Theorem relelrnb 5361

Description: Membership in a range. (Contributed by Mario Carneiro, 5-Nov-2015.)

Assertion

Ref	Expression
relelrnb	|- ( Rel R -> ( A e. ran R <-> E. x x R A ) )

Distinct variable groups:   x, A   x, R

Proof of Theorem relelrnb

Step	Hyp	Ref	Expression
1		elrng 5314	. . 3 |- ( A e. ran R -> ( A e. ran R <-> E. x x R A ) )
2	1	ibi 256	. 2 |- ( A e. ran R -> E. x x R A )
3		relelrn 5359	. . . 4 |- ( ( Rel R /\ x R A ) -> A e. ran R )
4	3	ex 450	. . 3 |- ( Rel R -> ( x R A -> A e. ran R ) )
5	4	exlimdv 1861	. 2 |- ( Rel R -> ( E. x x R A -> A e. ran R ) )
6	2, 5	impbid2 216	1 |- ( Rel R -> ( A e. ran R <-> E. x x R A ) )

Syntax hints:   -> wi 4   <-> wb 196   E.wex 1704   e. wcel 1990   class class class wbr 4653   ran crn 5115   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-rex 2918  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  df-dm 5124  df-rn 5125

This theorem is referenced by: (None)