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Theorem erssxp 6581
Description: An equivalence relation is a subset of the cartesian product of the field. (Contributed by Mario Carneiro, 12-Aug-2015.)
Assertion
Ref Expression
erssxp  |-  ( R  Er  A  ->  R  C_  ( A  X.  A
) )

Proof of Theorem erssxp
StepHypRef Expression
1 errel 6567 . . 3  |-  ( R  Er  A  ->  Rel  R )
2 relssdmrn 5167 . . 3  |-  ( Rel 
R  ->  R  C_  ( dom  R  X.  ran  R
) )
31, 2syl 14 . 2  |-  ( R  Er  A  ->  R  C_  ( dom  R  X.  ran  R ) )
4 erdm 6568 . . 3  |-  ( R  Er  A  ->  dom  R  =  A )
5 errn 6580 . . 3  |-  ( R  Er  A  ->  ran  R  =  A )
64, 5xpeq12d 4669 . 2  |-  ( R  Er  A  ->  ( dom  R  X.  ran  R
)  =  ( A  X.  A ) )
73, 6sseqtrd 3208 1  |-  ( R  Er  A  ->  R  C_  ( A  X.  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    C_ wss 3144    X. cxp 4642   dom cdm 4644   ran crn 4645   Rel wrel 4649    Er wer 6555
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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-br 4019  df-opab 4080  df-xp 4650  df-rel 4651  df-cnv 4652  df-dm 4654  df-rn 4655  df-er 6558
This theorem is referenced by:  erex  6582  riinerm  6633
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