Theorem rescnvcnv 5317

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 5308	. . 3 |- `' `' A = ( A |` _V )
2	1	reseq1i 5120	. 2 |- ( `' `' A |` B ) = ( ( A |` _V ) |` B )
3		resres 5136	. 2 |- ( ( A |` _V ) |` B ) = ( A |` ( _V i^i B ) )
4		ssv 3464	. . . 4 |- B C_ _V
5		sseqin2 3663	. . . 4 |- ( B C_ _V <-> ( _V i^i B ) = B )
6	4, 5	mpbi 213	. . 3 |- ( _V i^i B ) = B
7	6	reseq2i 5121	. 2 |- ( A |` ( _V i^i B ) ) = ( A |` B )
8	2, 3, 7	3eqtri 2488	1 |- ( `' `' A |` B ) = ( A |` B )

Syntax hints:   = wceq 1455   _Vcvv 3057   i^i cin 3415   C_ wss 3416   `'ccnv 4852   |` cres 4855

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4539  ax-nul 4548  ax-pr 4653

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-rab 2758  df-v 3059  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-sn 3981  df-pr 3983  df-op 3987  df-br 4417  df-opab 4476  df-xp 4859  df-rel 4860  df-cnv 4861  df-res 4865

This theorem is referenced by:  cnvcnvres  5318  imacnvcnv  5319  resdm2  5344  resdmres  5345  coires1  5372  f1oresrab  6079