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Theorem rrgss 16126
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 2358 . . 3  |-  ( .r
`  R )  =  ( .r `  R
)
4 eqid 2358 . . 3  |-  ( 0g
`  R )  =  ( 0g `  R
)
51, 2, 3, 4rrgval 16121 . 2  |-  E  =  { x  e.  B  |  A. y  e.  B  ( ( x ( .r `  R ) y )  =  ( 0g `  R )  ->  y  =  ( 0g `  R ) ) }
6 ssrab2 3334 . 2  |-  { x  e.  B  |  A. y  e.  B  (
( x ( .r
`  R ) y )  =  ( 0g
`  R )  -> 
y  =  ( 0g
`  R ) ) }  C_  B
75, 6eqsstri 3284 1  |-  E  C_  B
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1642   A.wral 2619   {crab 2623    C_ wss 3228   ` cfv 5334  (class class class)co 5942   Basecbs 13239   .rcmulr 13300   0gc0g 13493  RLRegcrlreg 16113
This theorem is referenced by:  znrrg  16619  mdegvsca  19560  deg1mul3  19599
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3907  df-br 4103  df-opab 4157  df-mpt 4158  df-id 4388  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-iota 5298  df-fun 5336  df-fv 5342  df-ov 5945  df-rlreg 16117
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