Theorem ercl 6319

Description: Elementhood in the field of an equivalence relation. (Contributed by Mario Carneiro, 12-Aug-2015.)

Hypotheses

Ref	Expression
ersym.1	|- ( ph -> R Er X )
ersym.2	|- ( ph -> A R B )

Assertion

Ref	Expression
ercl	|- ( ph -> A e. X )

Proof of Theorem ercl

Step	Hyp	Ref	Expression
1		ersym.1	. . . 4 |- ( ph -> R Er X )
2		errel 6317	. . . 4 |- ( R Er X -> Rel R )
3	1, 2	syl 14	. . 3 |- ( ph -> Rel R )
4		ersym.2	. . 3 |- ( ph -> A R B )
5		releldm 4685	. . 3 |- ( ( Rel R /\ A R B ) -> A e. dom R )
6	3, 4, 5	syl2anc 404	. 2 |- ( ph -> A e. dom R )
7		erdm 6318	. . 3 |- ( R Er X -> dom R = X )
8	1, 7	syl 14	. 2 |- ( ph -> dom R = X )
9	6, 8	eleqtrd 2167	1 |- ( ph -> A e. X )

Syntax hints:   -> wi 4   = wceq 1290   e. wcel 1439   class class class wbr 3853   dom cdm 4454   Rel wrel 4459   Er wer 6305

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3965  ax-pow 4017  ax-pr 4047

This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2624  df-un 3006  df-in 3008  df-ss 3015  df-pw 3437  df-sn 3458  df-pr 3459  df-op 3461  df-br 3854  df-opab 3908  df-xp 4460  df-rel 4461  df-dm 4464  df-er 6308

This theorem is referenced by:  ercl2  6321  erthi  6354  qliftfun  6390