Theorem ecelqsdm 6295

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 6293	. . 3 |- ( ( dom R = A /\ [ B ] R e. ( A /. R ) ) -> E. x x e. [ B ] R )
2		ecdmn0m 6267	. . 3 |- ( B e. dom R <-> E. x x e. [ B ] R )
3	1, 2	sylibr 132	. 2 |- ( ( dom R = A /\ [ B ] R e. ( A /. R ) ) -> B e. dom R )
4		simpl 107	. 2 |- ( ( dom R = A /\ [ B ] R e. ( A /. R ) ) -> dom R = A )
5	3, 4	eleqtrd 2163	1 |- ( ( dom R = A /\ [ B ] R e. ( A /. R ) ) -> B e. A )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1287   E.wex 1424   e. wcel 1436   dom cdm 4404   [cec 6223   /.cqs 6224

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3925  ax-pow 3977  ax-pr 4003

This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-v 2616  df-sbc 2829  df-un 2990  df-in 2992  df-ss 2999  df-pw 3411  df-sn 3431  df-pr 3432  df-op 3434  df-br 3815  df-opab 3869  df-xp 4410  df-cnv 4412  df-dm 4414  df-rn 4415  df-res 4416  df-ima 4417  df-ec 6227  df-qs 6231

This theorem is referenced by:  th3qlem1  6327  nnnq0lem1  6926  prsrlem1  7209  gt0srpr  7215