Theorem cnvcnv 4793

Description: The double converse of a class strips out all elements that are not ordered pairs. (Contributed by NM, 8-Dec-2003.)

Assertion

Ref	Expression
cnvcnv	|- `' `' A = ( A i^i ( _V X. _V ) )

Proof of Theorem cnvcnv

Step	Hyp	Ref	Expression
1		relcnv 4723	. . . . 5 |- Rel `' `' A
2		df-rel 4370	. . . . 5 |- ( Rel `' `' A <-> `' `' A C_ ( _V X. _V ) )
3	1, 2	mpbi 143	. . . 4 |- `' `' A C_ ( _V X. _V )
4		relxp 4465	. . . . 5 |- Rel ( _V X. _V )
5		dfrel2 4791	. . . . 5 |- ( Rel ( _V X. _V ) <-> `' `' ( _V X. _V ) = ( _V X. _V ) )
6	4, 5	mpbi 143	. . . 4 |- `' `' ( _V X. _V ) = ( _V X. _V )
7	3, 6	sseqtr4i 3032	. . 3 |- `' `' A C_ `' `' ( _V X. _V )
8		dfss 2987	. . 3 |- ( `' `' A C_ `' `' ( _V X. _V ) <-> `' `' A = ( `' `' A i^i `' `' ( _V X. _V ) ) )
9	7, 8	mpbi 143	. 2 |- `' `' A = ( `' `' A i^i `' `' ( _V X. _V ) )
10		cnvin 4751	. 2 |- `' ( `' A i^i `' ( _V X. _V ) ) = ( `' `' A i^i `' `' ( _V X. _V ) )
11		cnvin 4751	. . . 4 |- `' ( A i^i ( _V X. _V ) ) = ( `' A i^i `' ( _V X. _V ) )
12	11	cnveqi 4528	. . 3 |- `' `' ( A i^i ( _V X. _V ) ) = `' ( `' A i^i `' ( _V X. _V ) )
13		inss2 3187	. . . . 5 |- ( A i^i ( _V X. _V ) ) C_ ( _V X. _V )
14		df-rel 4370	. . . . 5 |- ( Rel ( A i^i ( _V X. _V ) ) <-> ( A i^i ( _V X. _V ) ) C_ ( _V X. _V ) )
15	13, 14	mpbir 144	. . . 4 |- Rel ( A i^i ( _V X. _V ) )
16		dfrel2 4791	. . . 4 |- ( Rel ( A i^i ( _V X. _V ) ) <-> `' `' ( A i^i ( _V X. _V ) ) = ( A i^i ( _V X. _V ) ) )
17	15, 16	mpbi 143	. . 3 |- `' `' ( A i^i ( _V X. _V ) ) = ( A i^i ( _V X. _V ) )
18	12, 17	eqtr3i 2103	. 2 |- `' ( `' A i^i `' ( _V X. _V ) ) = ( A i^i ( _V X. _V ) )
19	9, 10, 18	3eqtr2i 2107	1 |- `' `' A = ( A i^i ( _V X. _V ) )

Syntax hints:   = wceq 1284   _Vcvv 2601   i^i cin 2972   C_ wss 2973   X. cxp 4361   `'ccnv 4362   Rel wrel 4368

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786  df-opab 3840  df-xp 4369  df-rel 4370  df-cnv 4371

This theorem is referenced by:  cnvcnv2  4794  cnvcnvss  4795