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Theorem rescnvcnv 4726
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 4717 . . 3 A = (A ↾ V)
21reseq1i 4551 . 2 (AB) = ((A ↾ V) ↾ B)
3 resres 4567 . 2 ((A ↾ V) ↾ B) = (A ↾ (V ∩ B))
4 ssv 2959 . . . 4 B ⊆ V
5 sseqin2 3150 . . . 4 (B ⊆ V ↔ (V ∩ B) = B)
64, 5mpbi 133 . . 3 (V ∩ B) = B
76reseq2i 4552 . 2 (A ↾ (V ∩ B)) = (AB)
82, 3, 73eqtri 2061 1 (AB) = (AB)
Colors of variables: wff set class
Syntax hints:   = wceq 1242  Vcvv 2551  cin 2910  wss 2911  ccnv 4287  cres 4290
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-cnv 4296  df-res 4300
This theorem is referenced by:  cnvcnvres  4727  imacnvcnv  4728  resdm2  4754  resdmres  4755  coires1  4781  f1oresrab  5272
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