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Theorem igenidl2 30702
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 2454 . . . 4  |-  ( 1st `  R )  =  ( 1st `  R )
2 eqid 2454 . . . 4  |-  ran  ( 1st `  R )  =  ran  ( 1st `  R
)
31, 2idlss 30653 . . 3  |-  ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R
) )  ->  I  C_ 
ran  ( 1st `  R
) )
41, 2igenval 30698 . . 3  |-  ( ( R  e.  RingOps  /\  I  C_ 
ran  ( 1st `  R
) )  ->  ( R  IdlGen  I )  = 
|^| { j  e.  ( Idl `  R )  |  I  C_  j } )
53, 4syldan 468 . 2  |-  ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R
) )  ->  ( R  IdlGen  I )  = 
|^| { j  e.  ( Idl `  R )  |  I  C_  j } )
6 intmin 4291 . . 3  |-  ( I  e.  ( Idl `  R
)  ->  |^| { j  e.  ( Idl `  R
)  |  I  C_  j }  =  I
)
76adantl 464 . 2  |-  ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R
) )  ->  |^| { j  e.  ( Idl `  R
)  |  I  C_  j }  =  I
)
85, 7eqtrd 2495 1  |-  ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R
) )  ->  ( R  IdlGen  I )  =  I )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823   {crab 2808    C_ wss 3461   |^|cint 4271   ran crn 4989   ` cfv 5570  (class class class)co 6270   1stc1st 6771   RingOpscrngo 25575   Idlcidl 30644    IdlGen cigen 30696
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-fo 5576  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-1st 6773  df-2nd 6774  df-grpo 25391  df-gid 25392  df-ablo 25482  df-rngo 25576  df-idl 30647  df-igen 30697
This theorem is referenced by: (None)
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