Theorem rngqiprngimfv 21292

Description: The value of the function 𝐹 at an element of (the base set of) a non-unital ring. (Contributed by AV, 24-Feb-2025.)

Hypotheses

Ref	Expression
rng2idlring.r	⊢ (𝜑 → 𝑅 ∈ Rng)
rng2idlring.i	⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))
rng2idlring.j	⊢ 𝐽 = (𝑅 ↾s 𝐼)
rng2idlring.u	⊢ (𝜑 → 𝐽 ∈ Ring)
rng2idlring.b	⊢ 𝐵 = (Base‘𝑅)
rng2idlring.t	⊢ · = (.r‘𝑅)
rng2idlring.1	⊢ 1 = (1r‘𝐽)
rngqiprngim.g	⊢ ∼ = (𝑅 ~QG 𝐼)
rngqiprngim.q	⊢ 𝑄 = (𝑅 /s ∼ )
rngqiprngim.c	⊢ 𝐶 = (Base‘𝑄)
rngqiprngim.p	⊢ 𝑃 = (𝑄 ×s 𝐽)
rngqiprngim.f	⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ ⟨[𝑥] ∼ , ( 1 · 𝑥)⟩)

Assertion

Ref	Expression
rngqiprngimfv	⊢ ((𝜑 ∧ 𝐴 ∈ 𝐵) → (𝐹‘𝐴) = ⟨[𝐴] ∼ , ( 1 · 𝐴)⟩)

Distinct variable groups:   𝑥,𝐶   𝑥,𝐼   𝑥,𝐵   𝜑,𝑥   𝑥,𝐴   𝑥, ∼   𝑥, 1   𝑥, ·

Allowed substitution hints:   𝑃(𝑥)   𝑄(𝑥)   𝑅(𝑥)   𝐹(𝑥)   𝐽(𝑥)

Proof of Theorem rngqiprngimfv

Step	Hyp	Ref	Expression
1		rngqiprngim.f	. . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ ⟨[𝑥] ∼ , ( 1 · 𝑥)⟩)
2	1	a1i 11	. 2 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝐵) → 𝐹 = (𝑥 ∈ 𝐵 ↦ ⟨[𝑥] ∼ , ( 1 · 𝑥)⟩))
3		eceq1 8674	. . . 4 ⊢ (𝑥 = 𝐴 → [𝑥] ∼ = [𝐴] ∼ )
4		oveq2 7365	. . . 4 ⊢ (𝑥 = 𝐴 → ( 1 · 𝑥) = ( 1 · 𝐴))
5	3, 4	opeq12d 4813	. . 3 ⊢ (𝑥 = 𝐴 → ⟨[𝑥] ∼ , ( 1 · 𝑥)⟩ = ⟨[𝐴] ∼ , ( 1 · 𝐴)⟩)
6	5	adantl 482	. 2 ⊢ (((𝜑 ∧ 𝐴 ∈ 𝐵) ∧ 𝑥 = 𝐴) → ⟨[𝑥] ∼ , ( 1 · 𝑥)⟩ = ⟨[𝐴] ∼ , ( 1 · 𝐴)⟩)
7		simpr 485	. 2 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝐵) → 𝐴 ∈ 𝐵)
8		opex 5404	. . 3 ⊢ ⟨[𝐴] ∼ , ( 1 · 𝐴)⟩ ∈ V
9	8	a1i 11	. 2 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝐵) → ⟨[𝐴] ∼ , ( 1 · 𝐴)⟩ ∈ V)
10	2, 6, 7, 9	fvmptd 6944	1 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝐵) → (𝐹‘𝐴) = ⟨[𝐴] ∼ , ( 1 · 𝐴)⟩)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119  Vcvv 3431  ⟨cop 4562   ↦ cmpt 5154  ‘cfv 6486  (class class class)co 7357  [cec 8632  Basecbs 17171   ↾_s cress 17192  ._rcmulr 17213   /_s cqus 17461   ×_s cxps 17462   ~_QG cqg 19090  Rngcrng 20125  1_rcur 20154  Ringcrg 20206  2Idealc2idl 21243

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5219  ax-pr 5363

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4263  df-if 4456  df-sn 4557  df-pr 4559  df-op 4563  df-uni 4840  df-br 5074  df-opab 5136  df-mpt 5155  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6442  df-fun 6488  df-fv 6494  df-ov 7360  df-ec 8636

This theorem is referenced by:  rngqiprngghm  21293  rngqiprngimf1  21294  rngqiprngimfo  21295  rngqiprnglin  21296  rngqiprngfu  21311