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Theorem igenss 26687
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 3880 . 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 26686 . 2  |-  ( ( R  e.  RingOps  /\  S  C_  X )  ->  ( R  IdlGen  S )  = 
|^| { j  e.  ( Idl `  R )  |  S  C_  j } )
51, 4syl5sseqr 3227 1  |-  ( ( R  e.  RingOps  /\  S  C_  X )  ->  S  C_  ( R  IdlGen  S ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   {crab 2547    C_ wss 3152   |^|cint 3862   ran crn 4690   ` cfv 5255  (class class class)co 5858   1stc1st 6120   RingOpscrngo 21042   Idlcidl 26632    IdlGen cigen 26684
This theorem is referenced by:  igenval2  26691  isfldidl  26693  ispridlc  26695
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fo 5261  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-grpo 20858  df-gid 20859  df-ablo 20949  df-rngo 21043  df-idl 26635  df-igen 26685
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