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Theorem rescnvcnv 4969
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 4960 . . 3 𝐴 = (𝐴 ↾ V)
21reseq1i 4783 . 2 (𝐴𝐵) = ((𝐴 ↾ V) ↾ 𝐵)
3 resres 4799 . 2 ((𝐴 ↾ V) ↾ 𝐵) = (𝐴 ↾ (V ∩ 𝐵))
4 ssv 3087 . . . 4 𝐵 ⊆ V
5 sseqin2 3263 . . . 4 (𝐵 ⊆ V ↔ (V ∩ 𝐵) = 𝐵)
64, 5mpbi 144 . . 3 (V ∩ 𝐵) = 𝐵
76reseq2i 4784 . 2 (𝐴 ↾ (V ∩ 𝐵)) = (𝐴𝐵)
82, 3, 73eqtri 2140 1 (𝐴𝐵) = (𝐴𝐵)
Colors of variables: wff set class
Syntax hints:   = wceq 1314  Vcvv 2658  cin 3038  wss 3039  ccnv 4506  cres 4509
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-br 3898  df-opab 3958  df-xp 4513  df-rel 4514  df-cnv 4515  df-res 4519
This theorem is referenced by:  cnvcnvres  4970  imacnvcnv  4971  resdm2  4997  resdmres  4998  coires1  5024  cocnvres  5031  f1oresrab  5551
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