Theorem ecelqsi 7803

Description: Membership of an equivalence class in a quotient set. (Contributed by NM, 25-Jul-1995.) (Revised by Mario Carneiro, 9-Jul-2014.)

Hypothesis

Ref	Expression
ecelqsi.1	|- R e. _V

Assertion

Ref	Expression
ecelqsi	|- ( B e. A -> [ B ] R e. ( A /. R ) )

Proof of Theorem ecelqsi

Step	Hyp	Ref	Expression
1		ecelqsi.1	. 2 |- R e. _V
2		ecelqsg 7802	. 2 |- ( ( R e. _V /\ B e. A ) -> [ B ] R e. ( A /. R ) )
3	1, 2	mpan 706	1 |- ( B e. A -> [ B ] R e. ( A /. R ) )

Syntax hints:   -> wi 4   e. wcel 1990   _Vcvv 3200   [cec 7740   /.cqs 7741

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-xp 5120  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ec 7744  df-qs 7748

This theorem is referenced by:  ecopqsi  7804  addsrpr  9896  mulsrpr  9897  0r  9901  1sr  9902  m1r  9903  addclsr  9904  mulclsr  9905  quseccl  17650  orbsta  17746  frgpeccl  18174  qustgphaus  21926  vitalilem2  23378  vitalilem3  23379  pstmfval  29939