorem rrgmex 13757

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.

Step	Hyp	Ref	Expression
1		mptrel 4790	. . . 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 13754	. . . . 5 |- RLReg = ( r e. _V |-> { x e. ( Base ` r ) | A. y e. ( Base ` r ) ( ( x ( .r ` r ) y ) = ( 0g ` r ) -> y = ( 0g ` r ) ) } )
3	2	releqi 4742	. . . 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 ) ) } ) )
4	1, 3	mpbir 146	. . 3 |- Rel RLReg
5		rrgmex.e	. . . . 5 |- E = (RLReg ` R )
6	5	eleq2i 2260	. . . 4 |- ( A e. E <-> A e. (RLReg ` R ) )
7	6	biimpi 120	. . 3 |- ( A e. E -> A e. (RLReg ` R ) )
8		relelfvdm 5586	. . 3 |- ( ( Rel RLReg /\ A e. (RLReg ` R ) ) -> R e. dom RLReg )
9	4, 7, 8	sylancr 414	. 2 |- ( A e. E -> R e. dom RLReg )
10	9	elexd 2773	1 |- ( A e. E -> R e. _V )

Syntax hints:   -> wi 4   = wceq 1364   e. wcel 2164   A.wral 2472   {crab 2476   _Vcvv 2760   |-> cmpt 4090   dom cdm 4659   Rel wrel 4664   ` cfv 5254  (class class class)co 5918   Basecbs 12618   .rcmulr 12696   0gc0g 12867  RLRegcrlreg 13751

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 4665  df-rel 4666  df-dm 4669  df-iota 5215  df-fv 5262  df-rlreg 13754

This theorem is referenced by:  rrgval  13758