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Theorem pridlval 36495
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 6843 . . 3 (π‘Ÿ = 𝑅 β†’ (Idlβ€˜π‘Ÿ) = (Idlβ€˜π‘…))
2 fveq2 6843 . . . . . . . 8 (π‘Ÿ = 𝑅 β†’ (1st β€˜π‘Ÿ) = (1st β€˜π‘…))
3 pridlval.1 . . . . . . . 8 𝐺 = (1st β€˜π‘…)
42, 3eqtr4di 2795 . . . . . . 7 (π‘Ÿ = 𝑅 β†’ (1st β€˜π‘Ÿ) = 𝐺)
54rneqd 5894 . . . . . 6 (π‘Ÿ = 𝑅 β†’ ran (1st β€˜π‘Ÿ) = ran 𝐺)
6 pridlval.3 . . . . . 6 𝑋 = ran 𝐺
75, 6eqtr4di 2795 . . . . 5 (π‘Ÿ = 𝑅 β†’ ran (1st β€˜π‘Ÿ) = 𝑋)
87neeq2d 3005 . . . 4 (π‘Ÿ = 𝑅 β†’ (𝑖 β‰  ran (1st β€˜π‘Ÿ) ↔ 𝑖 β‰  𝑋))
9 fveq2 6843 . . . . . . . . . . 11 (π‘Ÿ = 𝑅 β†’ (2nd β€˜π‘Ÿ) = (2nd β€˜π‘…))
10 pridlval.2 . . . . . . . . . . 11 𝐻 = (2nd β€˜π‘…)
119, 10eqtr4di 2795 . . . . . . . . . 10 (π‘Ÿ = 𝑅 β†’ (2nd β€˜π‘Ÿ) = 𝐻)
1211oveqd 7375 . . . . . . . . 9 (π‘Ÿ = 𝑅 β†’ (π‘₯(2nd β€˜π‘Ÿ)𝑦) = (π‘₯𝐻𝑦))
1312eleq1d 2823 . . . . . . . 8 (π‘Ÿ = 𝑅 β†’ ((π‘₯(2nd β€˜π‘Ÿ)𝑦) ∈ 𝑖 ↔ (π‘₯𝐻𝑦) ∈ 𝑖))
14132ralbidv 3213 . . . . . . 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 342 . . . . . 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 3320 . . . . 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 3320 . . . 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 632 . . 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 3425 . 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 36473 . 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 6856 . . 3 (Idlβ€˜π‘…) ∈ V
2221rabex 5290 . 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 6949 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 397   ∨ wo 846   = wceq 1542   ∈ wcel 2107   β‰  wne 2944  βˆ€wral 3065  {crab 3408   βŠ† wss 3911  ran crn 5635  β€˜cfv 6497  (class class class)co 7358  1st c1st 7920  2nd c2nd 7921  RingOpscrngo 36356  Idlcidl 36469  PrIdlcpridl 36470
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-iota 6449  df-fun 6499  df-fv 6505  df-ov 7361  df-pridl 36473
This theorem is referenced by:  ispridl  36496
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