Theorem rrgmex 14073

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 4811	. . . 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 14070	. . . . 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 4763	. . . 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 2273	. . . 4 |- ( A e. E <-> A e. (RLReg ` R ) )
7	6	biimpi 120	. . 3 |- ( A e. E -> A e. (RLReg ` R ) )
8		relelfvdm 5618	. . 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 2787	1 |- ( A e. E -> R e. _V )

Syntax hints:   -> wi 4   = wceq 1373   e. wcel 2177   A.wral 2485   {crab 2489   _Vcvv 2773   |-> cmpt 4110   dom cdm 4680   Rel wrel 4685   ` cfv 5277  (class class class)co 5954   Basecbs 12882   .rcmulr 12960   0gc0g 13138  RLRegcrlreg 14067

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2180  ax-ext 2188  ax-sep 4167  ax-pow 4223  ax-pr 4258

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-v 2775  df-un 3172  df-in 3174  df-ss 3181  df-pw 3620  df-sn 3641  df-pr 3642  df-op 3644  df-uni 3854  df-br 4049  df-opab 4111  df-mpt 4112  df-xp 4686  df-rel 4687  df-dm 4690  df-iota 5238  df-fv 5285  df-rlreg 14070

This theorem is referenced by:  rrgval  14074