Theorem rspecval 34496

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 6885	. . 3 ⊢ (𝑟 = 𝑅 → (IDLsrg‘𝑟) = (IDLsrg‘𝑅))
2		fveq2 6885	. . 3 ⊢ (𝑟 = 𝑅 → (PrmIdeal‘𝑟) = (PrmIdeal‘𝑅))
3	1, 2	oveq12d 7438	. 2 ⊢ (𝑟 = 𝑅 → ((IDLsrg‘𝑟) ↾s (PrmIdeal‘𝑟)) = ((IDLsrg‘𝑅) ↾s (PrmIdeal‘𝑅)))
4		df-rspec 34495	. 2 ⊢ Spec = (𝑟 ∈ Ring ↦ ((IDLsrg‘𝑟) ↾s (PrmIdeal‘𝑟)))
5		ovex 7453	. 2 ⊢ ((IDLsrg‘𝑅) ↾s (PrmIdeal‘𝑅)) ∈ V
6	3, 4, 5	fvmpt 6993	1 ⊢ (𝑅 ∈ Ring → (Spec‘𝑅) = ((IDLsrg‘𝑅) ↾s (PrmIdeal‘𝑅)))

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2144  ‘cfv 6538  (class class class)co 7420   ↾_s cress 17408  Ringcrg 20459  PrmIdealcprmidl 21616  IDLsrgcidlsrg 34032  Speccrspec 34494

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-iota 6494  df-fun 6540  df-fv 6546  df-ov 7423  df-rspec 34495

This theorem is used by:  rspecbas  34497  rspectset  34498  rspectopn  34499