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Theorem rngqiprngimfv 21187
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 (πœ‘ β†’ 𝑅 ∈ Rng)
rng2idlring.i (πœ‘ β†’ 𝐼 ∈ (2Idealβ€˜π‘…))
rng2idlring.j 𝐽 = (𝑅 β†Ύs 𝐼)
rng2idlring.u (πœ‘ β†’ 𝐽 ∈ Ring)
rng2idlring.b 𝐡 = (Baseβ€˜π‘…)
rng2idlring.t Β· = (.rβ€˜π‘…)
rng2idlring.1 1 = (1rβ€˜π½)
rngqiprngim.g ∼ = (𝑅 ~QG 𝐼)
rngqiprngim.q 𝑄 = (𝑅 /s ∼ )
rngqiprngim.c 𝐢 = (Baseβ€˜π‘„)
rngqiprngim.p 𝑃 = (𝑄 Γ—s 𝐽)
rngqiprngim.f 𝐹 = (π‘₯ ∈ 𝐡 ↦ ⟨[π‘₯] ∼ , ( 1 Β· π‘₯)⟩)
Assertion
Ref Expression
rngqiprngimfv ((πœ‘ ∧ 𝐴 ∈ 𝐡) β†’ (πΉβ€˜π΄) = ⟨[𝐴] ∼ , ( 1 Β· 𝐴)⟩)
Distinct variable groups:   π‘₯,𝐢   π‘₯,𝐼   π‘₯,𝐡   πœ‘,π‘₯   π‘₯,𝐴   π‘₯, ∼   π‘₯, 1   π‘₯, Β·
Allowed substitution hints:   𝑃(π‘₯)   𝑄(π‘₯)   𝑅(π‘₯)   𝐹(π‘₯)   𝐽(π‘₯)

Proof of Theorem rngqiprngimfv
StepHypRef Expression
1 rngqiprngim.f . . 3 𝐹 = (π‘₯ ∈ 𝐡 ↦ ⟨[π‘₯] ∼ , ( 1 Β· π‘₯)⟩)
21a1i 11 . 2 ((πœ‘ ∧ 𝐴 ∈ 𝐡) β†’ 𝐹 = (π‘₯ ∈ 𝐡 ↦ ⟨[π‘₯] ∼ , ( 1 Β· π‘₯)⟩))
3 eceq1 8756 . . . 4 (π‘₯ = 𝐴 β†’ [π‘₯] ∼ = [𝐴] ∼ )
4 oveq2 7421 . . . 4 (π‘₯ = 𝐴 β†’ ( 1 Β· π‘₯) = ( 1 Β· 𝐴))
53, 4opeq12d 4878 . . 3 (π‘₯ = 𝐴 β†’ ⟨[π‘₯] ∼ , ( 1 Β· π‘₯)⟩ = ⟨[𝐴] ∼ , ( 1 Β· 𝐴)⟩)
65adantl 480 . 2 (((πœ‘ ∧ 𝐴 ∈ 𝐡) ∧ π‘₯ = 𝐴) β†’ ⟨[π‘₯] ∼ , ( 1 Β· π‘₯)⟩ = ⟨[𝐴] ∼ , ( 1 Β· 𝐴)⟩)
7 simpr 483 . 2 ((πœ‘ ∧ 𝐴 ∈ 𝐡) β†’ 𝐴 ∈ 𝐡)
8 opex 5461 . . 3 ⟨[𝐴] ∼ , ( 1 Β· 𝐴)⟩ ∈ V
98a1i 11 . 2 ((πœ‘ ∧ 𝐴 ∈ 𝐡) β†’ ⟨[𝐴] ∼ , ( 1 Β· 𝐴)⟩ ∈ V)
102, 6, 7, 9fvmptd 7005 1 ((πœ‘ ∧ 𝐴 ∈ 𝐡) β†’ (πΉβ€˜π΄) = ⟨[𝐴] ∼ , ( 1 Β· 𝐴)⟩)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098  Vcvv 3463  βŸ¨cop 4631   ↦ cmpt 5227  β€˜cfv 6543  (class class class)co 7413  [cec 8716  Basecbs 17174   β†Ύs cress 17203  .rcmulr 17228   /s cqus 17481   Γ—s cxps 17482   ~QG cqg 19076  Rngcrng 20091  1rcur 20120  Ringcrg 20172  2Idealc2idl 21142
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5295  ax-nul 5302  ax-pr 5424
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4320  df-if 4526  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-br 5145  df-opab 5207  df-mpt 5228  df-id 5571  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-iota 6495  df-fun 6545  df-fv 6551  df-ov 7416  df-ec 8720
This theorem is referenced by:  rngqiprngghm  21188  rngqiprngimf1  21189  rngqiprngimfo  21190  rngqiprnglin  21191  rngqiprngfu  21206
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