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Theorem igenidl2 28888
Description: The ideal generated by an ideal is that ideal. (Contributed by Jeff Madsen, 10-Jun-2010.)
Assertion
Ref Expression
igenidl2  |-  ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R
) )  ->  ( R  IdlGen  I )  =  I )

Proof of Theorem igenidl2
Dummy variable  j is distinct from all other variables.
StepHypRef Expression
1 eqid 2443 . . . 4  |-  ( 1st `  R )  =  ( 1st `  R )
2 eqid 2443 . . . 4  |-  ran  ( 1st `  R )  =  ran  ( 1st `  R
)
31, 2idlss 28839 . . 3  |-  ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R
) )  ->  I  C_ 
ran  ( 1st `  R
) )
41, 2igenval 28884 . . 3  |-  ( ( R  e.  RingOps  /\  I  C_ 
ran  ( 1st `  R
) )  ->  ( R  IdlGen  I )  = 
|^| { j  e.  ( Idl `  R )  |  I  C_  j } )
53, 4syldan 470 . 2  |-  ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R
) )  ->  ( R  IdlGen  I )  = 
|^| { j  e.  ( Idl `  R )  |  I  C_  j } )
6 intmin 4167 . . 3  |-  ( I  e.  ( Idl `  R
)  ->  |^| { j  e.  ( Idl `  R
)  |  I  C_  j }  =  I
)
76adantl 466 . 2  |-  ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R
) )  ->  |^| { j  e.  ( Idl `  R
)  |  I  C_  j }  =  I
)
85, 7eqtrd 2475 1  |-  ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R
) )  ->  ( R  IdlGen  I )  =  I )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   {crab 2738    C_ wss 3347   |^|cint 4147   ran crn 4860   ` cfv 5437  (class class class)co 6110   1stc1st 6594   RingOpscrngo 23881   Idlcidl 28830    IdlGen cigen 28882
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4432  ax-nul 4440  ax-pow 4489  ax-pr 4550  ax-un 6391
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2739  df-rex 2740  df-reu 2741  df-rab 2743  df-v 2993  df-sbc 3206  df-csb 3308  df-dif 3350  df-un 3352  df-in 3354  df-ss 3361  df-nul 3657  df-if 3811  df-pw 3881  df-sn 3897  df-pr 3899  df-op 3903  df-uni 4111  df-int 4148  df-iun 4192  df-br 4312  df-opab 4370  df-mpt 4371  df-id 4655  df-xp 4865  df-rel 4866  df-cnv 4867  df-co 4868  df-dm 4869  df-rn 4870  df-iota 5400  df-fun 5439  df-fn 5440  df-f 5441  df-fo 5443  df-fv 5445  df-riota 6071  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-1st 6596  df-2nd 6597  df-grpo 23697  df-gid 23698  df-ablo 23788  df-rngo 23882  df-idl 28833  df-igen 28883
This theorem is referenced by: (None)
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