Theorem eldmcnv 34113

Description: Elementhood in a domain of a converse. (Contributed by Peter Mazsa, 25-May-2018.)

Assertion

Ref	Expression
eldmcnv	|- ( A e. V -> ( A e. dom `' R <-> E. u u R A ) )

Distinct variable groups:   u, A   u, R   u, V

Proof of Theorem eldmcnv

Step	Hyp	Ref	Expression
1		eldmg 5319	. 2 |- ( A e. V -> ( A e. dom `' R <-> E. u A `' R u ) )
2		brcnvg 5303	. . . 4 |- ( ( A e. V /\ u e. _V ) -> ( A `' R u <-> u R A ) )
3	2	el2v2 33986	. . 3 |- ( A e. V -> ( A `' R u <-> u R A ) )
4	3	exbidv 1850	. 2 |- ( A e. V -> ( E. u A `' R u <-> E. u u R A ) )
5	1, 4	bitrd 268	1 |- ( A e. V -> ( A e. dom `' R <-> E. u u R A ) )

Syntax hints:   -> wi 4   <-> wb 196   E.wex 1704   e. wcel 1990   _Vcvv 3200   class class class wbr 4653   `'ccnv 5113   dom cdm 5114

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-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-cnv 5122  df-dm 5124

This theorem is referenced by: (None)