Theorem ecdmn0 7789

Description: A representative of a nonempty equivalence class belongs to the domain of the equivalence relation. (Contributed by NM, 15-Feb-1996.) (Revised by Mario Carneiro, 9-Jul-2014.)

Assertion

Ref	Expression
ecdmn0	|- ( A e. dom R <-> [ A ] R =/= (/) )

Proof of Theorem ecdmn0

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3212	. 2 |- ( A e. dom R -> A e. _V )
2		n0 3931	. . 3 |- ( [ A ] R =/= (/) <-> E. x x e. [ A ] R )
3		ecexr 7747	. . . 4 |- ( x e. [ A ] R -> A e. _V )
4	3	exlimiv 1858	. . 3 |- ( E. x x e. [ A ] R -> A e. _V )
5	2, 4	sylbi 207	. 2 |- ( [ A ] R =/= (/) -> A e. _V )
6		vex 3203	. . . . 5 |- x e. _V
7		elecg 7785	. . . . 5 |- ( ( x e. _V /\ A e. _V ) -> ( x e. [ A ] R <-> A R x ) )
8	6, 7	mpan 706	. . . 4 |- ( A e. _V -> ( x e. [ A ] R <-> A R x ) )
9	8	exbidv 1850	. . 3 |- ( A e. _V -> ( E. x x e. [ A ] R <-> E. x A R x ) )
10	2	a1i 11	. . 3 |- ( A e. _V -> ( [ A ] R =/= (/) <-> E. x x e. [ A ] R ) )
11		eldmg 5319	. . 3 |- ( A e. _V -> ( A e. dom R <-> E. x A R x ) )
12	9, 10, 11	3bitr4rd 301	. 2 |- ( A e. _V -> ( A e. dom R <-> [ A ] R =/= (/) ) )
13	1, 5, 12	pm5.21nii 368	1 |- ( A e. dom R <-> [ A ] R =/= (/) )

Syntax hints:   <-> wb 196   E.wex 1704   e. wcel 1990   =/= wne 2794   _Vcvv 3200   (/)c0 3915   class class class wbr 4653   dom cdm 5114   [cec 7740

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-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ec 7744

This theorem is referenced by:  ereldm  7790  elqsn0  7816  ecelqsdm  7817  eceqoveq  7853  divsfval  16207  sylow1lem5  18017  vitalilem2  23378  vitalilem3  23379  dfdm6  34071  dmecd  34074  n0elqs  34098