Theorem ecdmn0m 6666

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 2783	. 2 |- ( A e. dom R -> A e. _V )
2		ecexr 6627	. . 3 |- ( x e. [ A ] R -> A e. _V )
3	2	exlimiv 1621	. 2 |- ( E. x x e. [ A ] R -> A e. _V )
4		eldmg 4874	. . 3 |- ( A e. _V -> ( A e. dom R <-> E. x A R x ) )
5		vex 2775	. . . . 5 |- x e. _V
6		elecg 6662	. . . . 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 1848	. . 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 706	1 |- ( A e. dom R <-> E. x x e. [ A ] R )

Syntax hints:   <-> wb 105   E.wex 1515   e. wcel 2176   _Vcvv 2772   class class class wbr 4045   dom cdm 4676   [cec 6620

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4163  ax-pow 4219  ax-pr 4254

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-sbc 2999  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-br 4046  df-opab 4107  df-xp 4682  df-cnv 4684  df-dm 4686  df-rn 4687  df-res 4688  df-ima 4689  df-ec 6624

This theorem is referenced by:  ereldm  6667  elqsn0m  6692  ecelqsdm  6694  divsfval  13193