Theorem ereldm 6580

Description: Equality of equivalence classes implies equivalence of domain membership. (Contributed by NM, 28-Jan-1996.) (Revised by Mario Carneiro, 12-Aug-2015.)

Hypotheses

Ref	Expression
ereldm.1	|- ( ph -> R Er X )
ereldm.2	|- ( ph -> [ A ] R = [ B ] R )

Assertion

Ref	Expression
ereldm	|- ( ph -> ( A e. X <-> B e. X ) )

Proof of Theorem ereldm

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ereldm.2	. . . . 5 |- ( ph -> [ A ] R = [ B ] R )
2	1	eleq2d 2247	. . . 4 |- ( ph -> ( x e. [ A ] R <-> x e. [ B ] R ) )
3	2	exbidv 1825	. . 3 |- ( ph -> ( E. x x e. [ A ] R <-> E. x x e. [ B ] R ) )
4		ecdmn0m 6579	. . 3 |- ( A e. dom R <-> E. x x e. [ A ] R )
5		ecdmn0m 6579	. . 3 |- ( B e. dom R <-> E. x x e. [ B ] R )
6	3, 4, 5	3bitr4g 223	. 2 |- ( ph -> ( A e. dom R <-> B e. dom R ) )
7		ereldm.1	. . . 4 |- ( ph -> R Er X )
8		erdm 6547	. . . 4 |- ( R Er X -> dom R = X )
9	7, 8	syl 14	. . 3 |- ( ph -> dom R = X )
10	9	eleq2d 2247	. 2 |- ( ph -> ( A e. dom R <-> A e. X ) )
11	9	eleq2d 2247	. 2 |- ( ph -> ( B e. dom R <-> B e. X ) )
12	6, 10, 11	3bitr3d 218	1 |- ( ph -> ( A e. X <-> B e. X ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1353   E.wex 1492   e. wcel 2148   dom cdm 4628   Er wer 6534   [cec 6535

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-sbc 2965  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-br 4006  df-opab 4067  df-xp 4634  df-cnv 4636  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-er 6537  df-ec 6539

This theorem is referenced by:  erth  6581  brecop  6627