Theorem rrgss 14548

Description: Left-regular elements are a subset of the base set. (Contributed by Stefan O'Rear, 22-Mar-2015.)

Hypotheses

Ref	Expression
rrgss.e	|- E = (RLReg ` R )
rrgss.b	|- B = ( Base ` R )

Assertion

Ref	Expression
rrgss	|- E C_ B

Proof of Theorem rrgss

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rrgss.e	. . 3 |- E = (RLReg ` R )
2		rrgss.b	. . 3 |- B = ( Base ` R )
3		eqid 2238	. . 3 |- ( .r ` R ) = ( .r ` R )
4		eqid 2238	. . 3 |- ( 0g ` R ) = ( 0g ` R )
5	1, 2, 3, 4	rrgval 14543	. 2 |- E = { x e. B | A. y e. B ( ( x ( .r ` R ) y ) = ( 0g ` R ) -> y = ( 0g ` R ) ) }
6	5	ssrab3 3334	1 |- E C_ B

Syntax hints:   -> wi 4   = wceq 1402   A.wral 2528   C_ wss 3220   ` cfv 5372  (class class class)co 6075   Basecbs 13330   .rcmulr 13409   0gc0g 13587  RLRegcrlreg 14536

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 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-cnex 8260  ax-resscn 8261  ax-1re 8263  ax-addrcl 8266

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-iota 5332  df-fun 5374  df-fn 5375  df-fv 5380  df-ov 6078  df-inn 9284  df-ndx 13333  df-slot 13334  df-base 13336  df-rlreg 14539

This theorem is referenced by:  znrrg  14967