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Theorem igenidl2 32832
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 2604 . . . 4 (1st𝑅) = (1st𝑅)
2 eqid 2604 . . . 4 ran (1st𝑅) = ran (1st𝑅)
31, 2idlss 32783 . . 3 ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → 𝐼 ⊆ ran (1st𝑅))
41, 2igenval 32828 . . 3 ((𝑅 ∈ RingOps ∧ 𝐼 ⊆ ran (1st𝑅)) → (𝑅 IdlGen 𝐼) = {𝑗 ∈ (Idl‘𝑅) ∣ 𝐼𝑗})
53, 4syldan 485 . 2 ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → (𝑅 IdlGen 𝐼) = {𝑗 ∈ (Idl‘𝑅) ∣ 𝐼𝑗})
6 intmin 4421 . . 3 (𝐼 ∈ (Idl‘𝑅) → {𝑗 ∈ (Idl‘𝑅) ∣ 𝐼𝑗} = 𝐼)
76adantl 480 . 2 ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → {𝑗 ∈ (Idl‘𝑅) ∣ 𝐼𝑗} = 𝐼)
85, 7eqtrd 2638 1 ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → (𝑅 IdlGen 𝐼) = 𝐼)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1975  {crab 2894  wss 3534   cint 4399  ran crn 5024  cfv 5785  (class class class)co 6522  1st c1st 7029  RingOpscrngo 32661  Idlcidl 32774   IdlGen cigen 32826
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2227  ax-ext 2584  ax-sep 4698  ax-nul 4707  ax-pow 4759  ax-pr 4823  ax-un 6819
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2456  df-mo 2457  df-clab 2591  df-cleq 2597  df-clel 2600  df-nfc 2734  df-ne 2776  df-ral 2895  df-rex 2896  df-reu 2897  df-rab 2899  df-v 3169  df-sbc 3397  df-csb 3494  df-dif 3537  df-un 3539  df-in 3541  df-ss 3548  df-nul 3869  df-if 4031  df-pw 4104  df-sn 4120  df-pr 4122  df-op 4126  df-uni 4362  df-int 4400  df-iun 4446  df-br 4573  df-opab 4633  df-mpt 4634  df-id 4938  df-xp 5029  df-rel 5030  df-cnv 5031  df-co 5032  df-dm 5033  df-rn 5034  df-iota 5749  df-fun 5787  df-fn 5788  df-f 5789  df-fo 5791  df-fv 5793  df-riota 6484  df-ov 6525  df-oprab 6526  df-mpt2 6527  df-1st 7031  df-2nd 7032  df-grpo 26492  df-gid 26493  df-ablo 26547  df-rngo 32662  df-idl 32777  df-igen 32827
This theorem is referenced by: (None)
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