Theorem ecss 6147

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 6108	. . 3 |- [ A ] R = ( R " { A } )
2		imassrn 4679	. . 3 |- ( R " { A } ) C_ ran R
3	1, 2	eqsstri 2975	. 2 |- [ A ] R C_ ran R
4		ecss.1	. . 3 |- ( ph -> R Er X )
5		errn 6128	. . 3 |- ( R Er X -> ran R = X )
6	4, 5	syl 14	. 2 |- ( ph -> ran R = X )
7	3, 6	syl5sseq 2993	1 |- ( ph -> [ A ] R C_ X )

Syntax hints:   -> wi 4   = wceq 1243   C_ wss 2917   {csn 3375   ran crn 4346   "cima 4348   Er wer 6103   [cec 6104

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-xp 4351  df-rel 4352  df-cnv 4353  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-er 6106  df-ec 6108

This theorem is referenced by:  qsss  6165