Theorem eqrelriiv 4844

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 4843	. 2 |- ( ( Rel A /\ Rel B ) -> A = B )
5	1, 2, 4	mp2an 426	1 |- A = B

Syntax hints:   <-> wb 105   = wceq 1398   e. wcel 2203   <.cop 3692   Rel wrel 4754

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 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2815  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-opab 4172  df-xp 4755  df-rel 4756

This theorem is referenced by:  eqbrriv  4845  inopab  4887  difopab  4888  dfres2  5090  restidsing  5094  cnvopab  5164  cnv0  5166  cnvdif  5169  cnvcnvsn  5239  dfco2  5262  coiun  5272  co02  5276  coass  5281  ressn  5303