Theorem rspecval 33862

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 6865	. . 3 ⊢ (𝑟 = 𝑅 → (IDLsrg‘𝑟) = (IDLsrg‘𝑅))
2		fveq2 6865	. . 3 ⊢ (𝑟 = 𝑅 → (PrmIdeal‘𝑟) = (PrmIdeal‘𝑅))
3	1, 2	oveq12d 7412	. 2 ⊢ (𝑟 = 𝑅 → ((IDLsrg‘𝑟) ↾s (PrmIdeal‘𝑟)) = ((IDLsrg‘𝑅) ↾s (PrmIdeal‘𝑅)))
4		df-rspec 33861	. 2 ⊢ Spec = (𝑟 ∈ Ring ↦ ((IDLsrg‘𝑟) ↾s (PrmIdeal‘𝑟)))
5		ovex 7427	. 2 ⊢ ((IDLsrg‘𝑅) ↾s (PrmIdeal‘𝑅)) ∈ V
6	3, 4, 5	fvmpt 6975	1 ⊢ (𝑅 ∈ Ring → (Spec‘𝑅) = ((IDLsrg‘𝑅) ↾s (PrmIdeal‘𝑅)))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  ‘cfv 6519  (class class class)co 7394   ↾_s cress 17206  Ringcrg 20148  PrmIdealcprmidl 33414  IDLsrgcidlsrg 33479  Speccrspec 33860

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5259  ax-nul 5269  ax-pr 5395

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2880  df-ne 2928  df-ral 3047  df-rex 3056  df-rab 3412  df-v 3457  df-dif 3925  df-un 3927  df-ss 3939  df-nul 4305  df-if 4497  df-sn 4598  df-pr 4600  df-op 4604  df-uni 4880  df-br 5116  df-opab 5178  df-mpt 5197  df-id 5541  df-xp 5652  df-rel 5653  df-cnv 5654  df-co 5655  df-dm 5656  df-iota 6472  df-fun 6521  df-fv 6527  df-ov 7397  df-rspec 33861

This theorem is referenced by:  rspecbas  33863  rspectset  33864  rspectopn  33865