Theorem rrgmex 14552

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 4906	. . . 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 14549	. . . . 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 4856	. . . 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 2305	. . . 4 |- ( A e. E <-> A e. (RLReg ` R ) )
7	6	biimpi 120	. . 3 |- ( A e. E -> A e. (RLReg ` R ) )
8		relelfvdm 5725	. . 3 |- ( ( Rel RLReg /\ A e. (RLReg ` R ) ) -> R e. dom RLReg )
9	4, 7, 8	sylancr 418	. 2 |- ( A e. E -> R e. dom RLReg )
10	9	elexd 2835	1 |- ( A e. E -> R e. _V )

Syntax hints:   -> wi 4   = wceq 1402   e. wcel 2209   A.wral 2528   {crab 2532   _Vcvv 2821   |-> cmpt 4190   dom cdm 4772   Rel wrel 4777   ` cfv 5375  (class class class)co 6079   Basecbs 13335   .rcmulr 13415   0gc0g 13593  RLRegcrlreg 14546

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-pow 4309  ax-pr 4344

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-br 4129  df-opab 4191  df-mpt 4192  df-xp 4778  df-rel 4779  df-dm 4782  df-iota 5335  df-fv 5383  df-rlreg 14549

This theorem is referenced by:  rrgval  14553