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Theorem imain 6438
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 6437 . . 3 (Fun 𝐹 → (𝐹 “ (𝐴 ∖ (𝐴𝐵))) = ((𝐹𝐴) ∖ (𝐹 “ (𝐴𝐵))))
2 imadif 6437 . . . 4 (Fun 𝐹 → (𝐹 “ (𝐴𝐵)) = ((𝐹𝐴) ∖ (𝐹𝐵)))
32difeq2d 4103 . . 3 (Fun 𝐹 → ((𝐹𝐴) ∖ (𝐹 “ (𝐴𝐵))) = ((𝐹𝐴) ∖ ((𝐹𝐴) ∖ (𝐹𝐵))))
41, 3eqtrd 2861 . 2 (Fun 𝐹 → (𝐹 “ (𝐴 ∖ (𝐴𝐵))) = ((𝐹𝐴) ∖ ((𝐹𝐴) ∖ (𝐹𝐵))))
5 dfin4 4248 . . 3 (𝐴𝐵) = (𝐴 ∖ (𝐴𝐵))
65imaeq2i 5926 . 2 (𝐹 “ (𝐴𝐵)) = (𝐹 “ (𝐴 ∖ (𝐴𝐵)))
7 dfin4 4248 . 2 ((𝐹𝐴) ∩ (𝐹𝐵)) = ((𝐹𝐴) ∖ ((𝐹𝐴) ∖ (𝐹𝐵)))
84, 6, 73eqtr4g 2886 1 (Fun 𝐹 → (𝐹 “ (𝐴𝐵)) = ((𝐹𝐴) ∩ (𝐹𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1530  cdif 3937  cin 3939  ccnv 5553  cima 5557  Fun wfun 6348
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pr 5326
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-op 4571  df-br 5064  df-opab 5126  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-fun 6356
This theorem is referenced by:  inpreima  6832  rnelfmlem  22495  fmfnfmlem3  22499  spthispth  27440  swrdrndisj  30564  ballotlemfrc  31689  poimirlem1  34779  poimirlem2  34780  poimirlem3  34781  poimirlem4  34782  poimirlem6  34784  poimirlem7  34785  poimirlem11  34789  poimirlem12  34790  poimirlem16  34794  poimirlem17  34795  poimirlem19  34797  poimirlem20  34798  poimirlem23  34801  poimirlem24  34802  poimirlem25  34803  poimirlem29  34807  poimirlem31  34809
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