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Theorem iscyggen 15492
 Description: The property of being a cyclic generator for a group. (Contributed by Mario Carneiro, 21-Apr-2016.)
Hypotheses
Ref Expression
iscyg.1
iscyg.2 .g
iscyg3.e
Assertion
Ref Expression
iscyggen
Distinct variable groups:   ,,   ,,   ,,   ,,
Allowed substitution hints:   (,)

Proof of Theorem iscyggen
StepHypRef Expression
1 simpl 445 . . . . . 6
21oveq2d 6099 . . . . 5
32mpteq2dva 4297 . . . 4
43rneqd 5099 . . 3
54eqeq1d 2446 . 2
6 iscyg3.e . 2
75, 6elrab2 3096 1
 Colors of variables: wff set class Syntax hints:   wb 178   wa 360   wceq 1653   wcel 1726  crab 2711   cmpt 4268   crn 4881  cfv 5456  (class class class)co 6083  cz 10284  cbs 13471  .gcmg 14691 This theorem is referenced by:  iscyggen2  15493  cyggenod  15496  cyggenod2  15497  cygznlem1  16849  cygznlem3  16852 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419 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-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-mpt 4270  df-cnv 4888  df-dm 4890  df-rn 4891  df-iota 5420  df-fv 5464  df-ov 6086
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