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Theorem fimacnvinrn 7072
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 7064 . 2 (Fun 𝐹 β†’ (◑𝐹 β€œ (𝐴 ∩ ran 𝐹)) = ((◑𝐹 β€œ 𝐴) ∩ (◑𝐹 β€œ ran 𝐹)))
2 funforn 6811 . . . . 5 (Fun 𝐹 ↔ 𝐹:dom 𝐹–ontoβ†’ran 𝐹)
3 fof 6804 . . . . 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 6738 . . . 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 4211 . 2 (Fun 𝐹 β†’ ((◑𝐹 β€œ 𝐴) ∩ (◑𝐹 β€œ ran 𝐹)) = ((◑𝐹 β€œ 𝐴) ∩ dom 𝐹))
8 cnvresima 6228 . . 3 (β—‘(𝐹 β†Ύ dom 𝐹) β€œ 𝐴) = ((◑𝐹 β€œ 𝐴) ∩ dom 𝐹)
9 resdm2 6229 . . . . . 6 (𝐹 β†Ύ dom 𝐹) = ◑◑𝐹
10 funrel 6564 . . . . . . 7 (Fun 𝐹 β†’ Rel 𝐹)
11 dfrel2 6187 . . . . . . 7 (Rel 𝐹 ↔ ◑◑𝐹 = 𝐹)
1210, 11sylib 217 . . . . . 6 (Fun 𝐹 β†’ ◑◑𝐹 = 𝐹)
139, 12eqtrid 2782 . . . . 5 (Fun 𝐹 β†’ (𝐹 β†Ύ dom 𝐹) = 𝐹)
1413cnveqd 5874 . . . 4 (Fun 𝐹 β†’ β—‘(𝐹 β†Ύ dom 𝐹) = ◑𝐹)
1514imaeq1d 6057 . . 3 (Fun 𝐹 β†’ (β—‘(𝐹 β†Ύ dom 𝐹) β€œ 𝐴) = (◑𝐹 β€œ 𝐴))
168, 15eqtr3id 2784 . 2 (Fun 𝐹 β†’ ((◑𝐹 β€œ 𝐴) ∩ dom 𝐹) = (◑𝐹 β€œ 𝐴))
171, 7, 163eqtrrd 2775 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 1539   ∩ cin 3946  β—‘ccnv 5674  dom cdm 5675  ran crn 5676   β†Ύ cres 5677   β€œ cima 5678  Rel wrel 5680  Fun wfun 6536  βŸΆwf 6538  β€“ontoβ†’wfo 6540
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-clab 2708  df-cleq 2722  df-clel 2808  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-br 5148  df-opab 5210  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-fun 6544  df-fn 6545  df-f 6546  df-fo 6548
This theorem is referenced by:  fimacnvinrn2  7073  preiman0  32198
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