Theorem ecdmn0m 6572

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 2748	. 2 |- ( A e. dom R -> A e. _V )
2		ecexr 6535	. . 3 |- ( x e. [ A ] R -> A e. _V )
3	2	exlimiv 1598	. 2 |- ( E. x x e. [ A ] R -> A e. _V )
4		eldmg 4819	. . 3 |- ( A e. _V -> ( A e. dom R <-> E. x A R x ) )
5		vex 2740	. . . . 5 |- x e. _V
6		elecg 6568	. . . . 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 1825	. . 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 704	1 |- ( A e. dom R <-> E. x x e. [ A ] R )

Syntax hints:   <-> wb 105   E.wex 1492   e. wcel 2148   _Vcvv 2737   class class class wbr 4001   dom cdm 4624   [cec 6528

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 4119  ax-pow 4172  ax-pr 4207

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-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-br 4002  df-opab 4063  df-xp 4630  df-cnv 4632  df-dm 4634  df-rn 4635  df-res 4636  df-ima 4637  df-ec 6532

This theorem is referenced by:  ereldm  6573  elqsn0m  6598  ecelqsdm  6600