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Theorem imain 6651
Description: The image of an intersection is the intersection of images. (Contributed by Paul Chapman, 11-Apr-2009.)
Assertion
Ref Expression
imain (Fun 𝐹 → (𝐹 “ (𝐴𝐵)) = ((𝐹𝐴) ∩ (𝐹𝐵)))

Proof of Theorem imain
StepHypRef Expression
1 imadif 6650 . . 3 (Fun 𝐹 → (𝐹 “ (𝐴 ∖ (𝐴𝐵))) = ((𝐹𝐴) ∖ (𝐹 “ (𝐴𝐵))))
2 imadif 6650 . . . 4 (Fun 𝐹 → (𝐹 “ (𝐴𝐵)) = ((𝐹𝐴) ∖ (𝐹𝐵)))
32difeq2d 4126 . . 3 (Fun 𝐹 → ((𝐹𝐴) ∖ (𝐹 “ (𝐴𝐵))) = ((𝐹𝐴) ∖ ((𝐹𝐴) ∖ (𝐹𝐵))))
41, 3eqtrd 2777 . 2 (Fun 𝐹 → (𝐹 “ (𝐴 ∖ (𝐴𝐵))) = ((𝐹𝐴) ∖ ((𝐹𝐴) ∖ (𝐹𝐵))))
5 dfin4 4278 . . 3 (𝐴𝐵) = (𝐴 ∖ (𝐴𝐵))
65imaeq2i 6076 . 2 (𝐹 “ (𝐴𝐵)) = (𝐹 “ (𝐴 ∖ (𝐴𝐵)))
7 dfin4 4278 . 2 ((𝐹𝐴) ∩ (𝐹𝐵)) = ((𝐹𝐴) ∖ ((𝐹𝐴) ∖ (𝐹𝐵)))
84, 6, 73eqtr4g 2802 1 (Fun 𝐹 → (𝐹 “ (𝐴𝐵)) = ((𝐹𝐴) ∩ (𝐹𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  cdif 3948  cin 3950  ccnv 5684  cima 5688  Fun wfun 6555
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-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
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-clab 2715  df-cleq 2729  df-clel 2816  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-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-id 5578  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-fun 6563
This theorem is referenced by:  inpreima  7084  rnelfmlem  23960  fmfnfmlem3  23964  spthispth  29744  swrdrndisj  32942  ballotlemfrc  34529  poimirlem1  37628  poimirlem2  37629  poimirlem3  37630  poimirlem4  37631  poimirlem6  37633  poimirlem7  37634  poimirlem11  37638  poimirlem12  37639  poimirlem16  37643  poimirlem17  37644  poimirlem19  37646  poimirlem20  37647  poimirlem23  37650  poimirlem24  37651  poimirlem25  37652  poimirlem29  37656  poimirlem31  37658
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