Theorem eqrelriiv 4864

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 4863	. 2 |- ( ( Rel A /\ Rel B ) -> A = B )
5	1, 2, 4	mp2an 430	1 |- A = B

Syntax hints:   <-> wb 105   = wceq 1402   e. wcel 2209   <.cop 3708   Rel wrel 4774

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-opab 4188  df-xp 4775  df-rel 4776

This theorem is referenced by:  eqbrriv  4865  inopab  4907  difopab  4908  dfres2  5110  restidsing  5114  cnvopab  5184  cnv0  5186  cnvdif  5189  cnvcnvsn  5259  dfco2  5282  coiun  5292  co02  5296  coass  5301  ressn  5323