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Theorem igenss 26010
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 3961 . 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 26009 . 2  |-  ( ( R  e.  RingOps  /\  S  C_  X )  ->  ( R  IdlGen  S )  = 
|^| { j  e.  ( Idl `  R )  |  S  C_  j } )
51, 4syl5sseqr 3303 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 1642    e. wcel 1710   {crab 2623    C_ wss 3228   |^|cint 3943   ran crn 4772   ` cfv 5337  (class class class)co 5945   1stc1st 6207   RingOpscrngo 21154   Idlcidl 25955    IdlGen cigen 26007
This theorem is referenced by:  igenval2  26014  isfldidl  26016  ispridlc  26018
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-int 3944  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-id 4391  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-fo 5343  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-1st 6209  df-2nd 6210  df-riota 6391  df-grpo 20970  df-gid 20971  df-ablo 21061  df-rngo 21155  df-idl 25958  df-igen 26008
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