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Theorem fvimage 35932
Description: Value of the image functor. (Contributed by Scott Fenton, 4-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Assertion
Ref Expression
fvimage ((𝐴𝑉 ∧ (𝑅𝐴) ∈ 𝑊) → (Image𝑅𝐴) = (𝑅𝐴))

Proof of Theorem fvimage
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 elex 3501 . 2 (𝐴𝑉𝐴 ∈ V)
2 imaeq2 6074 . . 3 (𝑥 = 𝐴 → (𝑅𝑥) = (𝑅𝐴))
3 imageval 35931 . . 3 Image𝑅 = (𝑥 ∈ V ↦ (𝑅𝑥))
42, 3fvmptg 7014 . 2 ((𝐴 ∈ V ∧ (𝑅𝐴) ∈ 𝑊) → (Image𝑅𝐴) = (𝑅𝐴))
51, 4sylan 580 1 ((𝐴𝑉 ∧ (𝑅𝐴) ∈ 𝑊) → (Image𝑅𝐴) = (𝑅𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  Vcvv 3480  cima 5688  cfv 6561  Imagecimage 35841
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-symdif 4253  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-eprel 5584  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fo 6567  df-fv 6569  df-1st 8014  df-2nd 8015  df-txp 35855  df-image 35865
This theorem is referenced by: (None)
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