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Theorem fimacnvinrn 6256
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 6250 . 2 (Fun 𝐹 → (𝐹 “ (𝐴 ∩ ran 𝐹)) = ((𝐹𝐴) ∩ (𝐹 “ ran 𝐹)))
2 funforn 6035 . . . . 5 (Fun 𝐹𝐹:dom 𝐹onto→ran 𝐹)
3 fof 6028 . . . . 5 (𝐹:dom 𝐹onto→ran 𝐹𝐹:dom 𝐹⟶ran 𝐹)
42, 3sylbi 206 . . . 4 (Fun 𝐹𝐹:dom 𝐹⟶ran 𝐹)
5 fimacnv 6255 . . . 4 (𝐹:dom 𝐹⟶ran 𝐹 → (𝐹 “ ran 𝐹) = dom 𝐹)
64, 5syl 17 . . 3 (Fun 𝐹 → (𝐹 “ ran 𝐹) = dom 𝐹)
76ineq2d 3776 . 2 (Fun 𝐹 → ((𝐹𝐴) ∩ (𝐹 “ ran 𝐹)) = ((𝐹𝐴) ∩ dom 𝐹))
8 cnvresima 5541 . . 3 ((𝐹 ↾ dom 𝐹) “ 𝐴) = ((𝐹𝐴) ∩ dom 𝐹)
9 resdm2 5542 . . . . . 6 (𝐹 ↾ dom 𝐹) = 𝐹
10 funrel 5821 . . . . . . 7 (Fun 𝐹 → Rel 𝐹)
11 dfrel2 5502 . . . . . . 7 (Rel 𝐹𝐹 = 𝐹)
1210, 11sylib 207 . . . . . 6 (Fun 𝐹𝐹 = 𝐹)
139, 12syl5eq 2656 . . . . 5 (Fun 𝐹 → (𝐹 ↾ dom 𝐹) = 𝐹)
1413cnveqd 5220 . . . 4 (Fun 𝐹(𝐹 ↾ dom 𝐹) = 𝐹)
1514imaeq1d 5384 . . 3 (Fun 𝐹 → ((𝐹 ↾ dom 𝐹) “ 𝐴) = (𝐹𝐴))
168, 15syl5eqr 2658 . 2 (Fun 𝐹 → ((𝐹𝐴) ∩ dom 𝐹) = (𝐹𝐴))
171, 7, 163eqtrrd 2649 1 (Fun 𝐹 → (𝐹𝐴) = (𝐹 “ (𝐴 ∩ ran 𝐹)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1475  cin 3539  ccnv 5037  dom cdm 5038  ran crn 5039  cres 5040  cima 5041  Rel wrel 5043  Fun wfun 5798  wf 5800  ontowfo 5802
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fo 5810  df-fv 5812
This theorem is referenced by:  fimacnvinrn2  6257
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