Theorem ercl 6117

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 6115	. . . 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 4569	. . 3 |- ( ( Rel R /\ A R B ) -> A e. dom R )
6	3, 4, 5	syl2anc 391	. 2 |- ( ph -> A e. dom R )
7		erdm 6116	. . 3 |- ( R Er X -> dom R = X )
8	1, 7	syl 14	. 2 |- ( ph -> dom R = X )
9	6, 8	eleqtrd 2116	1 |- ( ph -> A e. X )

Syntax hints:   -> wi 4   = wceq 1243   e. wcel 1393   class class class wbr 3764   dom cdm 4345   Rel wrel 4350   Er wer 6103

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-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-dm 4355  df-er 6106

This theorem is referenced by:  ercl2  6119  erthi  6152  qliftfun  6188