Theorem elrn3 31652

Description: Quantifier-free definition of membership in a range. (Contributed by Scott Fenton, 21-Jan-2017.)

Assertion

Ref	Expression
elrn3	|- ( A e. ran B <-> ( B i^i ( _V X. { A } ) ) =/= (/) )

Proof of Theorem elrn3

Step	Hyp	Ref	Expression
1		df-rn 5125	. . 3 |- ran B = dom `' B
2	1	eleq2i 2693	. 2 |- ( A e. ran B <-> A e. dom `' B )
3		eldm3 31651	. 2 |- ( A e. dom `' B <-> ( `' B |` { A } ) =/= (/) )
4		cnvxp 5551	. . . . . . 7 |- `' ( _V X. { A } ) = ( { A } X. _V )
5	4	ineq2i 3811	. . . . . 6 |- ( `' B i^i `' ( _V X. { A } ) ) = ( `' B i^i ( { A } X. _V ) )
6		cnvin 5540	. . . . . 6 |- `' ( B i^i ( _V X. { A } ) ) = ( `' B i^i `' ( _V X. { A } ) )
7		df-res 5126	. . . . . 6 |- ( `' B |` { A } ) = ( `' B i^i ( { A } X. _V ) )
8	5, 6, 7	3eqtr4ri 2655	. . . . 5 |- ( `' B |` { A } ) = `' ( B i^i ( _V X. { A } ) )
9	8	eqeq1i 2627	. . . 4 |- ( ( `' B |` { A } ) = (/) <-> `' ( B i^i ( _V X. { A } ) ) = (/) )
10		inss2 3834	. . . . . 6 |- ( B i^i ( _V X. { A } ) ) C_ ( _V X. { A } )
11		relxp 5227	. . . . . 6 |- Rel ( _V X. { A } )
12		relss 5206	. . . . . 6 |- ( ( B i^i ( _V X. { A } ) ) C_ ( _V X. { A } ) -> ( Rel ( _V X. { A } ) -> Rel ( B i^i ( _V X. { A } ) ) ) )
13	10, 11, 12	mp2 9	. . . . 5 |- Rel ( B i^i ( _V X. { A } ) )
14		cnveq0 5591	. . . . 5 |- ( Rel ( B i^i ( _V X. { A } ) ) -> ( ( B i^i ( _V X. { A } ) ) = (/) <-> `' ( B i^i ( _V X. { A } ) ) = (/) ) )
15	13, 14	ax-mp 5	. . . 4 |- ( ( B i^i ( _V X. { A } ) ) = (/) <-> `' ( B i^i ( _V X. { A } ) ) = (/) )
16	9, 15	bitr4i 267	. . 3 |- ( ( `' B |` { A } ) = (/) <-> ( B i^i ( _V X. { A } ) ) = (/) )
17	16	necon3bii 2846	. 2 |- ( ( `' B |` { A } ) =/= (/) <-> ( B i^i ( _V X. { A } ) ) =/= (/) )
18	2, 3, 17	3bitri 286	1 |- ( A e. ran B <-> ( B i^i ( _V X. { A } ) ) =/= (/) )

Syntax hints:   <-> wb 196   = wceq 1483   e. wcel 1990   =/= wne 2794   _Vcvv 3200   i^i cin 3573   C_ wss 3574   (/)c0 3915   {csn 4177   X. cxp 5112   `'ccnv 5113   dom cdm 5114   ran crn 5115   |` cres 5116   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-ne 2795  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-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126

This theorem is referenced by: (None)