Theorem rngqiprngimfv 21184

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 8687	. . . 4 ⊢ (𝑥 = 𝐴 → [𝑥] ∼ = [𝐴] ∼ )
4		oveq2 7377	. . . 4 ⊢ (𝑥 = 𝐴 → ( 1 · 𝑥) = ( 1 · 𝐴))
5	3, 4	opeq12d 4841	. . 3 ⊢ (𝑥 = 𝐴 → ⟨[𝑥] ∼ , ( 1 · 𝑥)⟩ = ⟨[𝐴] ∼ , ( 1 · 𝐴)⟩)
6	5	adantl 481	. 2 ⊢ (((𝜑 ∧ 𝐴 ∈ 𝐵) ∧ 𝑥 = 𝐴) → ⟨[𝑥] ∼ , ( 1 · 𝑥)⟩ = ⟨[𝐴] ∼ , ( 1 · 𝐴)⟩)
7		simpr 484	. 2 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝐵) → 𝐴 ∈ 𝐵)
8		opex 5419	. . 3 ⊢ ⟨[𝐴] ∼ , ( 1 · 𝐴)⟩ ∈ V
9	8	a1i 11	. 2 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝐵) → ⟨[𝐴] ∼ , ( 1 · 𝐴)⟩ ∈ V)
10	2, 6, 7, 9	fvmptd 6957	1 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝐵) → (𝐹‘𝐴) = ⟨[𝐴] ∼ , ( 1 · 𝐴)⟩)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3444  ⟨cop 4591   ↦ cmpt 5183  ‘cfv 6499  (class class class)co 7369  [cec 8646  Basecbs 17155   ↾_s cress 17176  ._rcmulr 17197   /_s cqus 17444   ×_s cxps 17445   ~_QG cqg 19030  Rngcrng 20037  1_rcur 20066  Ringcrg 20118  2Idealc2idl 21135

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 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fun 6501  df-fv 6507  df-ov 7372  df-ec 8650

This theorem is referenced by:  rngqiprngghm  21185  rngqiprngimf1  21186  rngqiprngimfo  21187  rngqiprnglin  21188  rngqiprngfu  21203