Theorem cnvcnv 5063

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 4989	. . . . 5 |- Rel `' `' A
2		df-rel 4618	. . . . 5 |- ( Rel `' `' A <-> `' `' A C_ ( _V X. _V ) )
3	1, 2	mpbi 144	. . . 4 |- `' `' A C_ ( _V X. _V )
4		relxp 4720	. . . . 5 |- Rel ( _V X. _V )
5		dfrel2 5061	. . . . 5 |- ( Rel ( _V X. _V ) <-> `' `' ( _V X. _V ) = ( _V X. _V ) )
6	4, 5	mpbi 144	. . . 4 |- `' `' ( _V X. _V ) = ( _V X. _V )
7	3, 6	sseqtrri 3182	. . 3 |- `' `' A C_ `' `' ( _V X. _V )
8		dfss 3135	. . 3 |- ( `' `' A C_ `' `' ( _V X. _V ) <-> `' `' A = ( `' `' A i^i `' `' ( _V X. _V ) ) )
9	7, 8	mpbi 144	. 2 |- `' `' A = ( `' `' A i^i `' `' ( _V X. _V ) )
10		cnvin 5018	. 2 |- `' ( `' A i^i `' ( _V X. _V ) ) = ( `' `' A i^i `' `' ( _V X. _V ) )
11		cnvin 5018	. . . 4 |- `' ( A i^i ( _V X. _V ) ) = ( `' A i^i `' ( _V X. _V ) )
12	11	cnveqi 4786	. . 3 |- `' `' ( A i^i ( _V X. _V ) ) = `' ( `' A i^i `' ( _V X. _V ) )
13		inss2 3348	. . . . 5 |- ( A i^i ( _V X. _V ) ) C_ ( _V X. _V )
14		df-rel 4618	. . . . 5 |- ( Rel ( A i^i ( _V X. _V ) ) <-> ( A i^i ( _V X. _V ) ) C_ ( _V X. _V ) )
15	13, 14	mpbir 145	. . . 4 |- Rel ( A i^i ( _V X. _V ) )
16		dfrel2 5061	. . . 4 |- ( Rel ( A i^i ( _V X. _V ) ) <-> `' `' ( A i^i ( _V X. _V ) ) = ( A i^i ( _V X. _V ) ) )
17	15, 16	mpbi 144	. . 3 |- `' `' ( A i^i ( _V X. _V ) ) = ( A i^i ( _V X. _V ) )
18	12, 17	eqtr3i 2193	. 2 |- `' ( `' A i^i `' ( _V X. _V ) ) = ( A i^i ( _V X. _V ) )
19	9, 10, 18	3eqtr2i 2197	1 |- `' `' A = ( A i^i ( _V X. _V ) )

Syntax hints:   = wceq 1348   _Vcvv 2730   i^i cin 3120   C_ wss 3121   X. cxp 4609   `'ccnv 4610   Rel wrel 4616

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-xp 4617  df-rel 4618  df-cnv 4619

This theorem is referenced by:  cnvcnv2  5064  cnvcnvss  5065  structcnvcnv  12432  strslfv2d  12458