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Theorem fimacnvinrn 6839
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 6833 . 2 (Fun 𝐹 → (𝐹 “ (𝐴 ∩ ran 𝐹)) = ((𝐹𝐴) ∩ (𝐹 “ ran 𝐹)))
2 funforn 6596 . . . . 5 (Fun 𝐹𝐹:dom 𝐹onto→ran 𝐹)
3 fof 6589 . . . . 5 (𝐹:dom 𝐹onto→ran 𝐹𝐹:dom 𝐹⟶ran 𝐹)
42, 3sylbi 219 . . . 4 (Fun 𝐹𝐹:dom 𝐹⟶ran 𝐹)
5 fimacnv 6838 . . . 4 (𝐹:dom 𝐹⟶ran 𝐹 → (𝐹 “ ran 𝐹) = dom 𝐹)
64, 5syl 17 . . 3 (Fun 𝐹 → (𝐹 “ ran 𝐹) = dom 𝐹)
76ineq2d 4188 . 2 (Fun 𝐹 → ((𝐹𝐴) ∩ (𝐹 “ ran 𝐹)) = ((𝐹𝐴) ∩ dom 𝐹))
8 cnvresima 6086 . . 3 ((𝐹 ↾ dom 𝐹) “ 𝐴) = ((𝐹𝐴) ∩ dom 𝐹)
9 resdm2 6087 . . . . . 6 (𝐹 ↾ dom 𝐹) = 𝐹
10 funrel 6371 . . . . . . 7 (Fun 𝐹 → Rel 𝐹)
11 dfrel2 6045 . . . . . . 7 (Rel 𝐹𝐹 = 𝐹)
1210, 11sylib 220 . . . . . 6 (Fun 𝐹𝐹 = 𝐹)
139, 12syl5eq 2868 . . . . 5 (Fun 𝐹 → (𝐹 ↾ dom 𝐹) = 𝐹)
1413cnveqd 5745 . . . 4 (Fun 𝐹(𝐹 ↾ dom 𝐹) = 𝐹)
1514imaeq1d 5927 . . 3 (Fun 𝐹 → ((𝐹 ↾ dom 𝐹) “ 𝐴) = (𝐹𝐴))
168, 15syl5eqr 2870 . 2 (Fun 𝐹 → ((𝐹𝐴) ∩ dom 𝐹) = (𝐹𝐴))
171, 7, 163eqtrrd 2861 1 (Fun 𝐹 → (𝐹𝐴) = (𝐹 “ (𝐴 ∩ ran 𝐹)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  cin 3934  ccnv 5553  dom cdm 5554  ran crn 5555  cres 5556  cima 5557  Rel wrel 5559  Fun wfun 6348  wf 6350  ontowfo 6352
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pr 5329
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4838  df-br 5066  df-opab 5128  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-fo 6360  df-fv 6362
This theorem is referenced by:  fimacnvinrn2  6840
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