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Theorem igenidl 33992
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 33990 . 2 ((𝑅 ∈ RingOps ∧ 𝑆𝑋) → (𝑅 IdlGen 𝑆) = {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆𝑗})
41, 2rngoidl 33953 . . . . 5 (𝑅 ∈ RingOps → 𝑋 ∈ (Idl‘𝑅))
5 sseq2 3660 . . . . . 6 (𝑗 = 𝑋 → (𝑆𝑗𝑆𝑋))
65rspcev 3340 . . . . 5 ((𝑋 ∈ (Idl‘𝑅) ∧ 𝑆𝑋) → ∃𝑗 ∈ (Idl‘𝑅)𝑆𝑗)
74, 6sylan 487 . . . 4 ((𝑅 ∈ RingOps ∧ 𝑆𝑋) → ∃𝑗 ∈ (Idl‘𝑅)𝑆𝑗)
8 rabn0 3991 . . . 4 ({𝑗 ∈ (Idl‘𝑅) ∣ 𝑆𝑗} ≠ ∅ ↔ ∃𝑗 ∈ (Idl‘𝑅)𝑆𝑗)
97, 8sylibr 224 . . 3 ((𝑅 ∈ RingOps ∧ 𝑆𝑋) → {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆𝑗} ≠ ∅)
10 ssrab2 3720 . . . 4 {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆𝑗} ⊆ (Idl‘𝑅)
11 intidl 33958 . . . 4 ((𝑅 ∈ RingOps ∧ {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆𝑗} ≠ ∅ ∧ {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆𝑗} ⊆ (Idl‘𝑅)) → {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆𝑗} ∈ (Idl‘𝑅))
1210, 11mp3an3 1453 . . 3 ((𝑅 ∈ RingOps ∧ {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆𝑗} ≠ ∅) → {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆𝑗} ∈ (Idl‘𝑅))
139, 12syldan 486 . 2 ((𝑅 ∈ RingOps ∧ 𝑆𝑋) → {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆𝑗} ∈ (Idl‘𝑅))
143, 13eqeltrd 2730 1 ((𝑅 ∈ RingOps ∧ 𝑆𝑋) → (𝑅 IdlGen 𝑆) ∈ (Idl‘𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  wne 2823  wrex 2942  {crab 2945  wss 3607  c0 3948   cint 4507  ran crn 5144  cfv 5926  (class class class)co 6690  1st c1st 7208  RingOpscrngo 33823  Idlcidl 33936   IdlGen cigen 33988
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fo 5932  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-1st 7210  df-2nd 7211  df-grpo 27475  df-gid 27476  df-ablo 27527  df-rngo 33824  df-idl 33939  df-igen 33989
This theorem is referenced by:  igenval2  33995  isfldidl  33997  ispridlc  33999
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