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Theorem isrrg 16340
 Description: Membership in the set of left-regular elements. (Contributed by Stefan O'Rear, 22-Mar-2015.)
Hypotheses
Ref Expression
rrgval.e RLReg
rrgval.b
rrgval.t
rrgval.z
Assertion
Ref Expression
isrrg
Distinct variable groups:   ,   ,   ,
Allowed substitution hints:   ()   ()   ()

Proof of Theorem isrrg
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 oveq1 6080 . . . . 5
21eqeq1d 2443 . . . 4
32imbi1d 309 . . 3
43ralbidv 2717 . 2
5 rrgval.e . . 3 RLReg
6 rrgval.b . . 3
7 rrgval.t . . 3
8 rrgval.z . . 3
95, 6, 7, 8rrgval 16339 . 2
104, 9elrab2 3086 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wa 359   wceq 1652   wcel 1725  wral 2697  cfv 5446  (class class class)co 6073  cbs 13461  cmulr 13522  c0g 13715  RLRegcrlreg 16331 This theorem is referenced by:  rrgeq0i  16341  unitrrg  16345  isdomn2  16351 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-iota 5410  df-fun 5448  df-fv 5454  df-ov 6076  df-rlreg 16335
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