Theorem rrgmex 14337

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 4864	. . . 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 14334	. . . . 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 4815	. . . 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 2298	. . . 4 |- ( A e. E <-> A e. (RLReg ` R ) )
7	6	biimpi 120	. . 3 |- ( A e. E -> A e. (RLReg ` R ) )
8		relelfvdm 5680	. . 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 2817	1 |- ( A e. E -> R e. _V )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2202   A.wral 2511   {crab 2515   _Vcvv 2803   |-> cmpt 4155   dom cdm 4731   Rel wrel 4736   ` cfv 5333  (class class class)co 6028   Basecbs 13143   .rcmulr 13222   0gc0g 13400  RLRegcrlreg 14331

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-mpt 4157  df-xp 4737  df-rel 4738  df-dm 4741  df-iota 5293  df-fv 5341  df-rlreg 14334

This theorem is referenced by:  rrgval  14338