Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  rrgss Structured version   Visualization version   GIF version

Theorem rrgss 19113
 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 2610 . . 3 (.r𝑅) = (.r𝑅)
4 eqid 2610 . . 3 (0g𝑅) = (0g𝑅)
51, 2, 3, 4rrgval 19108 . 2 𝐸 = {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥(.r𝑅)𝑦) = (0g𝑅) → 𝑦 = (0g𝑅))}
6 ssrab2 3650 . 2 {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥(.r𝑅)𝑦) = (0g𝑅) → 𝑦 = (0g𝑅))} ⊆ 𝐵
75, 6eqsstri 3598 1 𝐸𝐵
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475  ∀wral 2896  {crab 2900   ⊆ wss 3540  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  .rcmulr 15769  0gc0g 15923  RLRegcrlreg 19100 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-rlreg 19104 This theorem is referenced by:  znrrg  19733  mdegvsca  23640  deg1mul3  23679
 Copyright terms: Public domain W3C validator