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Theorem igenidl 35807
 Description: The ideal generated by a set is an ideal. (Contributed by Jeff Madsen, 10-Jun-2010.)
Hypotheses
Ref Expression
igenval.1 𝐺 = (1st𝑅)
igenval.2 𝑋 = ran 𝐺
Assertion
Ref Expression
igenidl ((𝑅 ∈ RingOps ∧ 𝑆𝑋) → (𝑅 IdlGen 𝑆) ∈ (Idl‘𝑅))

Proof of Theorem igenidl
Dummy variable 𝑗 is distinct from all other variables.
StepHypRef Expression
1 igenval.1 . . 3 𝐺 = (1st𝑅)
2 igenval.2 . . 3 𝑋 = ran 𝐺
31, 2igenval 35805 . 2 ((𝑅 ∈ RingOps ∧ 𝑆𝑋) → (𝑅 IdlGen 𝑆) = {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆𝑗})
41, 2rngoidl 35768 . . . . 5 (𝑅 ∈ RingOps → 𝑋 ∈ (Idl‘𝑅))
5 sseq2 3920 . . . . . 6 (𝑗 = 𝑋 → (𝑆𝑗𝑆𝑋))
65rspcev 3543 . . . . 5 ((𝑋 ∈ (Idl‘𝑅) ∧ 𝑆𝑋) → ∃𝑗 ∈ (Idl‘𝑅)𝑆𝑗)
74, 6sylan 583 . . . 4 ((𝑅 ∈ RingOps ∧ 𝑆𝑋) → ∃𝑗 ∈ (Idl‘𝑅)𝑆𝑗)
8 rabn0 4284 . . . 4 ({𝑗 ∈ (Idl‘𝑅) ∣ 𝑆𝑗} ≠ ∅ ↔ ∃𝑗 ∈ (Idl‘𝑅)𝑆𝑗)
97, 8sylibr 237 . . 3 ((𝑅 ∈ RingOps ∧ 𝑆𝑋) → {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆𝑗} ≠ ∅)
10 ssrab2 3986 . . . 4 {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆𝑗} ⊆ (Idl‘𝑅)
11 intidl 35773 . . . 4 ((𝑅 ∈ RingOps ∧ {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆𝑗} ≠ ∅ ∧ {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆𝑗} ⊆ (Idl‘𝑅)) → {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆𝑗} ∈ (Idl‘𝑅))
1210, 11mp3an3 1447 . . 3 ((𝑅 ∈ RingOps ∧ {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆𝑗} ≠ ∅) → {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆𝑗} ∈ (Idl‘𝑅))
139, 12syldan 594 . 2 ((𝑅 ∈ RingOps ∧ 𝑆𝑋) → {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆𝑗} ∈ (Idl‘𝑅))
143, 13eqeltrd 2852 1 ((𝑅 ∈ RingOps ∧ 𝑆𝑋) → (𝑅 IdlGen 𝑆) ∈ (Idl‘𝑅))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111   ≠ wne 2951  ∃wrex 3071  {crab 3074   ⊆ wss 3860  ∅c0 4227  ∩ cint 4841  ran crn 5528  ‘cfv 6339  (class class class)co 7155  1st c1st 7696  RingOpscrngo 35638  Idlcidl 35751   IdlGen cigen 35803 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5172  ax-nul 5179  ax-pow 5237  ax-pr 5301  ax-un 7464 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-int 4842  df-iun 4888  df-br 5036  df-opab 5098  df-mpt 5116  df-id 5433  df-xp 5533  df-rel 5534  df-cnv 5535  df-co 5536  df-dm 5537  df-rn 5538  df-iota 6298  df-fun 6341  df-fn 6342  df-f 6343  df-fo 6345  df-fv 6347  df-riota 7113  df-ov 7158  df-oprab 7159  df-mpo 7160  df-1st 7698  df-2nd 7699  df-grpo 28380  df-gid 28381  df-ablo 28432  df-rngo 35639  df-idl 35754  df-igen 35804 This theorem is referenced by:  igenval2  35810  isfldidl  35812  ispridlc  35814
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