Theorem eqrelriiv 4813

Description: Inference from extensionality principle for relations. (Contributed by NM, 17-Mar-1995.)

Hypotheses

Ref	Expression
eqreliiv.1	|- Rel A
eqreliiv.2	|- Rel B
eqreliiv.3	|- ( <. x , y >. e. A <-> <. x , y >. e. B )

Assertion

Ref	Expression
eqrelriiv	|- A = B

Distinct variable groups:   x, y, A   x, B, y

Proof of Theorem eqrelriiv

Step	Hyp	Ref	Expression
1		eqreliiv.1	. 2 |- Rel A
2		eqreliiv.2	. 2 |- Rel B
3		eqreliiv.3	. . 3 |- ( <. x , y >. e. A <-> <. x , y >. e. B )
4	3	eqrelriv 4812	. 2 |- ( ( Rel A /\ Rel B ) -> A = B )
5	1, 2, 4	mp2an 426	1 |- A = B

Syntax hints:   <-> wb 105   = wceq 1395   e. wcel 2200   <.cop 3669   Rel wrel 4724

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-opab 4146  df-xp 4725  df-rel 4726

This theorem is referenced by:  eqbrriv  4814  inopab  4854  difopab  4855  dfres2  5057  restidsing  5061  cnvopab  5130  cnv0  5132  cnvdif  5135  cnvcnvsn  5205  dfco2  5228  coiun  5238  co02  5242  coass  5247  ressn  5269