Theorem ecdmn0m 6813

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 2827	. 2 |- ( A e. dom R -> A e. _V )
2		ecexr 6774	. . 3 |- ( x e. [ A ] R -> A e. _V )
3	2	exlimiv 1647	. 2 |- ( E. x x e. [ A ] R -> A e. _V )
4		eldmg 4953	. . 3 |- ( A e. _V -> ( A e. dom R <-> E. x A R x ) )
5		vex 2818	. . . . 5 |- x e. _V
6		elecg 6809	. . . . 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 1874	. . 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 712	1 |- ( A e. dom R <-> E. x x e. [ A ] R )

Syntax hints:   <-> wb 105   E.wex 1541   e. wcel 2205   _Vcvv 2815   class class class wbr 4111   dom cdm 4751   [cec 6767

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-sbc 3045  df-un 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-br 4112  df-opab 4174  df-xp 4757  df-cnv 4759  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-ec 6771

This theorem is referenced by:  ereldm  6814  elqsn0m  6839  ecelqsdm  6841  divsfval  13558