Theorem rescnvcnv 5453

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 5444	. . 3 |- `' `' A = ( A |` _V )
2	1	reseq1i 5258	. 2 |- ( `' `' A |` B ) = ( ( A |` _V ) |` B )
3		resres 5274	. 2 |- ( ( A |` _V ) |` B ) = ( A |` ( _V i^i B ) )
4		ssv 3509	. . . 4 |- B C_ _V
5		sseqin2 3703	. . . 4 |- ( B C_ _V <-> ( _V i^i B ) = B )
6	4, 5	mpbi 208	. . 3 |- ( _V i^i B ) = B
7	6	reseq2i 5259	. 2 |- ( A |` ( _V i^i B ) ) = ( A |` B )
8	2, 3, 7	3eqtri 2487	1 |- ( `' `' A |` B ) = ( A |` B )

Syntax hints:   = wceq 1398   _Vcvv 3106   i^i cin 3460   C_ wss 3461   `'ccnv 4987   |` cres 4990

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-br 4440  df-opab 4498  df-xp 4994  df-rel 4995  df-cnv 4996  df-res 5000

This theorem is referenced by:  cnvcnvres  5454  imacnvcnv  5455  resdm2  5480  resdmres  5481  coires1  5508  f1oresrab  6039