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Theorem igenidl 34341
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 34339 . 2 ((𝑅 ∈ RingOps ∧ 𝑆𝑋) → (𝑅 IdlGen 𝑆) = {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆𝑗})
41, 2rngoidl 34302 . . . . 5 (𝑅 ∈ RingOps → 𝑋 ∈ (Idl‘𝑅))
5 sseq2 3821 . . . . . 6 (𝑗 = 𝑋 → (𝑆𝑗𝑆𝑋))
65rspcev 3495 . . . . 5 ((𝑋 ∈ (Idl‘𝑅) ∧ 𝑆𝑋) → ∃𝑗 ∈ (Idl‘𝑅)𝑆𝑗)
74, 6sylan 576 . . . 4 ((𝑅 ∈ RingOps ∧ 𝑆𝑋) → ∃𝑗 ∈ (Idl‘𝑅)𝑆𝑗)
8 rabn0 4156 . . . 4 ({𝑗 ∈ (Idl‘𝑅) ∣ 𝑆𝑗} ≠ ∅ ↔ ∃𝑗 ∈ (Idl‘𝑅)𝑆𝑗)
97, 8sylibr 226 . . 3 ((𝑅 ∈ RingOps ∧ 𝑆𝑋) → {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆𝑗} ≠ ∅)
10 ssrab2 3881 . . . 4 {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆𝑗} ⊆ (Idl‘𝑅)
11 intidl 34307 . . . 4 ((𝑅 ∈ RingOps ∧ {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆𝑗} ≠ ∅ ∧ {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆𝑗} ⊆ (Idl‘𝑅)) → {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆𝑗} ∈ (Idl‘𝑅))
1210, 11mp3an3 1575 . . 3 ((𝑅 ∈ RingOps ∧ {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆𝑗} ≠ ∅) → {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆𝑗} ∈ (Idl‘𝑅))
139, 12syldan 586 . 2 ((𝑅 ∈ RingOps ∧ 𝑆𝑋) → {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆𝑗} ∈ (Idl‘𝑅))
143, 13eqeltrd 2876 1 ((𝑅 ∈ RingOps ∧ 𝑆𝑋) → (𝑅 IdlGen 𝑆) ∈ (Idl‘𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385   = wceq 1653  wcel 2157  wne 2969  wrex 3088  {crab 3091  wss 3767  c0 4113   cint 4665  ran crn 5311  cfv 6099  (class class class)co 6876  1st c1st 7397  RingOpscrngo 34172  Idlcidl 34285   IdlGen cigen 34337
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354  ax-ext 2775  ax-sep 4973  ax-nul 4981  ax-pow 5033  ax-pr 5095  ax-un 7181
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2590  df-eu 2607  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ne 2970  df-ral 3092  df-rex 3093  df-reu 3094  df-rab 3096  df-v 3385  df-sbc 3632  df-csb 3727  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-nul 4114  df-if 4276  df-pw 4349  df-sn 4367  df-pr 4369  df-op 4373  df-uni 4627  df-int 4666  df-iun 4710  df-br 4842  df-opab 4904  df-mpt 4921  df-id 5218  df-xp 5316  df-rel 5317  df-cnv 5318  df-co 5319  df-dm 5320  df-rn 5321  df-iota 6062  df-fun 6101  df-fn 6102  df-f 6103  df-fo 6105  df-fv 6107  df-riota 6837  df-ov 6879  df-oprab 6880  df-mpt2 6881  df-1st 7399  df-2nd 7400  df-grpo 27865  df-gid 27866  df-ablo 27917  df-rngo 34173  df-idl 34288  df-igen 34338
This theorem is referenced by:  igenval2  34344  isfldidl  34346  ispridlc  34348
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