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Theorem imain 6630
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 6629 . . 3 (Fun 𝐹 → (𝐹 “ (𝐴 ∖ (𝐴𝐵))) = ((𝐹𝐴) ∖ (𝐹 “ (𝐴𝐵))))
2 imadif 6629 . . . 4 (Fun 𝐹 → (𝐹 “ (𝐴𝐵)) = ((𝐹𝐴) ∖ (𝐹𝐵)))
32difeq2d 4121 . . 3 (Fun 𝐹 → ((𝐹𝐴) ∖ (𝐹 “ (𝐴𝐵))) = ((𝐹𝐴) ∖ ((𝐹𝐴) ∖ (𝐹𝐵))))
41, 3eqtrd 2772 . 2 (Fun 𝐹 → (𝐹 “ (𝐴 ∖ (𝐴𝐵))) = ((𝐹𝐴) ∖ ((𝐹𝐴) ∖ (𝐹𝐵))))
5 dfin4 4266 . . 3 (𝐴𝐵) = (𝐴 ∖ (𝐴𝐵))
65imaeq2i 6055 . 2 (𝐹 “ (𝐴𝐵)) = (𝐹 “ (𝐴 ∖ (𝐴𝐵)))
7 dfin4 4266 . 2 ((𝐹𝐴) ∩ (𝐹𝐵)) = ((𝐹𝐴) ∖ ((𝐹𝐴) ∖ (𝐹𝐵)))
84, 6, 73eqtr4g 2797 1 (Fun 𝐹 → (𝐹 “ (𝐴𝐵)) = ((𝐹𝐴) ∩ (𝐹𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541  cdif 3944  cin 3946  ccnv 5674  cima 5678  Fun wfun 6534
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-br 5148  df-opab 5210  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-fun 6542
This theorem is referenced by:  inpreima  7062  rnelfmlem  23447  fmfnfmlem3  23451  spthispth  28972  swrdrndisj  32108  ballotlemfrc  33513  poimirlem1  36477  poimirlem2  36478  poimirlem3  36479  poimirlem4  36480  poimirlem6  36482  poimirlem7  36483  poimirlem11  36487  poimirlem12  36488  poimirlem16  36492  poimirlem17  36493  poimirlem19  36495  poimirlem20  36496  poimirlem23  36499  poimirlem24  36500  poimirlem25  36501  poimirlem29  36505  poimirlem31  36507
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