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Theorem fvimage 31733
Description: The 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 3202 . 2 (𝐴𝑉𝐴 ∈ V)
2 imaeq2 5431 . . 3 (𝑥 = 𝐴 → (𝑅𝑥) = (𝑅𝐴))
3 imageval 31732 . . 3 Image𝑅 = (𝑥 ∈ V ↦ (𝑅𝑥))
42, 3fvmptg 6247 . 2 ((𝐴 ∈ V ∧ (𝑅𝐴) ∈ 𝑊) → (Image𝑅𝐴) = (𝑅𝐴))
51, 4sylan 488 1 ((𝐴𝑉 ∧ (𝑅𝐴) ∈ 𝑊) → (Image𝑅𝐴) = (𝑅𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  Vcvv 3190  cima 5087  cfv 5857  Imagecimage 31641
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-symdif 3828  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-mpt 4685  df-eprel 4995  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-fo 5863  df-fv 5865  df-1st 7128  df-2nd 7129  df-txp 31655  df-image 31665
This theorem is referenced by: (None)
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