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Theorem rescnvcnv 4971
Description: The restriction of the double converse of a class. (Contributed by NM, 8-Apr-2007.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Assertion
Ref Expression
rescnvcnv (𝐴𝐵) = (𝐴𝐵)

Proof of Theorem rescnvcnv
StepHypRef Expression
1 cnvcnv2 4962 . . 3 𝐴 = (𝐴 ↾ V)
21reseq1i 4785 . 2 (𝐴𝐵) = ((𝐴 ↾ V) ↾ 𝐵)
3 resres 4801 . 2 ((𝐴 ↾ V) ↾ 𝐵) = (𝐴 ↾ (V ∩ 𝐵))
4 ssv 3089 . . . 4 𝐵 ⊆ V
5 sseqin2 3265 . . . 4 (𝐵 ⊆ V ↔ (V ∩ 𝐵) = 𝐵)
64, 5mpbi 144 . . 3 (V ∩ 𝐵) = 𝐵
76reseq2i 4786 . 2 (𝐴 ↾ (V ∩ 𝐵)) = (𝐴𝐵)
82, 3, 73eqtri 2142 1 (𝐴𝐵) = (𝐴𝐵)
Colors of variables: wff set class
Syntax hints:   = wceq 1316  Vcvv 2660  cin 3040  wss 3041  ccnv 4508  cres 4511
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-br 3900  df-opab 3960  df-xp 4515  df-rel 4516  df-cnv 4517  df-res 4521
This theorem is referenced by:  cnvcnvres  4972  imacnvcnv  4973  resdm2  4999  resdmres  5000  coires1  5026  cocnvres  5033  f1oresrab  5553
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