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Theorem erssxp 6445
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 6431 . . 3  |-  ( R  Er  A  ->  Rel  R )
2 relssdmrn 5054 . . 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 6432 . . 3  |-  ( R  Er  A  ->  dom  R  =  A )
5 errn 6444 . . 3  |-  ( R  Er  A  ->  ran  R  =  A )
64, 5xpeq12d 4559 . 2  |-  ( R  Er  A  ->  ( dom  R  X.  ran  R
)  =  ( A  X.  A ) )
73, 6sseqtrd 3130 1  |-  ( R  Er  A  ->  R  C_  ( A  X.  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    C_ wss 3066    X. cxp 4532   dom cdm 4534   ran crn 4535   Rel wrel 4539    Er wer 6419
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-br 3925  df-opab 3985  df-xp 4540  df-rel 4541  df-cnv 4542  df-dm 4544  df-rn 4545  df-er 6422
This theorem is referenced by:  erex  6446  riinerm  6495
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