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Theorem funcnvres2 6627
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 6614 . . 3 (Fun 𝐹 → Fun 𝐹)
2 funcnvres 6625 . . 3 (Fun 𝐹(𝐹𝐴) = (𝐹 ↾ (𝐹𝐴)))
31, 2syl 17 . 2 (Fun 𝐹(𝐹𝐴) = (𝐹 ↾ (𝐹𝐴)))
4 funrel 6564 . . . 4 (Fun 𝐹 → Rel 𝐹)
5 dfrel2 6188 . . . 4 (Rel 𝐹𝐹 = 𝐹)
64, 5sylib 217 . . 3 (Fun 𝐹𝐹 = 𝐹)
76reseq1d 5978 . 2 (Fun 𝐹 → (𝐹 ↾ (𝐹𝐴)) = (𝐹 ↾ (𝐹𝐴)))
83, 7eqtrd 2765 1 (Fun 𝐹(𝐹𝐴) = (𝐹 ↾ (𝐹𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  ccnv 5671  cres 5674  cima 5675  Rel wrel 5677  Fun wfun 6536
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5144  df-opab 5206  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-fun 6544
This theorem is referenced by:  funimacnv  6628  foimacnv  6850  unbenlem  16874  ofco2  22369  dvlog  26601  fresf1o  32459  fressupp  32511
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