MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rescnvcnv Structured version   Visualization version   GIF version

Theorem rescnvcnv 6054
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 6043 . . 3 𝐴 = (𝐴 ↾ V)
21reseq1i 5842 . 2 (𝐴𝐵) = ((𝐴 ↾ V) ↾ 𝐵)
3 resres 5859 . 2 ((𝐴 ↾ V) ↾ 𝐵) = (𝐴 ↾ (V ∩ 𝐵))
4 ssv 3988 . . . 4 𝐵 ⊆ V
5 sseqin2 4189 . . . 4 (𝐵 ⊆ V ↔ (V ∩ 𝐵) = 𝐵)
64, 5mpbi 231 . . 3 (V ∩ 𝐵) = 𝐵
76reseq2i 5843 . 2 (𝐴 ↾ (V ∩ 𝐵)) = (𝐴𝐵)
82, 3, 73eqtri 2845 1 (𝐴𝐵) = (𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1528  Vcvv 3492  cin 3932  wss 3933  ccnv 5547  cres 5550
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-opab 5120  df-xp 5554  df-rel 5555  df-cnv 5556  df-res 5560
This theorem is referenced by:  cnvcnvres  6055  imacnvcnv  6056  resdm2  6081  resdmres  6082  coires1  6110  f1oresrab  6881
  Copyright terms: Public domain W3C validator