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Theorem igenval 35498
Description: The ideal generated by a subset of a ring. (Contributed by Jeff Madsen, 10-Jun-2010.) (Proof shortened by Mario Carneiro, 20-Dec-2013.)
Hypotheses
Ref Expression
igenval.1 𝐺 = (1st𝑅)
igenval.2 𝑋 = ran 𝐺
Assertion
Ref Expression
igenval ((𝑅 ∈ RingOps ∧ 𝑆𝑋) → (𝑅 IdlGen 𝑆) = {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆𝑗})
Distinct variable groups:   𝑅,𝑗   𝑆,𝑗   𝑗,𝑋
Allowed substitution hint:   𝐺(𝑗)

Proof of Theorem igenval
Dummy variables 𝑟 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 igenval.1 . . . . . 6 𝐺 = (1st𝑅)
2 igenval.2 . . . . . 6 𝑋 = ran 𝐺
31, 2rngoidl 35461 . . . . 5 (𝑅 ∈ RingOps → 𝑋 ∈ (Idl‘𝑅))
4 sseq2 3944 . . . . . 6 (𝑗 = 𝑋 → (𝑆𝑗𝑆𝑋))
54rspcev 3574 . . . . 5 ((𝑋 ∈ (Idl‘𝑅) ∧ 𝑆𝑋) → ∃𝑗 ∈ (Idl‘𝑅)𝑆𝑗)
63, 5sylan 583 . . . 4 ((𝑅 ∈ RingOps ∧ 𝑆𝑋) → ∃𝑗 ∈ (Idl‘𝑅)𝑆𝑗)
7 rabn0 4296 . . . 4 ({𝑗 ∈ (Idl‘𝑅) ∣ 𝑆𝑗} ≠ ∅ ↔ ∃𝑗 ∈ (Idl‘𝑅)𝑆𝑗)
86, 7sylibr 237 . . 3 ((𝑅 ∈ RingOps ∧ 𝑆𝑋) → {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆𝑗} ≠ ∅)
9 intex 5207 . . 3 ({𝑗 ∈ (Idl‘𝑅) ∣ 𝑆𝑗} ≠ ∅ ↔ {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆𝑗} ∈ V)
108, 9sylib 221 . 2 ((𝑅 ∈ RingOps ∧ 𝑆𝑋) → {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆𝑗} ∈ V)
111fvexi 6663 . . . . . 6 𝐺 ∈ V
1211rnex 7603 . . . . 5 ran 𝐺 ∈ V
132, 12eqeltri 2889 . . . 4 𝑋 ∈ V
1413elpw2 5215 . . 3 (𝑆 ∈ 𝒫 𝑋𝑆𝑋)
15 simpl 486 . . . . . . 7 ((𝑟 = 𝑅𝑠 = 𝑆) → 𝑟 = 𝑅)
1615fveq2d 6653 . . . . . 6 ((𝑟 = 𝑅𝑠 = 𝑆) → (Idl‘𝑟) = (Idl‘𝑅))
17 sseq1 3943 . . . . . . 7 (𝑠 = 𝑆 → (𝑠𝑗𝑆𝑗))
1817adantl 485 . . . . . 6 ((𝑟 = 𝑅𝑠 = 𝑆) → (𝑠𝑗𝑆𝑗))
1916, 18rabeqbidv 3436 . . . . 5 ((𝑟 = 𝑅𝑠 = 𝑆) → {𝑗 ∈ (Idl‘𝑟) ∣ 𝑠𝑗} = {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆𝑗})
2019inteqd 4846 . . . 4 ((𝑟 = 𝑅𝑠 = 𝑆) → {𝑗 ∈ (Idl‘𝑟) ∣ 𝑠𝑗} = {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆𝑗})
21 fveq2 6649 . . . . . . . 8 (𝑟 = 𝑅 → (1st𝑟) = (1st𝑅))
2221, 1eqtr4di 2854 . . . . . . 7 (𝑟 = 𝑅 → (1st𝑟) = 𝐺)
2322rneqd 5776 . . . . . 6 (𝑟 = 𝑅 → ran (1st𝑟) = ran 𝐺)
2423, 2eqtr4di 2854 . . . . 5 (𝑟 = 𝑅 → ran (1st𝑟) = 𝑋)
2524pweqd 4519 . . . 4 (𝑟 = 𝑅 → 𝒫 ran (1st𝑟) = 𝒫 𝑋)
26 df-igen 35497 . . . 4 IdlGen = (𝑟 ∈ RingOps, 𝑠 ∈ 𝒫 ran (1st𝑟) ↦ {𝑗 ∈ (Idl‘𝑟) ∣ 𝑠𝑗})
2720, 25, 26ovmpox 7286 . . 3 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ 𝒫 𝑋 {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆𝑗} ∈ V) → (𝑅 IdlGen 𝑆) = {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆𝑗})
2814, 27syl3an2br 1404 . 2 ((𝑅 ∈ RingOps ∧ 𝑆𝑋 {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆𝑗} ∈ V) → (𝑅 IdlGen 𝑆) = {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆𝑗})
2910, 28mpd3an3 1459 1 ((𝑅 ∈ RingOps ∧ 𝑆𝑋) → (𝑅 IdlGen 𝑆) = {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆𝑗})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2112  wne 2990  wrex 3110  {crab 3113  Vcvv 3444  wss 3884  c0 4246  𝒫 cpw 4500   cint 4841  ran crn 5524  cfv 6328  (class class class)co 7139  1st c1st 7673  RingOpscrngo 35331  Idlcidl 35444   IdlGen cigen 35496
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 28280  df-gid 28281  df-ablo 28332  df-rngo 35332  df-idl 35447  df-igen 35497
This theorem is referenced by:  igenss  35499  igenidl  35500  igenmin  35501  igenidl2  35502  igenval2  35503
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