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Theorem pridlval 37547
Description: The class of prime ideals of a ring 𝑅. (Contributed by Jeff Madsen, 10-Jun-2010.)
Hypotheses
Ref Expression
pridlval.1 𝐺 = (1st β€˜π‘…)
pridlval.2 𝐻 = (2nd β€˜π‘…)
pridlval.3 𝑋 = ran 𝐺
Assertion
Ref Expression
pridlval (𝑅 ∈ RingOps β†’ (PrIdlβ€˜π‘…) = {𝑖 ∈ (Idlβ€˜π‘…) ∣ (𝑖 β‰  𝑋 ∧ βˆ€π‘Ž ∈ (Idlβ€˜π‘…)βˆ€π‘ ∈ (Idlβ€˜π‘…)(βˆ€π‘₯ ∈ π‘Ž βˆ€π‘¦ ∈ 𝑏 (π‘₯𝐻𝑦) ∈ 𝑖 β†’ (π‘Ž βŠ† 𝑖 ∨ 𝑏 βŠ† 𝑖)))})
Distinct variable groups:   𝑅,𝑖,π‘₯,𝑦,π‘Ž,𝑏   𝑖,𝑋   𝑖,𝐻
Allowed substitution hints:   𝐺(π‘₯,𝑦,𝑖,π‘Ž,𝑏)   𝐻(π‘₯,𝑦,π‘Ž,𝑏)   𝑋(π‘₯,𝑦,π‘Ž,𝑏)

Proof of Theorem pridlval
Dummy variable π‘Ÿ is distinct from all other variables.
StepHypRef Expression
1 fveq2 6902 . . 3 (π‘Ÿ = 𝑅 β†’ (Idlβ€˜π‘Ÿ) = (Idlβ€˜π‘…))
2 fveq2 6902 . . . . . . . 8 (π‘Ÿ = 𝑅 β†’ (1st β€˜π‘Ÿ) = (1st β€˜π‘…))
3 pridlval.1 . . . . . . . 8 𝐺 = (1st β€˜π‘…)
42, 3eqtr4di 2786 . . . . . . 7 (π‘Ÿ = 𝑅 β†’ (1st β€˜π‘Ÿ) = 𝐺)
54rneqd 5944 . . . . . 6 (π‘Ÿ = 𝑅 β†’ ran (1st β€˜π‘Ÿ) = ran 𝐺)
6 pridlval.3 . . . . . 6 𝑋 = ran 𝐺
75, 6eqtr4di 2786 . . . . 5 (π‘Ÿ = 𝑅 β†’ ran (1st β€˜π‘Ÿ) = 𝑋)
87neeq2d 2998 . . . 4 (π‘Ÿ = 𝑅 β†’ (𝑖 β‰  ran (1st β€˜π‘Ÿ) ↔ 𝑖 β‰  𝑋))
9 fveq2 6902 . . . . . . . . . . 11 (π‘Ÿ = 𝑅 β†’ (2nd β€˜π‘Ÿ) = (2nd β€˜π‘…))
10 pridlval.2 . . . . . . . . . . 11 𝐻 = (2nd β€˜π‘…)
119, 10eqtr4di 2786 . . . . . . . . . 10 (π‘Ÿ = 𝑅 β†’ (2nd β€˜π‘Ÿ) = 𝐻)
1211oveqd 7443 . . . . . . . . 9 (π‘Ÿ = 𝑅 β†’ (π‘₯(2nd β€˜π‘Ÿ)𝑦) = (π‘₯𝐻𝑦))
1312eleq1d 2814 . . . . . . . 8 (π‘Ÿ = 𝑅 β†’ ((π‘₯(2nd β€˜π‘Ÿ)𝑦) ∈ 𝑖 ↔ (π‘₯𝐻𝑦) ∈ 𝑖))
14132ralbidv 3216 . . . . . . 7 (π‘Ÿ = 𝑅 β†’ (βˆ€π‘₯ ∈ π‘Ž βˆ€π‘¦ ∈ 𝑏 (π‘₯(2nd β€˜π‘Ÿ)𝑦) ∈ 𝑖 ↔ βˆ€π‘₯ ∈ π‘Ž βˆ€π‘¦ ∈ 𝑏 (π‘₯𝐻𝑦) ∈ 𝑖))
1514imbi1d 340 . . . . . 6 (π‘Ÿ = 𝑅 β†’ ((βˆ€π‘₯ ∈ π‘Ž βˆ€π‘¦ ∈ 𝑏 (π‘₯(2nd β€˜π‘Ÿ)𝑦) ∈ 𝑖 β†’ (π‘Ž βŠ† 𝑖 ∨ 𝑏 βŠ† 𝑖)) ↔ (βˆ€π‘₯ ∈ π‘Ž βˆ€π‘¦ ∈ 𝑏 (π‘₯𝐻𝑦) ∈ 𝑖 β†’ (π‘Ž βŠ† 𝑖 ∨ 𝑏 βŠ† 𝑖))))
161, 15raleqbidv 3340 . . . . 5 (π‘Ÿ = 𝑅 β†’ (βˆ€π‘ ∈ (Idlβ€˜π‘Ÿ)(βˆ€π‘₯ ∈ π‘Ž βˆ€π‘¦ ∈ 𝑏 (π‘₯(2nd β€˜π‘Ÿ)𝑦) ∈ 𝑖 β†’ (π‘Ž βŠ† 𝑖 ∨ 𝑏 βŠ† 𝑖)) ↔ βˆ€π‘ ∈ (Idlβ€˜π‘…)(βˆ€π‘₯ ∈ π‘Ž βˆ€π‘¦ ∈ 𝑏 (π‘₯𝐻𝑦) ∈ 𝑖 β†’ (π‘Ž βŠ† 𝑖 ∨ 𝑏 βŠ† 𝑖))))
171, 16raleqbidv 3340 . . . 4 (π‘Ÿ = 𝑅 β†’ (βˆ€π‘Ž ∈ (Idlβ€˜π‘Ÿ)βˆ€π‘ ∈ (Idlβ€˜π‘Ÿ)(βˆ€π‘₯ ∈ π‘Ž βˆ€π‘¦ ∈ 𝑏 (π‘₯(2nd β€˜π‘Ÿ)𝑦) ∈ 𝑖 β†’ (π‘Ž βŠ† 𝑖 ∨ 𝑏 βŠ† 𝑖)) ↔ βˆ€π‘Ž ∈ (Idlβ€˜π‘…)βˆ€π‘ ∈ (Idlβ€˜π‘…)(βˆ€π‘₯ ∈ π‘Ž βˆ€π‘¦ ∈ 𝑏 (π‘₯𝐻𝑦) ∈ 𝑖 β†’ (π‘Ž βŠ† 𝑖 ∨ 𝑏 βŠ† 𝑖))))
188, 17anbi12d 630 . . 3 (π‘Ÿ = 𝑅 β†’ ((𝑖 β‰  ran (1st β€˜π‘Ÿ) ∧ βˆ€π‘Ž ∈ (Idlβ€˜π‘Ÿ)βˆ€π‘ ∈ (Idlβ€˜π‘Ÿ)(βˆ€π‘₯ ∈ π‘Ž βˆ€π‘¦ ∈ 𝑏 (π‘₯(2nd β€˜π‘Ÿ)𝑦) ∈ 𝑖 β†’ (π‘Ž βŠ† 𝑖 ∨ 𝑏 βŠ† 𝑖))) ↔ (𝑖 β‰  𝑋 ∧ βˆ€π‘Ž ∈ (Idlβ€˜π‘…)βˆ€π‘ ∈ (Idlβ€˜π‘…)(βˆ€π‘₯ ∈ π‘Ž βˆ€π‘¦ ∈ 𝑏 (π‘₯𝐻𝑦) ∈ 𝑖 β†’ (π‘Ž βŠ† 𝑖 ∨ 𝑏 βŠ† 𝑖)))))
191, 18rabeqbidv 3448 . 2 (π‘Ÿ = 𝑅 β†’ {𝑖 ∈ (Idlβ€˜π‘Ÿ) ∣ (𝑖 β‰  ran (1st β€˜π‘Ÿ) ∧ βˆ€π‘Ž ∈ (Idlβ€˜π‘Ÿ)βˆ€π‘ ∈ (Idlβ€˜π‘Ÿ)(βˆ€π‘₯ ∈ π‘Ž βˆ€π‘¦ ∈ 𝑏 (π‘₯(2nd β€˜π‘Ÿ)𝑦) ∈ 𝑖 β†’ (π‘Ž βŠ† 𝑖 ∨ 𝑏 βŠ† 𝑖)))} = {𝑖 ∈ (Idlβ€˜π‘…) ∣ (𝑖 β‰  𝑋 ∧ βˆ€π‘Ž ∈ (Idlβ€˜π‘…)βˆ€π‘ ∈ (Idlβ€˜π‘…)(βˆ€π‘₯ ∈ π‘Ž βˆ€π‘¦ ∈ 𝑏 (π‘₯𝐻𝑦) ∈ 𝑖 β†’ (π‘Ž βŠ† 𝑖 ∨ 𝑏 βŠ† 𝑖)))})
20 df-pridl 37525 . 2 PrIdl = (π‘Ÿ ∈ RingOps ↦ {𝑖 ∈ (Idlβ€˜π‘Ÿ) ∣ (𝑖 β‰  ran (1st β€˜π‘Ÿ) ∧ βˆ€π‘Ž ∈ (Idlβ€˜π‘Ÿ)βˆ€π‘ ∈ (Idlβ€˜π‘Ÿ)(βˆ€π‘₯ ∈ π‘Ž βˆ€π‘¦ ∈ 𝑏 (π‘₯(2nd β€˜π‘Ÿ)𝑦) ∈ 𝑖 β†’ (π‘Ž βŠ† 𝑖 ∨ 𝑏 βŠ† 𝑖)))})
21 fvex 6915 . . 3 (Idlβ€˜π‘…) ∈ V
2221rabex 5338 . 2 {𝑖 ∈ (Idlβ€˜π‘…) ∣ (𝑖 β‰  𝑋 ∧ βˆ€π‘Ž ∈ (Idlβ€˜π‘…)βˆ€π‘ ∈ (Idlβ€˜π‘…)(βˆ€π‘₯ ∈ π‘Ž βˆ€π‘¦ ∈ 𝑏 (π‘₯𝐻𝑦) ∈ 𝑖 β†’ (π‘Ž βŠ† 𝑖 ∨ 𝑏 βŠ† 𝑖)))} ∈ V
2319, 20, 22fvmpt 7010 1 (𝑅 ∈ RingOps β†’ (PrIdlβ€˜π‘…) = {𝑖 ∈ (Idlβ€˜π‘…) ∣ (𝑖 β‰  𝑋 ∧ βˆ€π‘Ž ∈ (Idlβ€˜π‘…)βˆ€π‘ ∈ (Idlβ€˜π‘…)(βˆ€π‘₯ ∈ π‘Ž βˆ€π‘¦ ∈ 𝑏 (π‘₯𝐻𝑦) ∈ 𝑖 β†’ (π‘Ž βŠ† 𝑖 ∨ 𝑏 βŠ† 𝑖)))})
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   ∨ wo 845   = wceq 1533   ∈ wcel 2098   β‰  wne 2937  βˆ€wral 3058  {crab 3430   βŠ† wss 3949  ran crn 5683  β€˜cfv 6553  (class class class)co 7426  1st c1st 7999  2nd c2nd 8000  RingOpscrngo 37408  Idlcidl 37521  PrIdlcpridl 37522
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-iota 6505  df-fun 6555  df-fv 6561  df-ov 7429  df-pridl 37525
This theorem is referenced by:  ispridl  37548
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