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Theorem rrgss 20553
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 𝐸𝐵

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 2740 . . 3 (.r𝑅) = (.r𝑅)
4 eqid 2740 . . 3 (0g𝑅) = (0g𝑅)
51, 2, 3, 4rrgval 20548 . 2 𝐸 = {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥(.r𝑅)𝑦) = (0g𝑅) → 𝑦 = (0g𝑅))}
65ssrab3 4020 1 𝐸𝐵
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wral 3066  wss 3892  cfv 6431  (class class class)co 7269  Basecbs 16902  .rcmulr 16953  0gc0g 17140  RLRegcrlreg 20540
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-sep 5227  ax-nul 5234  ax-pr 5356
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-ral 3071  df-rex 3072  df-rab 3075  df-v 3433  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-br 5080  df-opab 5142  df-mpt 5163  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-iota 6389  df-fun 6433  df-fv 6439  df-ov 7272  df-rlreg 20544
This theorem is referenced by:  znrrg  20763  mdegvsca  25231  deg1mul3  25270
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