Theorem ereldm 6172

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 2148	. . . 4 |- ( ph -> ( x e. [ A ] R <-> x e. [ B ] R ) )
3	2	exbidv 1746	. . 3 |- ( ph -> ( E. x x e. [ A ] R <-> E. x x e. [ B ] R ) )
4		ecdmn0m 6171	. . 3 |- ( A e. dom R <-> E. x x e. [ A ] R )
5		ecdmn0m 6171	. . 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 6139	. . . 4 |- ( R Er X -> dom R = X )
9	7, 8	syl 14	. . 3 |- ( ph -> dom R = X )
10	9	eleq2d 2148	. 2 |- ( ph -> ( A e. dom R <-> A e. X ) )
11	9	eleq2d 2148	. 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 1284   E.wex 1421   e. wcel 1433   dom cdm 4363   Er wer 6126   [cec 6127

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 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-sbc 2816  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786  df-opab 3840  df-xp 4369  df-cnv 4371  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-er 6129  df-ec 6131

This theorem is referenced by:  erth  6173  brecop  6219