Theorem ecss 6670

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 6629	. . 3 |- [ A ] R = ( R " { A } )
2		imassrn 5038	. . 3 |- ( R " { A } ) C_ ran R
3	1, 2	eqsstri 3226	. 2 |- [ A ] R C_ ran R
4		ecss.1	. . 3 |- ( ph -> R Er X )
5		errn 6649	. . 3 |- ( R Er X -> ran R = X )
6	4, 5	syl 14	. 2 |- ( ph -> ran R = X )
7	3, 6	sseqtrid 3244	1 |- ( ph -> [ A ] R C_ X )

Syntax hints:   -> wi 4   = wceq 1373   C_ wss 3167   {csn 3634   ran crn 4680   "cima 4682   Er wer 6624   [cec 6625

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2180  ax-ext 2188  ax-sep 4166  ax-pow 4222  ax-pr 4257

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-v 2775  df-un 3171  df-in 3173  df-ss 3180  df-pw 3619  df-sn 3640  df-pr 3641  df-op 3643  df-br 4048  df-opab 4110  df-xp 4685  df-rel 4686  df-cnv 4687  df-dm 4689  df-rn 4690  df-res 4691  df-ima 4692  df-er 6627  df-ec 6629

This theorem is referenced by:  qsss  6688  divsfval  13204  divsfvalg  13205