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Theorem igenidl2 36209
Description: The ideal generated by an ideal is that ideal. (Contributed by Jeff Madsen, 10-Jun-2010.)
Assertion
Ref Expression
igenidl2 ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → (𝑅 IdlGen 𝐼) = 𝐼)

Proof of Theorem igenidl2
Dummy variable 𝑗 is distinct from all other variables.
StepHypRef Expression
1 eqid 2738 . . . 4 (1st𝑅) = (1st𝑅)
2 eqid 2738 . . . 4 ran (1st𝑅) = ran (1st𝑅)
31, 2idlss 36160 . . 3 ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → 𝐼 ⊆ ran (1st𝑅))
41, 2igenval 36205 . . 3 ((𝑅 ∈ RingOps ∧ 𝐼 ⊆ ran (1st𝑅)) → (𝑅 IdlGen 𝐼) = {𝑗 ∈ (Idl‘𝑅) ∣ 𝐼𝑗})
53, 4syldan 591 . 2 ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → (𝑅 IdlGen 𝐼) = {𝑗 ∈ (Idl‘𝑅) ∣ 𝐼𝑗})
6 intmin 4900 . . 3 (𝐼 ∈ (Idl‘𝑅) → {𝑗 ∈ (Idl‘𝑅) ∣ 𝐼𝑗} = 𝐼)
76adantl 482 . 2 ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → {𝑗 ∈ (Idl‘𝑅) ∣ 𝐼𝑗} = 𝐼)
85, 7eqtrd 2778 1 ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → (𝑅 IdlGen 𝐼) = 𝐼)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1539  wcel 2106  {crab 3068  wss 3887   cint 4880  ran crn 5586  cfv 6427  (class class class)co 7268  1st c1st 7819  RingOpscrngo 36038  Idlcidl 36151   IdlGen cigen 36203
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5222  ax-nul 5229  ax-pow 5287  ax-pr 5351  ax-un 7579
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3071  df-rab 3073  df-v 3432  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4258  df-if 4461  df-pw 4536  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4841  df-int 4881  df-iun 4927  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5485  df-xp 5591  df-rel 5592  df-cnv 5593  df-co 5594  df-dm 5595  df-rn 5596  df-iota 6385  df-fun 6429  df-fn 6430  df-f 6431  df-fo 6433  df-fv 6435  df-riota 7225  df-ov 7271  df-oprab 7272  df-mpo 7273  df-1st 7821  df-2nd 7822  df-grpo 28841  df-gid 28842  df-ablo 28893  df-rngo 36039  df-idl 36154  df-igen 36204
This theorem is referenced by: (None)
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