Theorem rngqiprngimfv 21061

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 8747	. . . 4 ⊢ (𝑥 = 𝐴 → [𝑥] ∼ = [𝐴] ∼ )
4		oveq2 7420	. . . 4 ⊢ (𝑥 = 𝐴 → ( 1 · 𝑥) = ( 1 · 𝐴))
5	3, 4	opeq12d 4881	. . 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 5464	. . 3 ⊢ ⟨[𝐴] ∼ , ( 1 · 𝐴)⟩ ∈ V
9	8	a1i 11	. 2 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝐵) → ⟨[𝐴] ∼ , ( 1 · 𝐴)⟩ ∈ V)
10	2, 6, 7, 9	fvmptd 7005	1 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝐵) → (𝐹‘𝐴) = ⟨[𝐴] ∼ , ( 1 · 𝐴)⟩)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2105  Vcvv 3473  ⟨cop 4634   ↦ cmpt 5231  ‘cfv 6543  (class class class)co 7412  [cec 8707  Basecbs 17151   ↾_s cress 17180  ._rcmulr 17205   /_s cqus 17458   ×_s cxps 17459   ~_QG cqg 19042  Rngcrng 20050  1_rcur 20079  Ringcrg 20131  2Idealc2idl 21009

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fv 6551  df-ov 7415  df-ec 8711

This theorem is referenced by:  rngqiprngghm  21062  rngqiprngimf1  21063  rngqiprngimfo  21064  rngqiprnglin  21065  rngqiprngfu  21080