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Theorem igenss 26566
Description: A set is a subset of the ideal it generates. (Contributed by Jeff Madsen, 10-Jun-2010.)
Hypotheses
Ref Expression
igenval.1  |-  G  =  ( 1st `  R
)
igenval.2  |-  X  =  ran  G
Assertion
Ref Expression
igenss  |-  ( ( R  e.  RingOps  /\  S  C_  X )  ->  S  C_  ( R  IdlGen  S ) )

Proof of Theorem igenss
Dummy variable  j is distinct from all other variables.
StepHypRef Expression
1 ssintub 4032 . 2  |-  S  C_  |^|
{ j  e.  ( Idl `  R )  |  S  C_  j }
2 igenval.1 . . 3  |-  G  =  ( 1st `  R
)
3 igenval.2 . . 3  |-  X  =  ran  G
42, 3igenval 26565 . 2  |-  ( ( R  e.  RingOps  /\  S  C_  X )  ->  ( R  IdlGen  S )  = 
|^| { j  e.  ( Idl `  R )  |  S  C_  j } )
51, 4syl5sseqr 3361 1  |-  ( ( R  e.  RingOps  /\  S  C_  X )  ->  S  C_  ( R  IdlGen  S ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   {crab 2674    C_ wss 3284   |^|cint 4014   ran crn 4842   ` cfv 5417  (class class class)co 6044   1stc1st 6310   RingOpscrngo 21920   Idlcidl 26511    IdlGen cigen 26563
This theorem is referenced by:  igenval2  26570  isfldidl  26572  ispridlc  26574
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-reu 2677  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-int 4015  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-fo 5423  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-1st 6312  df-2nd 6313  df-riota 6512  df-grpo 21736  df-gid 21737  df-ablo 21827  df-rngo 21921  df-idl 26514  df-igen 26564
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