Theorem ereldm 6814

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 2304	. . . 4 |- ( ph -> ( x e. [ A ] R <-> x e. [ B ] R ) )
3	2	exbidv 1874	. . 3 |- ( ph -> ( E. x x e. [ A ] R <-> E. x x e. [ B ] R ) )
4		ecdmn0m 6813	. . 3 |- ( A e. dom R <-> E. x x e. [ A ] R )
5		ecdmn0m 6813	. . 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 6779	. . . 4 |- ( R Er X -> dom R = X )
9	7, 8	syl 14	. . 3 |- ( ph -> dom R = X )
10	9	eleq2d 2304	. 2 |- ( ph -> ( A e. dom R <-> A e. X ) )
11	9	eleq2d 2304	. 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 1398   E.wex 1541   e. wcel 2205   dom cdm 4751   Er wer 6766   [cec 6767

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-sbc 3045  df-un 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-br 4112  df-opab 4174  df-xp 4757  df-cnv 4759  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-er 6769  df-ec 6771

This theorem is referenced by:  erth  6815  brecop  6861