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Theorem rrgss 16383
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 2442 . . 3  |-  ( .r
`  R )  =  ( .r `  R
)
4 eqid 2442 . . 3  |-  ( 0g
`  R )  =  ( 0g `  R
)
51, 2, 3, 4rrgval 16378 . 2  |-  E  =  { x  e.  B  |  A. y  e.  B  ( ( x ( .r `  R ) y )  =  ( 0g `  R )  ->  y  =  ( 0g `  R ) ) }
6 ssrab2 3414 . 2  |-  { x  e.  B  |  A. y  e.  B  (
( x ( .r
`  R ) y )  =  ( 0g
`  R )  -> 
y  =  ( 0g
`  R ) ) }  C_  B
75, 6eqsstri 3364 1  |-  E  C_  B
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653   A.wral 2711   {crab 2715    C_ wss 3306   ` cfv 5483  (class class class)co 6110   Basecbs 13500   .rcmulr 13561   0gc0g 13754  RLRegcrlreg 16370
This theorem is referenced by:  znrrg  16877  mdegvsca  20030  deg1mul3  20069
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-rab 2720  df-v 2964  df-sbc 3168  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-sn 3844  df-pr 3845  df-op 3847  df-uni 4040  df-br 4238  df-opab 4292  df-mpt 4293  df-id 4527  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-iota 5447  df-fun 5485  df-fv 5491  df-ov 6113  df-rlreg 16374
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