Theorem ecdmn0m 6724

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 2811	. 2 |- ( A e. dom R -> A e. _V )
2		ecexr 6685	. . 3 |- ( x e. [ A ] R -> A e. _V )
3	2	exlimiv 1644	. 2 |- ( E. x x e. [ A ] R -> A e. _V )
4		eldmg 4918	. . 3 |- ( A e. _V -> ( A e. dom R <-> E. x A R x ) )
5		vex 2802	. . . . 5 |- x e. _V
6		elecg 6720	. . . . 5 |- ( ( x e. _V /\ A e. _V ) -> ( x e. [ A ] R <-> A R x ) )
7	5, 6	mpan 424	. . . 4 |- ( A e. _V -> ( x e. [ A ] R <-> A R x ) )
8	7	exbidv 1871	. . 3 |- ( A e. _V -> ( E. x x e. [ A ] R <-> E. x A R x ) )
9	4, 8	bitr4d 191	. 2 |- ( A e. _V -> ( A e. dom R <-> E. x x e. [ A ] R ) )
10	1, 3, 9	pm5.21nii 709	1 |- ( A e. dom R <-> E. x x e. [ A ] R )

Syntax hints:   <-> wb 105   E.wex 1538   e. wcel 2200   _Vcvv 2799   class class class wbr 4083   dom cdm 4719   [cec 6678

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-sbc 3029  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-br 4084  df-opab 4146  df-xp 4725  df-cnv 4727  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-ec 6682

This theorem is referenced by:  ereldm  6725  elqsn0m  6750  ecelqsdm  6752  divsfval  13361