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Theorem imain 6536
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 6535 . . 3 (Fun 𝐹 → (𝐹 “ (𝐴 ∖ (𝐴𝐵))) = ((𝐹𝐴) ∖ (𝐹 “ (𝐴𝐵))))
2 imadif 6535 . . . 4 (Fun 𝐹 → (𝐹 “ (𝐴𝐵)) = ((𝐹𝐴) ∖ (𝐹𝐵)))
32difeq2d 4060 . . 3 (Fun 𝐹 → ((𝐹𝐴) ∖ (𝐹 “ (𝐴𝐵))) = ((𝐹𝐴) ∖ ((𝐹𝐴) ∖ (𝐹𝐵))))
41, 3eqtrd 2773 . 2 (Fun 𝐹 → (𝐹 “ (𝐴 ∖ (𝐴𝐵))) = ((𝐹𝐴) ∖ ((𝐹𝐴) ∖ (𝐹𝐵))))
5 dfin4 4204 . . 3 (𝐴𝐵) = (𝐴 ∖ (𝐴𝐵))
65imaeq2i 5968 . 2 (𝐹 “ (𝐴𝐵)) = (𝐹 “ (𝐴 ∖ (𝐴𝐵)))
7 dfin4 4204 . 2 ((𝐹𝐴) ∩ (𝐹𝐵)) = ((𝐹𝐴) ∖ ((𝐹𝐴) ∖ (𝐹𝐵)))
84, 6, 73eqtr4g 2798 1 (Fun 𝐹 → (𝐹 “ (𝐴𝐵)) = ((𝐹𝐴) ∩ (𝐹𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  cdif 3886  cin 3888  ccnv 5590  cima 5594  Fun wfun 6441
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2103  ax-9 2111  ax-10 2132  ax-11 2149  ax-12 2166  ax-ext 2704  ax-sep 5226  ax-nul 5233  ax-pr 5355
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2063  df-mo 2535  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3060  df-rex 3069  df-rab 3224  df-v 3436  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4260  df-if 4463  df-sn 4565  df-pr 4567  df-op 4571  df-br 5078  df-opab 5140  df-id 5491  df-xp 5597  df-rel 5598  df-cnv 5599  df-co 5600  df-dm 5601  df-rn 5602  df-res 5603  df-ima 5604  df-fun 6449
This theorem is referenced by:  inpreima  6961  rnelfmlem  23131  fmfnfmlem3  23135  spthispth  28122  swrdrndisj  31257  ballotlemfrc  32521  poimirlem1  35806  poimirlem2  35807  poimirlem3  35808  poimirlem4  35809  poimirlem6  35811  poimirlem7  35812  poimirlem11  35816  poimirlem12  35817  poimirlem16  35821  poimirlem17  35822  poimirlem19  35824  poimirlem20  35825  poimirlem23  35828  poimirlem24  35829  poimirlem25  35830  poimirlem29  35834  poimirlem31  35836
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