Theorem ecelqsdm 6499

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 6497	. . 3 |- ( ( dom R = A /\ [ B ] R e. ( A /. R ) ) -> E. x x e. [ B ] R )
2		ecdmn0m 6471	. . 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 2218	1 |- ( ( dom R = A /\ [ B ] R e. ( A /. R ) ) -> B e. A )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1331   E.wex 1468   e. wcel 1480   dom cdm 4539   [cec 6427   /.cqs 6428

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-opab 3990  df-xp 4545  df-cnv 4547  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-ec 6431  df-qs 6435

This theorem is referenced by:  th3qlem1  6531  nnnq0lem1  7254  prsrlem1  7550  gt0srpr  7556