Theorem ecelqsdm 6608

Description: Membership of an equivalence class in a quotient set. (Contributed by NM, 30-Jul-1995.)

Assertion

Ref	Expression
ecelqsdm	|- ( ( dom R = A /\ [ B ] R e. ( A /. R ) ) -> B e. A )

Proof of Theorem ecelqsdm

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elqsn0m 6606	. . 3 |- ( ( dom R = A /\ [ B ] R e. ( A /. R ) ) -> E. x x e. [ B ] R )
2		ecdmn0m 6580	. . 3 |- ( B e. dom R <-> E. x x e. [ B ] R )
3	1, 2	sylibr 134	. 2 |- ( ( dom R = A /\ [ B ] R e. ( A /. R ) ) -> B e. dom R )
4		simpl 109	. 2 |- ( ( dom R = A /\ [ B ] R e. ( A /. R ) ) -> dom R = A )
5	3, 4	eleqtrd 2256	1 |- ( ( dom R = A /\ [ B ] R e. ( A /. R ) ) -> B e. A )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1353   E.wex 1492   e. wcel 2148   dom cdm 4628   [cec 6536   /.cqs 6537

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 4123  ax-pow 4176  ax-pr 4211

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 2741  df-sbc 2965  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-br 4006  df-opab 4067  df-xp 4634  df-cnv 4636  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-ec 6540  df-qs 6544

This theorem is referenced by:  th3qlem1  6640  nnnq0lem1  7448  prsrlem1  7744  gt0srpr  7750