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Theorem rrgss 13746
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.
StepHypRef Expression
1 rrgss.e . . 3  |-  E  =  (RLReg `  R )
2 rrgss.b . . 3  |-  B  =  ( Base `  R
)
3 eqid 2193 . . 3  |-  ( .r
`  R )  =  ( .r `  R
)
4 eqid 2193 . . 3  |-  ( 0g
`  R )  =  ( 0g `  R
)
51, 2, 3, 4rrgval 13742 . 2  |-  E  =  { x  e.  B  |  A. y  e.  B  ( ( x ( .r `  R ) y )  =  ( 0g `  R )  ->  y  =  ( 0g `  R ) ) }
65ssrab3 3265 1  |-  E  C_  B
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1364   A.wral 2472    C_ wss 3153   ` cfv 5246  (class class class)co 5910   Basecbs 12608   .rcmulr 12686   0gc0g 12857  RLRegcrlreg 13735
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-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4462  ax-cnex 7953  ax-resscn 7954  ax-1re 7956  ax-addrcl 7959
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  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-int 3871  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4322  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-iota 5207  df-fun 5248  df-fn 5249  df-fv 5254  df-ov 5913  df-inn 8973  df-ndx 12611  df-slot 12612  df-base 12614  df-rlreg 13738
This theorem is referenced by:  znrrg  14125
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