Theorem ecdmn0m 6534

Description: A representative of an inhabited equivalence class belongs to the domain of the equivalence relation. (Contributed by Jim Kingdon, 21-Aug-2019.)

Assertion

Ref	Expression
ecdmn0m	|- ( A e. dom R <-> E. x x e. [ A ] R )

Distinct variable groups:   x, R   x, A

Proof of Theorem ecdmn0m

Step	Hyp	Ref	Expression
1		elex 2732	. 2 |- ( A e. dom R -> A e. _V )
2		ecexr 6497	. . 3 |- ( x e. [ A ] R -> A e. _V )
3	2	exlimiv 1585	. 2 |- ( E. x x e. [ A ] R -> A e. _V )
4		eldmg 4793	. . 3 |- ( A e. _V -> ( A e. dom R <-> E. x A R x ) )
5		vex 2724	. . . . 5 |- x e. _V
6		elecg 6530	. . . . 5 |- ( ( x e. _V /\ A e. _V ) -> ( x e. [ A ] R <-> A R x ) )
7	5, 6	mpan 421	. . . 4 |- ( A e. _V -> ( x e. [ A ] R <-> A R x ) )
8	7	exbidv 1812	. . 3 |- ( A e. _V -> ( E. x x e. [ A ] R <-> E. x A R x ) )
9	4, 8	bitr4d 190	. 2 |- ( A e. _V -> ( A e. dom R <-> E. x x e. [ A ] R ) )
10	1, 3, 9	pm5.21nii 694	1 |- ( A e. dom R <-> E. x x e. [ A ] R )

Syntax hints:   <-> wb 104   E.wex 1479   e. wcel 2135   _Vcvv 2721   class class class wbr 3976   dom cdm 4598   [cec 6490

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-14 2138  ax-ext 2146  ax-sep 4094  ax-pow 4147  ax-pr 4181

This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ral 2447  df-rex 2448  df-v 2723  df-sbc 2947  df-un 3115  df-in 3117  df-ss 3124  df-pw 3555  df-sn 3576  df-pr 3577  df-op 3579  df-br 3977  df-opab 4038  df-xp 4604  df-cnv 4606  df-dm 4608  df-rn 4609  df-res 4610  df-ima 4611  df-ec 6494

This theorem is referenced by:  ereldm  6535  elqsn0m  6560  ecelqsdm  6562