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Theorem fimacnvinrn 6499
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 6493 . 2 (Fun 𝐹 → (𝐹 “ (𝐴 ∩ ran 𝐹)) = ((𝐹𝐴) ∩ (𝐹 “ ran 𝐹)))
2 funforn 6271 . . . . 5 (Fun 𝐹𝐹:dom 𝐹onto→ran 𝐹)
3 fof 6264 . . . . 5 (𝐹:dom 𝐹onto→ran 𝐹𝐹:dom 𝐹⟶ran 𝐹)
42, 3sylbi 207 . . . 4 (Fun 𝐹𝐹:dom 𝐹⟶ran 𝐹)
5 fimacnv 6498 . . . 4 (𝐹:dom 𝐹⟶ran 𝐹 → (𝐹 “ ran 𝐹) = dom 𝐹)
64, 5syl 17 . . 3 (Fun 𝐹 → (𝐹 “ ran 𝐹) = dom 𝐹)
76ineq2d 3945 . 2 (Fun 𝐹 → ((𝐹𝐴) ∩ (𝐹 “ ran 𝐹)) = ((𝐹𝐴) ∩ dom 𝐹))
8 cnvresima 5772 . . 3 ((𝐹 ↾ dom 𝐹) “ 𝐴) = ((𝐹𝐴) ∩ dom 𝐹)
9 resdm2 5773 . . . . . 6 (𝐹 ↾ dom 𝐹) = 𝐹
10 funrel 6054 . . . . . . 7 (Fun 𝐹 → Rel 𝐹)
11 dfrel2 5729 . . . . . . 7 (Rel 𝐹𝐹 = 𝐹)
1210, 11sylib 208 . . . . . 6 (Fun 𝐹𝐹 = 𝐹)
139, 12syl5eq 2794 . . . . 5 (Fun 𝐹 → (𝐹 ↾ dom 𝐹) = 𝐹)
1413cnveqd 5441 . . . 4 (Fun 𝐹(𝐹 ↾ dom 𝐹) = 𝐹)
1514imaeq1d 5611 . . 3 (Fun 𝐹 → ((𝐹 ↾ dom 𝐹) “ 𝐴) = (𝐹𝐴))
168, 15syl5eqr 2796 . 2 (Fun 𝐹 → ((𝐹𝐴) ∩ dom 𝐹) = (𝐹𝐴))
171, 7, 163eqtrrd 2787 1 (Fun 𝐹 → (𝐹𝐴) = (𝐹 “ (𝐴 ∩ ran 𝐹)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1620  cin 3702  ccnv 5253  dom cdm 5254  ran crn 5255  cres 5256  cima 5257  Rel wrel 5259  Fun wfun 6031  wf 6033  ontowfo 6035
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728  ax-sep 4921  ax-nul 4929  ax-pr 5043
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-mo 2600  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ne 2921  df-ral 3043  df-rex 3044  df-rab 3047  df-v 3330  df-sbc 3565  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-nul 4047  df-if 4219  df-sn 4310  df-pr 4312  df-op 4316  df-uni 4577  df-br 4793  df-opab 4853  df-id 5162  df-xp 5260  df-rel 5261  df-cnv 5262  df-co 5263  df-dm 5264  df-rn 5265  df-res 5266  df-ima 5267  df-iota 6000  df-fun 6039  df-fn 6040  df-f 6041  df-fo 6043  df-fv 6045
This theorem is referenced by:  fimacnvinrn2  6500
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