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

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 RLReg
rrgss.b
Assertion
Ref Expression
rrgss

Proof of Theorem rrgss
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rrgss.e . . 3 RLReg
2 rrgss.b . . 3
3 eqid 2442 . . 3
4 eqid 2442 . . 3
51, 2, 3, 4rrgval 16378 . 2
6 ssrab2 3414 . 2
75, 6eqsstri 3364 1
 Colors of variables: wff set class Syntax hints:   wi 4   wceq 1653  wral 2711  crab 2715   wss 3306  cfv 5483  (class class class)co 6110  cbs 13500  cmulr 13561  c0g 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
 Copyright terms: Public domain W3C validator