Theorem elcnvcnvlem 37905

Description: Two ways to say a set is a member of the converse of the converse of a class. (Contributed by RP, 20-Aug-2020.)

Assertion

Ref	Expression
elcnvcnvlem	|- ( A e. `' `' B <-> ( A e. ( _V X. _V ) /\ ( _I ` A ) e. B ) )

Proof of Theorem elcnvcnvlem

Step	Hyp	Ref	Expression
1		cnvcnv 5586	. . . 4 |- `' `' B = ( B i^i ( _V X. _V ) )
2		incom 3805	. . . 4 |- ( B i^i ( _V X. _V ) ) = ( ( _V X. _V ) i^i B )
3	1, 2	eqtri 2644	. . 3 |- `' `' B = ( ( _V X. _V ) i^i B )
4	3	eleq2i 2693	. 2 |- ( A e. `' `' B <-> A e. ( ( _V X. _V ) i^i B ) )
5		elinlem 37904	. 2 |- ( A e. ( ( _V X. _V ) i^i B ) <-> ( A e. ( _V X. _V ) /\ ( _I ` A ) e. B ) )
6	4, 5	bitri 264	1 |- ( A e. `' `' B <-> ( A e. ( _V X. _V ) /\ ( _I ` A ) e. B ) )

Syntax hints:   <-> wb 196   /\ wa 384   e. wcel 1990   _Vcvv 3200   i^i cin 3573   _I cid 5023   X. cxp 5112   `'ccnv 5113   ` cfv 5888

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-sbc 3436  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-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896

This theorem is referenced by: (None)