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Theorem rrgss 16311
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 2408 . . 3  |-  ( .r
`  R )  =  ( .r `  R
)
4 eqid 2408 . . 3  |-  ( 0g
`  R )  =  ( 0g `  R
)
51, 2, 3, 4rrgval 16306 . 2  |-  E  =  { x  e.  B  |  A. y  e.  B  ( ( x ( .r `  R ) y )  =  ( 0g `  R )  ->  y  =  ( 0g `  R ) ) }
6 ssrab2 3392 . 2  |-  { x  e.  B  |  A. y  e.  B  (
( x ( .r
`  R ) y )  =  ( 0g
`  R )  -> 
y  =  ( 0g
`  R ) ) }  C_  B
75, 6eqsstri 3342 1  |-  E  C_  B
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649   A.wral 2670   {crab 2674    C_ wss 3284   ` cfv 5417  (class class class)co 6044   Basecbs 13428   .rcmulr 13489   0gc0g 13682  RLRegcrlreg 16298
This theorem is referenced by:  znrrg  16805  mdegvsca  19956  deg1mul3  19995
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-rab 2679  df-v 2922  df-sbc 3126  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-iota 5381  df-fun 5419  df-fv 5425  df-ov 6047  df-rlreg 16302
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