Theorem eqrelriiv 4934

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 4933	. 2 |- ( ( Rel A /\ Rel B ) -> A = B )
5	1, 2, 4	mp2an 686	1 |- A = B

Syntax hints:   <-> wb 189   = wceq 1452   e. wcel 1904   <.cop 3965   Rel wrel 4844

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-opab 4455  df-xp 4845  df-rel 4846

This theorem is referenced by:  eqbrriv  4935  inopab  4970  difopab  4971  dfres2  5163  restidsing  5167  cnvopab  5243  cnv0  5245  cnvdif  5248  difxp  5267  cnvcnvsn  5320  dfco2  5341  coiun  5352  co02  5356  coass  5361  ressn  5379  ovoliunlem1  22533  h2hlm  26714  cnvco1  30471  cnvco2  30472  cnviun  36313  coiun1  36315