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Theorem funimacnv 6648
Description: The image of the preimage of a function. (Contributed by NM, 25-May-2004.)
Assertion
Ref Expression
funimacnv (Fun 𝐹 → (𝐹 “ (𝐹𝐴)) = (𝐴 ∩ ran 𝐹))

Proof of Theorem funimacnv
StepHypRef Expression
1 df-ima 5701 . . 3 (𝐹 “ (𝐹𝐴)) = ran (𝐹 ↾ (𝐹𝐴))
2 funcnvres2 6647 . . . 4 (Fun 𝐹(𝐹𝐴) = (𝐹 ↾ (𝐹𝐴)))
32rneqd 5951 . . 3 (Fun 𝐹 → ran (𝐹𝐴) = ran (𝐹 ↾ (𝐹𝐴)))
41, 3eqtr4id 2793 . 2 (Fun 𝐹 → (𝐹 “ (𝐹𝐴)) = ran (𝐹𝐴))
5 df-rn 5699 . . . 4 ran 𝐹 = dom 𝐹
65ineq2i 4224 . . 3 (𝐴 ∩ ran 𝐹) = (𝐴 ∩ dom 𝐹)
7 dmres 6031 . . 3 dom (𝐹𝐴) = (𝐴 ∩ dom 𝐹)
8 dfdm4 5908 . . 3 dom (𝐹𝐴) = ran (𝐹𝐴)
96, 7, 83eqtr2ri 2769 . 2 ran (𝐹𝐴) = (𝐴 ∩ ran 𝐹)
104, 9eqtrdi 2790 1 (Fun 𝐹 → (𝐹 “ (𝐹𝐴)) = (𝐴 ∩ ran 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1536  cin 3961  ccnv 5687  dom cdm 5688  ran crn 5689  cres 5690  cima 5691  Fun wfun 6556
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5148  df-opab 5210  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-fun 6564
This theorem is referenced by:  funimass1  6649  funimass2  6650  rescnvimafod  7092  isercolllem2  15698  isercolllem3  15699  isercoll  15700  cncls  23297  preimane  32686  fnpreimac  32687  ffsrn  32746  gsumhashmul  33046  zarcmplem  33841  cvmliftlem15  35282  fcoreslem2  47013  imaelsetpreimafv  47319
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