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Theorem funcnvres2 6130
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 6117 . . 3 (Fun 𝐹 → Fun 𝐹)
2 funcnvres 6128 . . 3 (Fun 𝐹(𝐹𝐴) = (𝐹 ↾ (𝐹𝐴)))
31, 2syl 17 . 2 (Fun 𝐹(𝐹𝐴) = (𝐹 ↾ (𝐹𝐴)))
4 funrel 6066 . . . 4 (Fun 𝐹 → Rel 𝐹)
5 dfrel2 5741 . . . 4 (Rel 𝐹𝐹 = 𝐹)
64, 5sylib 208 . . 3 (Fun 𝐹𝐹 = 𝐹)
76reseq1d 5550 . 2 (Fun 𝐹 → (𝐹 ↾ (𝐹𝐴)) = (𝐹 ↾ (𝐹𝐴)))
83, 7eqtrd 2794 1 (Fun 𝐹(𝐹𝐴) = (𝐹 ↾ (𝐹𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1632  ccnv 5265  cres 5268  cima 5269  Rel wrel 5271  Fun wfun 6043
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-opab 4865  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-fun 6051
This theorem is referenced by:  funimacnv  6131  foimacnv  6315  unbenlem  15814  ofco2  20459  dvlog  24596  fresf1o  29742
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