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Theorem erssxp 6699
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 6685 . . 3  |-  ( R  Er  A  ->  Rel  R )
2 relssdmrn 5209 . . 3  |-  ( Rel 
R  ->  R  C_  ( dom  R  X.  ran  R
) )
31, 2syl 15 . 2  |-  ( R  Er  A  ->  R  C_  ( dom  R  X.  ran  R ) )
4 erdm 6686 . . 3  |-  ( R  Er  A  ->  dom  R  =  A )
5 errn 6698 . . 3  |-  ( R  Er  A  ->  ran  R  =  A )
64, 5xpeq12d 4730 . 2  |-  ( R  Er  A  ->  ( dom  R  X.  ran  R
)  =  ( A  X.  A ) )
73, 6sseqtrd 3227 1  |-  ( R  Er  A  ->  R  C_  ( A  X.  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    C_ wss 3165    X. cxp 4703   dom cdm 4705   ran crn 4706   Rel wrel 4710    Er wer 6673
This theorem is referenced by:  erex  6700  riiner  6748  efgval  15042
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-xp 4711  df-rel 4712  df-cnv 4713  df-dm 4715  df-rn 4716  df-er 6676
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