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Theorem fimacnvinrn 7090
Description: Taking the converse image of a set can be limited to the range of the function used. (Contributed by Thierry Arnoux, 21-Jan-2017.)
Assertion
Ref Expression
fimacnvinrn (Fun 𝐹 → (𝐹𝐴) = (𝐹 “ (𝐴 ∩ ran 𝐹)))

Proof of Theorem fimacnvinrn
StepHypRef Expression
1 inpreima 7083 . 2 (Fun 𝐹 → (𝐹 “ (𝐴 ∩ ran 𝐹)) = ((𝐹𝐴) ∩ (𝐹 “ ran 𝐹)))
2 funforn 6826 . . . . 5 (Fun 𝐹𝐹:dom 𝐹onto→ran 𝐹)
3 fof 6819 . . . . 5 (𝐹:dom 𝐹onto→ran 𝐹𝐹:dom 𝐹⟶ran 𝐹)
42, 3sylbi 217 . . . 4 (Fun 𝐹𝐹:dom 𝐹⟶ran 𝐹)
5 fimacnv 6757 . . . 4 (𝐹:dom 𝐹⟶ran 𝐹 → (𝐹 “ ran 𝐹) = dom 𝐹)
64, 5syl 17 . . 3 (Fun 𝐹 → (𝐹 “ ran 𝐹) = dom 𝐹)
76ineq2d 4219 . 2 (Fun 𝐹 → ((𝐹𝐴) ∩ (𝐹 “ ran 𝐹)) = ((𝐹𝐴) ∩ dom 𝐹))
8 cnvresima 6249 . . 3 ((𝐹 ↾ dom 𝐹) “ 𝐴) = ((𝐹𝐴) ∩ dom 𝐹)
9 resdm2 6250 . . . . . 6 (𝐹 ↾ dom 𝐹) = 𝐹
10 funrel 6582 . . . . . . 7 (Fun 𝐹 → Rel 𝐹)
11 dfrel2 6208 . . . . . . 7 (Rel 𝐹𝐹 = 𝐹)
1210, 11sylib 218 . . . . . 6 (Fun 𝐹𝐹 = 𝐹)
139, 12eqtrid 2788 . . . . 5 (Fun 𝐹 → (𝐹 ↾ dom 𝐹) = 𝐹)
1413cnveqd 5885 . . . 4 (Fun 𝐹(𝐹 ↾ dom 𝐹) = 𝐹)
1514imaeq1d 6076 . . 3 (Fun 𝐹 → ((𝐹 ↾ dom 𝐹) “ 𝐴) = (𝐹𝐴))
168, 15eqtr3id 2790 . 2 (Fun 𝐹 → ((𝐹𝐴) ∩ dom 𝐹) = (𝐹𝐴))
171, 7, 163eqtrrd 2781 1 (Fun 𝐹 → (𝐹𝐴) = (𝐹 “ (𝐴 ∩ ran 𝐹)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  cin 3949  ccnv 5683  dom cdm 5684  ran crn 5685  cres 5686  cima 5687  Rel wrel 5689  Fun wfun 6554  wf 6556  ontowfo 6558
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-br 5143  df-opab 5205  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-fun 6562  df-fn 6563  df-f 6564  df-fo 6566
This theorem is referenced by:  fimacnvinrn2  7091  preiman0  32720
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