Theorem cnvcnv 4773

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 4703	. . . . 5 |- Rel `' `' A
2		df-rel 4352	. . . . 5 |- ( Rel `' `' A <-> `' `' A C_ ( _V X. _V ) )
3	1, 2	mpbi 133	. . . 4 |- `' `' A C_ ( _V X. _V )
4		relxp 4447	. . . . 5 |- Rel ( _V X. _V )
5		dfrel2 4771	. . . . 5 |- ( Rel ( _V X. _V ) <-> `' `' ( _V X. _V ) = ( _V X. _V ) )
6	4, 5	mpbi 133	. . . 4 |- `' `' ( _V X. _V ) = ( _V X. _V )
7	3, 6	sseqtr4i 2978	. . 3 |- `' `' A C_ `' `' ( _V X. _V )
8		dfss 2932	. . 3 |- ( `' `' A C_ `' `' ( _V X. _V ) <-> `' `' A = ( `' `' A i^i `' `' ( _V X. _V ) ) )
9	7, 8	mpbi 133	. 2 |- `' `' A = ( `' `' A i^i `' `' ( _V X. _V ) )
10		cnvin 4731	. 2 |- `' ( `' A i^i `' ( _V X. _V ) ) = ( `' `' A i^i `' `' ( _V X. _V ) )
11		cnvin 4731	. . . 4 |- `' ( A i^i ( _V X. _V ) ) = ( `' A i^i `' ( _V X. _V ) )
12	11	cnveqi 4510	. . 3 |- `' `' ( A i^i ( _V X. _V ) ) = `' ( `' A i^i `' ( _V X. _V ) )
13		inss2 3158	. . . . 5 |- ( A i^i ( _V X. _V ) ) C_ ( _V X. _V )
14		df-rel 4352	. . . . 5 |- ( Rel ( A i^i ( _V X. _V ) ) <-> ( A i^i ( _V X. _V ) ) C_ ( _V X. _V ) )
15	13, 14	mpbir 134	. . . 4 |- Rel ( A i^i ( _V X. _V ) )
16		dfrel2 4771	. . . 4 |- ( Rel ( A i^i ( _V X. _V ) ) <-> `' `' ( A i^i ( _V X. _V ) ) = ( A i^i ( _V X. _V ) ) )
17	15, 16	mpbi 133	. . 3 |- `' `' ( A i^i ( _V X. _V ) ) = ( A i^i ( _V X. _V ) )
18	12, 17	eqtr3i 2062	. 2 |- `' ( `' A i^i `' ( _V X. _V ) ) = ( A i^i ( _V X. _V ) )
19	9, 10, 18	3eqtr2i 2066	1 |- `' `' A = ( A i^i ( _V X. _V ) )

Syntax hints:   = wceq 1243   _Vcvv 2557   i^i cin 2916   C_ wss 2917   X. cxp 4343   `'ccnv 4344   Rel wrel 4350

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-xp 4351  df-rel 4352  df-cnv 4353

This theorem is referenced by:  cnvcnv2  4774  cnvcnvss  4775