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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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