Theorem ereldm 6632

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 2263	. . . 4 |- ( ph -> ( x e. [ A ] R <-> x e. [ B ] R ) )
3	2	exbidv 1836	. . 3 |- ( ph -> ( E. x x e. [ A ] R <-> E. x x e. [ B ] R ) )
4		ecdmn0m 6631	. . 3 |- ( A e. dom R <-> E. x x e. [ A ] R )
5		ecdmn0m 6631	. . 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 6597	. . . 4 |- ( R Er X -> dom R = X )
9	7, 8	syl 14	. . 3 |- ( ph -> dom R = X )
10	9	eleq2d 2263	. 2 |- ( ph -> ( A e. dom R <-> A e. X ) )
11	9	eleq2d 2263	. 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 1364   E.wex 1503   e. wcel 2164   dom cdm 4659   Er wer 6584   [cec 6585

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2986  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-br 4030  df-opab 4091  df-xp 4665  df-cnv 4667  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-er 6587  df-ec 6589

This theorem is referenced by:  erth  6633  brecop  6679