Theorem ecss 6713

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 6672	. . 3 |- [ A ] R = ( R " { A } )
2		imassrn 5075	. . 3 |- ( R " { A } ) C_ ran R
3	1, 2	eqsstri 3256	. 2 |- [ A ] R C_ ran R
4		ecss.1	. . 3 |- ( ph -> R Er X )
5		errn 6692	. . 3 |- ( R Er X -> ran R = X )
6	4, 5	syl 14	. 2 |- ( ph -> ran R = X )
7	3, 6	sseqtrid 3274	1 |- ( ph -> [ A ] R C_ X )

Syntax hints:   -> wi 4   = wceq 1395   C_ wss 3197   {csn 3666   ran crn 4717   "cima 4719   Er wer 6667   [cec 6668

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-br 4083  df-opab 4145  df-xp 4722  df-rel 4723  df-cnv 4724  df-dm 4726  df-rn 4727  df-res 4728  df-ima 4729  df-er 6670  df-ec 6672

This theorem is referenced by:  qsss  6731  divsfval  13347  divsfvalg  13348