Theorem ecss 7788

Description: An equivalence class is a subset of the domain. (Contributed by NM, 6-Aug-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)

Hypothesis

Ref	Expression
ecss.1	|- ( ph -> R Er X )

Assertion

Ref	Expression
ecss	|- ( ph -> [ A ] R C_ X )

Proof of Theorem ecss

Step	Hyp	Ref	Expression
1		df-ec 7744	. . 3 |- [ A ] R = ( R " { A } )
2		imassrn 5477	. . 3 |- ( R " { A } ) C_ ran R
3	1, 2	eqsstri 3635	. 2 |- [ A ] R C_ ran R
4		ecss.1	. . 3 |- ( ph -> R Er X )
5		errn 7764	. . 3 |- ( R Er X -> ran R = X )
6	4, 5	syl 17	. 2 |- ( ph -> ran R = X )
7	3, 6	syl5sseq 3653	1 |- ( ph -> [ A ] R C_ X )

Syntax hints:   -> wi 4   = wceq 1483   C_ wss 3574   {csn 4177   ran crn 5115   "cima 5117   Er wer 7739   [cec 7740

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-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

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-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-er 7742  df-ec 7744

This theorem is referenced by:  qsss  7808  divsfval  16207  sylow1lem5  18017  sylow2alem2  18033  sylow2blem1  18035  sylow3lem3  18044  vitalilem2  23378