Theorem relelrnb 4864

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 4817	. . 3 |- ( A e. ran R -> ( A e. ran R <-> E. x x R A ) )
2	1	ibi 176	. 2 |- ( A e. ran R -> E. x x R A )
3		relelrn 4862	. . . 4 |- ( ( Rel R /\ x R A ) -> A e. ran R )
4	3	ex 115	. . 3 |- ( Rel R -> ( x R A -> A e. ran R ) )
5	4	exlimdv 1819	. 2 |- ( Rel R -> ( E. x x R A -> A e. ran R ) )
6	2, 5	impbid2 143	1 |- ( Rel R -> ( A e. ran R <-> E. x x R A ) )

Syntax hints:   -> wi 4   <-> wb 105   E.wex 1492   e. wcel 2148   class class class wbr 4002   ran crn 4626   Rel wrel 4630

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4120  ax-pow 4173  ax-pr 4208

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-br 4003  df-opab 4064  df-xp 4631  df-rel 4632  df-cnv 4633  df-dm 4635  df-rn 4636

This theorem is referenced by: (None)