Theorem ecelqsdm 6752

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 6750	. . 3 |- ( ( dom R = A /\ [ B ] R e. ( A /. R ) ) -> E. x x e. [ B ] R )
2		ecdmn0m 6724	. . 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 2308	1 |- ( ( dom R = A /\ [ B ] R e. ( A /. R ) ) -> B e. A )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   E.wex 1538   e. wcel 2200   dom cdm 4719   [cec 6678   /.cqs 6679

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  df-qs 6686

This theorem is referenced by:  th3qlem1  6784  nnnq0lem1  7633  prsrlem1  7929  gt0srpr  7935