Theorem cocnvcnv2 5647

Description: A composition is not affected by a double converse of its second argument. (Contributed by NM, 8-Oct-2007.)

Assertion

Ref	Expression
cocnvcnv2	|- ( A o. `' `' B ) = ( A o. B )

Proof of Theorem cocnvcnv2

Step	Hyp	Ref	Expression
1		cnvcnv2 5588	. . 3 |- `' `' B = ( B |` _V )
2	1	coeq2i 5282	. 2 |- ( A o. `' `' B ) = ( A o. ( B |` _V ) )
3		resco 5639	. 2 |- ( ( A o. B ) |` _V ) = ( A o. ( B |` _V ) )
4		relco 5633	. . 3 |- Rel ( A o. B )
5		dfrel3 5592	. . 3 |- ( Rel ( A o. B ) <-> ( ( A o. B ) |` _V ) = ( A o. B ) )
6	4, 5	mpbi 220	. 2 |- ( ( A o. B ) |` _V ) = ( A o. B )
7	2, 3, 6	3eqtr2i 2650	1 |- ( A o. `' `' B ) = ( A o. B )

Syntax hints:   = wceq 1483   _Vcvv 3200   `'ccnv 5113   |` cres 5116   o. ccom 5118   Rel wrel 5119

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-res 5126

This theorem is referenced by:  dfdm2  5667  cofunex2g  7131  trclubgNEW  37925  cnvtrrel  37962  trrelsuperrel2dg  37963