Theorem rspecval 34162

Description: Value of the spectrum of the ring 𝑅. Notation 1.1.1 of [EGA] p. 80. (Contributed by Thierry Arnoux, 2-Jun-2024.)

Assertion

Ref	Expression
rspecval	⊢ (𝑅 ∈ Ring → (Spec‘𝑅) = ((IDLsrg‘𝑅) ↾s (PrmIdeal‘𝑅)))

Proof of Theorem rspecval

Dummy variable 𝑟 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6868	. . 3 ⊢ (𝑟 = 𝑅 → (IDLsrg‘𝑟) = (IDLsrg‘𝑅))
2		fveq2 6868	. . 3 ⊢ (𝑟 = 𝑅 → (PrmIdeal‘𝑟) = (PrmIdeal‘𝑅))
3	1, 2	oveq12d 7415	. 2 ⊢ (𝑟 = 𝑅 → ((IDLsrg‘𝑟) ↾s (PrmIdeal‘𝑟)) = ((IDLsrg‘𝑅) ↾s (PrmIdeal‘𝑅)))
4		df-rspec 34161	. 2 ⊢ Spec = (𝑟 ∈ Ring ↦ ((IDLsrg‘𝑟) ↾s (PrmIdeal‘𝑟)))
5		ovex 7430	. 2 ⊢ ((IDLsrg‘𝑅) ↾s (PrmIdeal‘𝑅)) ∈ V
6	3, 4, 5	fvmpt 6976	1 ⊢ (𝑅 ∈ Ring → (Spec‘𝑅) = ((IDLsrg‘𝑅) ↾s (PrmIdeal‘𝑅)))

Syntax hints:   → wi 4   = wceq 1561   ∈ wcel 2143  ‘cfv 6522  (class class class)co 7397   ↾_s cress 17267  Ringcrg 20284  PrmIdealcprmidl 33622  IDLsrgcidlsrg 33697  Speccrspec 34160

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5247  ax-nul 5257  ax-pr 5391

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-nf 1805  df-sb 2092  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3078  df-rex 3088  df-rab 3416  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5102  df-opab 5164  df-mpt 5183  df-id 5543  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-iota 6478  df-fun 6524  df-fv 6530  df-ov 7400  df-rspec 34161

This theorem is referenced by:  rspecbas  34163  rspectset  34164  rspectopn  34165