Theorem ereldm 6333

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 2157	. . . 4 |- ( ph -> ( x e. [ A ] R <-> x e. [ B ] R ) )
3	2	exbidv 1753	. . 3 |- ( ph -> ( E. x x e. [ A ] R <-> E. x x e. [ B ] R ) )
4		ecdmn0m 6332	. . 3 |- ( A e. dom R <-> E. x x e. [ A ] R )
5		ecdmn0m 6332	. . 3 |- ( B e. dom R <-> E. x x e. [ B ] R )
6	3, 4, 5	3bitr4g 221	. 2 |- ( ph -> ( A e. dom R <-> B e. dom R ) )
7		ereldm.1	. . . 4 |- ( ph -> R Er X )
8		erdm 6300	. . . 4 |- ( R Er X -> dom R = X )
9	7, 8	syl 14	. . 3 |- ( ph -> dom R = X )
10	9	eleq2d 2157	. 2 |- ( ph -> ( A e. dom R <-> A e. X ) )
11	9	eleq2d 2157	. 2 |- ( ph -> ( B e. dom R <-> B e. X ) )
12	6, 10, 11	3bitr3d 216	1 |- ( ph -> ( A e. X <-> B e. X ) )

Syntax hints:   -> wi 4   <-> wb 103   = wceq 1289   E.wex 1426   e. wcel 1438   dom cdm 4438   Er wer 6287   [cec 6288

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2841  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-br 3846  df-opab 3900  df-xp 4444  df-cnv 4446  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-er 6290  df-ec 6292

This theorem is referenced by:  erth  6334  brecop  6380