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Theorem funcnvres2 5883
 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 5870 . . 3 (Fun 𝐹 → Fun 𝐹)
2 funcnvres 5881 . . 3 (Fun 𝐹(𝐹𝐴) = (𝐹 ↾ (𝐹𝐴)))
31, 2syl 17 . 2 (Fun 𝐹(𝐹𝐴) = (𝐹 ↾ (𝐹𝐴)))
4 funrel 5821 . . . 4 (Fun 𝐹 → Rel 𝐹)
5 dfrel2 5502 . . . 4 (Rel 𝐹𝐹 = 𝐹)
64, 5sylib 207 . . 3 (Fun 𝐹𝐹 = 𝐹)
76reseq1d 5316 . 2 (Fun 𝐹 → (𝐹 ↾ (𝐹𝐴)) = (𝐹 ↾ (𝐹𝐴)))
83, 7eqtrd 2644 1 (Fun 𝐹(𝐹𝐴) = (𝐹 ↾ (𝐹𝐴)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475  ◡ccnv 5037   ↾ cres 5040   “ cima 5041  Rel wrel 5043  Fun wfun 5798 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-fun 5806 This theorem is referenced by:  funimacnv  5884  foimacnv  6067  unbenlem  15450  ofco2  20076  dvlog  24197  fresf1o  28815
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