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Theorem igenss 32814
Description: A set is a subset of the ideal it generates. (Contributed by Jeff Madsen, 10-Jun-2010.)
Hypotheses
Ref Expression
igenval.1 𝐺 = (1st𝑅)
igenval.2 𝑋 = ran 𝐺
Assertion
Ref Expression
igenss ((𝑅 ∈ RingOps ∧ 𝑆𝑋) → 𝑆 ⊆ (𝑅 IdlGen 𝑆))

Proof of Theorem igenss
Dummy variable 𝑗 is distinct from all other variables.
StepHypRef Expression
1 ssintub 4424 . 2 𝑆 {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆𝑗}
2 igenval.1 . . 3 𝐺 = (1st𝑅)
3 igenval.2 . . 3 𝑋 = ran 𝐺
42, 3igenval 32813 . 2 ((𝑅 ∈ RingOps ∧ 𝑆𝑋) → (𝑅 IdlGen 𝑆) = {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆𝑗})
51, 4syl5sseqr 3616 1 ((𝑅 ∈ RingOps ∧ 𝑆𝑋) → 𝑆 ⊆ (𝑅 IdlGen 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1976  {crab 2899  wss 3539   cint 4404  ran crn 5028  cfv 5789  (class class class)co 6526  1st c1st 7034  RingOpscrngo 32646  Idlcidl 32759   IdlGen cigen 32811
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4711  ax-pow 4763  ax-pr 4827  ax-un 6824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-int 4405  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4942  df-xp 5033  df-rel 5034  df-cnv 5035  df-co 5036  df-dm 5037  df-rn 5038  df-iota 5753  df-fun 5791  df-fn 5792  df-f 5793  df-fo 5795  df-fv 5797  df-riota 6488  df-ov 6529  df-oprab 6530  df-mpt2 6531  df-1st 7036  df-2nd 7037  df-grpo 26524  df-gid 26525  df-ablo 26576  df-rngo 32647  df-idl 32762  df-igen 32812
This theorem is referenced by:  igenval2  32818  isfldidl  32820  ispridlc  32822
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