Theorem rescnvcnv 5133

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	|- ( `' `' A |` B ) = ( A |` B )

Proof of Theorem rescnvcnv

Step	Hyp	Ref	Expression
1		cnvcnv2 5124	. . 3 |- `' `' A = ( A |` _V )
2	1	reseq1i 4943	. 2 |- ( `' `' A |` B ) = ( ( A |` _V ) |` B )
3		resres 4959	. 2 |- ( ( A |` _V ) |` B ) = ( A |` ( _V i^i B ) )
4		ssv 3206	. . . 4 |- B C_ _V
5		sseqin2 3383	. . . 4 |- ( B C_ _V <-> ( _V i^i B ) = B )
6	4, 5	mpbi 145	. . 3 |- ( _V i^i B ) = B
7	6	reseq2i 4944	. 2 |- ( A |` ( _V i^i B ) ) = ( A |` B )
8	2, 3, 7	3eqtri 2221	1 |- ( `' `' A |` B ) = ( A |` B )

Syntax hints:   = wceq 1364   _Vcvv 2763   i^i cin 3156   C_ wss 3157   `'ccnv 4663   |` cres 4666

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-br 4035  df-opab 4096  df-xp 4670  df-rel 4671  df-cnv 4672  df-res 4676

This theorem is referenced by:  cnvcnvres  5134  imacnvcnv  5135  resdm2  5161  resdmres  5162  coires1  5188  cocnvres  5195  f1oresrab  5730