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Theorem igenidl2 35496
 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 2801 . . . 4 (1st𝑅) = (1st𝑅)
2 eqid 2801 . . . 4 ran (1st𝑅) = ran (1st𝑅)
31, 2idlss 35447 . . 3 ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → 𝐼 ⊆ ran (1st𝑅))
41, 2igenval 35492 . . 3 ((𝑅 ∈ RingOps ∧ 𝐼 ⊆ ran (1st𝑅)) → (𝑅 IdlGen 𝐼) = {𝑗 ∈ (Idl‘𝑅) ∣ 𝐼𝑗})
53, 4syldan 594 . 2 ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → (𝑅 IdlGen 𝐼) = {𝑗 ∈ (Idl‘𝑅) ∣ 𝐼𝑗})
6 intmin 4861 . . 3 (𝐼 ∈ (Idl‘𝑅) → {𝑗 ∈ (Idl‘𝑅) ∣ 𝐼𝑗} = 𝐼)
76adantl 485 . 2 ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → {𝑗 ∈ (Idl‘𝑅) ∣ 𝐼𝑗} = 𝐼)
85, 7eqtrd 2836 1 ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → (𝑅 IdlGen 𝐼) = 𝐼)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2112  {crab 3113   ⊆ wss 3884  ∩ cint 4841  ran crn 5524  ‘cfv 6328  (class class class)co 7139  1st c1st 7673  RingOpscrngo 35325  Idlcidl 35438   IdlGen cigen 35490 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 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-fo 6334  df-fv 6336  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-1st 7675  df-2nd 7676  df-grpo 28279  df-gid 28280  df-ablo 28331  df-rngo 35326  df-idl 35441  df-igen 35491 This theorem is referenced by: (None)
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