Theorem ecss 6809

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 6768	. . 3 |- [ A ] R = ( R " { A } )
2		imassrn 5111	. . 3 |- ( R " { A } ) C_ ran R
3	1, 2	eqsstri 3269	. 2 |- [ A ] R C_ ran R
4		ecss.1	. . 3 |- ( ph -> R Er X )
5		errn 6788	. . 3 |- ( R Er X -> ran R = X )
6	4, 5	syl 14	. 2 |- ( ph -> ran R = X )
7	3, 6	sseqtrid 3287	1 |- ( ph -> [ A ] R C_ X )

Syntax hints:   -> wi 4   = wceq 1398   C_ wss 3210   {csn 3688   ran crn 4749   "cima 4751   Er wer 6763   [cec 6764

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4227  ax-pow 4286  ax-pr 4321

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2814  df-un 3214  df-in 3216  df-ss 3223  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-br 4109  df-opab 4171  df-xp 4754  df-rel 4755  df-cnv 4756  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-er 6766  df-ec 6768

This theorem is referenced by:  qsss  6827  divsfval  13533  divsfvalg  13534