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Theorem fimacnvinrn 7074
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 β†’ (◑𝐹 β€œ 𝐴) = (◑𝐹 β€œ (𝐴 ∩ ran 𝐹)))

Proof of Theorem fimacnvinrn
StepHypRef Expression
1 inpreima 7066 . 2 (Fun 𝐹 β†’ (◑𝐹 β€œ (𝐴 ∩ ran 𝐹)) = ((◑𝐹 β€œ 𝐴) ∩ (◑𝐹 β€œ ran 𝐹)))
2 funforn 6813 . . . . 5 (Fun 𝐹 ↔ 𝐹:dom 𝐹–ontoβ†’ran 𝐹)
3 fof 6806 . . . . 5 (𝐹:dom 𝐹–ontoβ†’ran 𝐹 β†’ 𝐹:dom 𝐹⟢ran 𝐹)
42, 3sylbi 216 . . . 4 (Fun 𝐹 β†’ 𝐹:dom 𝐹⟢ran 𝐹)
5 fimacnv 6740 . . . 4 (𝐹:dom 𝐹⟢ran 𝐹 β†’ (◑𝐹 β€œ ran 𝐹) = dom 𝐹)
64, 5syl 17 . . 3 (Fun 𝐹 β†’ (◑𝐹 β€œ ran 𝐹) = dom 𝐹)
76ineq2d 4213 . 2 (Fun 𝐹 β†’ ((◑𝐹 β€œ 𝐴) ∩ (◑𝐹 β€œ ran 𝐹)) = ((◑𝐹 β€œ 𝐴) ∩ dom 𝐹))
8 cnvresima 6230 . . 3 (β—‘(𝐹 β†Ύ dom 𝐹) β€œ 𝐴) = ((◑𝐹 β€œ 𝐴) ∩ dom 𝐹)
9 resdm2 6231 . . . . . 6 (𝐹 β†Ύ dom 𝐹) = ◑◑𝐹
10 funrel 6566 . . . . . . 7 (Fun 𝐹 β†’ Rel 𝐹)
11 dfrel2 6189 . . . . . . 7 (Rel 𝐹 ↔ ◑◑𝐹 = 𝐹)
1210, 11sylib 217 . . . . . 6 (Fun 𝐹 β†’ ◑◑𝐹 = 𝐹)
139, 12eqtrid 2785 . . . . 5 (Fun 𝐹 β†’ (𝐹 β†Ύ dom 𝐹) = 𝐹)
1413cnveqd 5876 . . . 4 (Fun 𝐹 β†’ β—‘(𝐹 β†Ύ dom 𝐹) = ◑𝐹)
1514imaeq1d 6059 . . 3 (Fun 𝐹 β†’ (β—‘(𝐹 β†Ύ dom 𝐹) β€œ 𝐴) = (◑𝐹 β€œ 𝐴))
168, 15eqtr3id 2787 . 2 (Fun 𝐹 β†’ ((◑𝐹 β€œ 𝐴) ∩ dom 𝐹) = (◑𝐹 β€œ 𝐴))
171, 7, 163eqtrrd 2778 1 (Fun 𝐹 β†’ (◑𝐹 β€œ 𝐴) = (◑𝐹 β€œ (𝐴 ∩ ran 𝐹)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1542   ∩ cin 3948  β—‘ccnv 5676  dom cdm 5677  ran crn 5678   β†Ύ cres 5679   β€œ cima 5680  Rel wrel 5682  Fun wfun 6538  βŸΆwf 6540  β€“ontoβ†’wfo 6542
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-fun 6546  df-fn 6547  df-f 6548  df-fo 6550
This theorem is referenced by:  fimacnvinrn2  7075  preiman0  31931
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