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Theorem funimacnv 6215
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 funcnvres2 6214 . . . 4 (Fun 𝐹(𝐹𝐴) = (𝐹 ↾ (𝐹𝐴)))
21rneqd 5598 . . 3 (Fun 𝐹 → ran (𝐹𝐴) = ran (𝐹 ↾ (𝐹𝐴)))
3 df-ima 5368 . . 3 (𝐹 “ (𝐹𝐴)) = ran (𝐹 ↾ (𝐹𝐴))
42, 3syl6reqr 2832 . 2 (Fun 𝐹 → (𝐹 “ (𝐹𝐴)) = ran (𝐹𝐴))
5 df-rn 5366 . . . 4 ran 𝐹 = dom 𝐹
65ineq2i 4033 . . 3 (𝐴 ∩ ran 𝐹) = (𝐴 ∩ dom 𝐹)
7 dmres 5668 . . 3 dom (𝐹𝐴) = (𝐴 ∩ dom 𝐹)
8 dfdm4 5561 . . 3 dom (𝐹𝐴) = ran (𝐹𝐴)
96, 7, 83eqtr2ri 2808 . 2 ran (𝐹𝐴) = (𝐴 ∩ ran 𝐹)
104, 9syl6eq 2829 1 (Fun 𝐹 → (𝐹 “ (𝐹𝐴)) = (𝐴 ∩ ran 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1601  cin 3790  ccnv 5354  dom cdm 5355  ran crn 5356  cres 5357  cima 5358  Fun wfun 6129
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-sep 5017  ax-nul 5025  ax-pr 5138
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3399  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-nul 4141  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-br 4887  df-opab 4949  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-fun 6137
This theorem is referenced by:  funimass1  6216  funimass2  6217  isercolllem2  14804  isercolllem3  14805  isercoll  14806  cncls  21486  preimane  30035  fnpreimac  30036  ffsrn  30070  cvmliftlem15  31879
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