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

Theorem rrgss 20726
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 20721 . 2 𝐸 = {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥(.r𝑅)𝑦) = (0g𝑅) → 𝑦 = (0g𝑅))}
65ssrab3 4105 1 𝐸𝐵
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wral 3067  wss 3976  cfv 6575  (class class class)co 7450  Basecbs 17260  .rcmulr 17314  0gc0g 17501  RLRegcrlreg 20715
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-iota 6527  df-fun 6577  df-fv 6583  df-ov 7453  df-rlreg 20718
This theorem is referenced by:  isdomn6  20738  znrrg  21609  mdegvsca  26137  deg1mul3  26177  rrgsubm  33255  fracbas  33274  fracerl  33275  fracfld  33277  assalactf1o  33650  assarrginv  33651
  Copyright terms: Public domain W3C validator