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Theorem funcnvres2 6648
Description: The converse of a restriction of the converse of a function equals the function restricted to the image of its converse. (Contributed by NM, 4-May-2005.)
Assertion
Ref Expression
funcnvres2 (Fun 𝐹(𝐹𝐴) = (𝐹 ↾ (𝐹𝐴)))

Proof of Theorem funcnvres2
StepHypRef Expression
1 funcnvcnv 6635 . . 3 (Fun 𝐹 → Fun 𝐹)
2 funcnvres 6646 . . 3 (Fun 𝐹(𝐹𝐴) = (𝐹 ↾ (𝐹𝐴)))
31, 2syl 17 . 2 (Fun 𝐹(𝐹𝐴) = (𝐹 ↾ (𝐹𝐴)))
4 funrel 6585 . . . 4 (Fun 𝐹 → Rel 𝐹)
5 dfrel2 6211 . . . 4 (Rel 𝐹𝐹 = 𝐹)
64, 5sylib 218 . . 3 (Fun 𝐹𝐹 = 𝐹)
76reseq1d 5999 . 2 (Fun 𝐹 → (𝐹 ↾ (𝐹𝐴)) = (𝐹 ↾ (𝐹𝐴)))
83, 7eqtrd 2775 1 (Fun 𝐹(𝐹𝐴) = (𝐹 ↾ (𝐹𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  ccnv 5688  cres 5691  cima 5692  Rel wrel 5694  Fun wfun 6557
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 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-fun 6565
This theorem is referenced by:  funimacnv  6649  foimacnv  6866  unbenlem  16942  ofco2  22473  dvlog  26708  fresf1o  32648  fressupp  32703
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