Theorem ecelqsdm 6507

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 6505	. . 3 |- ( ( dom R = A /\ [ B ] R e. ( A /. R ) ) -> E. x x e. [ B ] R )
2		ecdmn0m 6479	. . 3 |- ( B e. dom R <-> E. x x e. [ B ] R )
3	1, 2	sylibr 133	. 2 |- ( ( dom R = A /\ [ B ] R e. ( A /. R ) ) -> B e. dom R )
4		simpl 108	. 2 |- ( ( dom R = A /\ [ B ] R e. ( A /. R ) ) -> dom R = A )
5	3, 4	eleqtrd 2219	1 |- ( ( dom R = A /\ [ B ] R e. ( A /. R ) ) -> B e. A )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1332   E.wex 1469   e. wcel 1481   dom cdm 4547   [cec 6435   /.cqs 6436

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-sbc 2914  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-br 3938  df-opab 3998  df-xp 4553  df-cnv 4555  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-ec 6439  df-qs 6443

This theorem is referenced by:  th3qlem1  6539  nnnq0lem1  7278  prsrlem1  7574  gt0srpr  7580