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Theorem difpreima 7073
Description: Preimage of a difference. (Contributed by Mario Carneiro, 14-Jun-2016.)
Assertion
Ref Expression
difpreima (Fun 𝐹 → (𝐹 “ (𝐴𝐵)) = ((𝐹𝐴) ∖ (𝐹𝐵)))

Proof of Theorem difpreima
StepHypRef Expression
1 funcnvcnv 6621 . 2 (Fun 𝐹 → Fun 𝐹)
2 imadif 6638 . 2 (Fun 𝐹 → (𝐹 “ (𝐴𝐵)) = ((𝐹𝐴) ∖ (𝐹𝐵)))
31, 2syl 17 1 (Fun 𝐹 → (𝐹 “ (𝐴𝐵)) = ((𝐹𝐴) ∖ (𝐹𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  cdif 3941  ccnv 5677  cima 5681  Fun wfun 6543
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5150  df-opab 5212  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-fun 6551
This theorem is referenced by:  gsumpropd2lem  18642  supppreima  32553  elrspunidl  33240  fsumcvg4  33682  zrhunitpreima  33710  imambfm  34013  carsggect  34069  sibfof  34091  eulerpartlemmf  34126  itg2addnclem  37275  itg2addnclem2  37276  smfresal  46314
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