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Theorem fimacnvinrn 6597
 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 6591 . 2 (Fun 𝐹 → (𝐹 “ (𝐴 ∩ ran 𝐹)) = ((𝐹𝐴) ∩ (𝐹 “ ran 𝐹)))
2 funforn 6360 . . . . 5 (Fun 𝐹𝐹:dom 𝐹onto→ran 𝐹)
3 fof 6353 . . . . 5 (𝐹:dom 𝐹onto→ran 𝐹𝐹:dom 𝐹⟶ran 𝐹)
42, 3sylbi 209 . . . 4 (Fun 𝐹𝐹:dom 𝐹⟶ran 𝐹)
5 fimacnv 6596 . . . 4 (𝐹:dom 𝐹⟶ran 𝐹 → (𝐹 “ ran 𝐹) = dom 𝐹)
64, 5syl 17 . . 3 (Fun 𝐹 → (𝐹 “ ran 𝐹) = dom 𝐹)
76ineq2d 4041 . 2 (Fun 𝐹 → ((𝐹𝐴) ∩ (𝐹 “ ran 𝐹)) = ((𝐹𝐴) ∩ dom 𝐹))
8 cnvresima 5864 . . 3 ((𝐹 ↾ dom 𝐹) “ 𝐴) = ((𝐹𝐴) ∩ dom 𝐹)
9 resdm2 5865 . . . . . 6 (𝐹 ↾ dom 𝐹) = 𝐹
10 funrel 6140 . . . . . . 7 (Fun 𝐹 → Rel 𝐹)
11 dfrel2 5824 . . . . . . 7 (Rel 𝐹𝐹 = 𝐹)
1210, 11sylib 210 . . . . . 6 (Fun 𝐹𝐹 = 𝐹)
139, 12syl5eq 2873 . . . . 5 (Fun 𝐹 → (𝐹 ↾ dom 𝐹) = 𝐹)
1413cnveqd 5530 . . . 4 (Fun 𝐹(𝐹 ↾ dom 𝐹) = 𝐹)
1514imaeq1d 5706 . . 3 (Fun 𝐹 → ((𝐹 ↾ dom 𝐹) “ 𝐴) = (𝐹𝐴))
168, 15syl5eqr 2875 . 2 (Fun 𝐹 → ((𝐹𝐴) ∩ dom 𝐹) = (𝐹𝐴))
171, 7, 163eqtrrd 2866 1 (Fun 𝐹 → (𝐹𝐴) = (𝐹 “ (𝐴 ∩ ran 𝐹)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1656   ∩ cin 3797  ◡ccnv 5341  dom cdm 5342  ran crn 5343   ↾ cres 5344   “ cima 5345  Rel wrel 5347  Fun wfun 6117  ⟶wf 6119  –onto→wfo 6121 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pr 5127 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-br 4874  df-opab 4936  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-fo 6129  df-fv 6131 This theorem is referenced by:  fimacnvinrn2  6598
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