Theorem cnvcnv 5586

Description: The double converse of a class strips out all elements that are not ordered pairs. (Contributed by NM, 8-Dec-2003.) (Proof shortened by BJ, 26-Nov-2021.)

Assertion

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

Proof of Theorem cnvcnv

Step	Hyp	Ref	Expression
1		cnvin 5540	. . 3 |- `' ( `' A i^i `' ( _V X. _V ) ) = ( `' `' A i^i `' `' ( _V X. _V ) )
2		cnvin 5540	. . . 4 |- `' ( A i^i ( _V X. _V ) ) = ( `' A i^i `' ( _V X. _V ) )
3	2	cnveqi 5297	. . 3 |- `' `' ( A i^i ( _V X. _V ) ) = `' ( `' A i^i `' ( _V X. _V ) )
4		relcnv 5503	. . . . . 6 |- Rel `' `' A
5		df-rel 5121	. . . . . 6 |- ( Rel `' `' A <-> `' `' A C_ ( _V X. _V ) )
6	4, 5	mpbi 220	. . . . 5 |- `' `' A C_ ( _V X. _V )
7		relxp 5227	. . . . . 6 |- Rel ( _V X. _V )
8		dfrel2 5583	. . . . . 6 |- ( Rel ( _V X. _V ) <-> `' `' ( _V X. _V ) = ( _V X. _V ) )
9	7, 8	mpbi 220	. . . . 5 |- `' `' ( _V X. _V ) = ( _V X. _V )
10	6, 9	sseqtr4i 3638	. . . 4 |- `' `' A C_ `' `' ( _V X. _V )
11		dfss 3589	. . . 4 |- ( `' `' A C_ `' `' ( _V X. _V ) <-> `' `' A = ( `' `' A i^i `' `' ( _V X. _V ) ) )
12	10, 11	mpbi 220	. . 3 |- `' `' A = ( `' `' A i^i `' `' ( _V X. _V ) )
13	1, 3, 12	3eqtr4ri 2655	. 2 |- `' `' A = `' `' ( A i^i ( _V X. _V ) )
14		inss2 3834	. . . 4 |- ( A i^i ( _V X. _V ) ) C_ ( _V X. _V )
15		df-rel 5121	. . . 4 |- ( Rel ( A i^i ( _V X. _V ) ) <-> ( A i^i ( _V X. _V ) ) C_ ( _V X. _V ) )
16	14, 15	mpbir 221	. . 3 |- Rel ( A i^i ( _V X. _V ) )
17		dfrel2 5583	. . 3 |- ( Rel ( A i^i ( _V X. _V ) ) <-> `' `' ( A i^i ( _V X. _V ) ) = ( A i^i ( _V X. _V ) ) )
18	16, 17	mpbi 220	. 2 |- `' `' ( A i^i ( _V X. _V ) ) = ( A i^i ( _V X. _V ) )
19	13, 18	eqtri 2644	1 |- `' `' A = ( A i^i ( _V X. _V ) )

Syntax hints:   = wceq 1483   _Vcvv 3200   i^i cin 3573   C_ wss 3574   X. cxp 5112   `'ccnv 5113   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-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

This theorem is referenced by:  cnvcnv2  5588  cnvcnvss  5589  structcnvcnv  15871  strfv2d  15905  elcnvcnvintab  37888  relintab  37889  nonrel  37890  elcnvcnvlem  37905  cnvcnvintabd  37906