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Theorem erssxp 6930
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 6916 . . 3  |-  ( R  Er  A  ->  Rel  R )
2 relssdmrn 5392 . . 3  |-  ( Rel 
R  ->  R  C_  ( dom  R  X.  ran  R
) )
31, 2syl 16 . 2  |-  ( R  Er  A  ->  R  C_  ( dom  R  X.  ran  R ) )
4 erdm 6917 . . 3  |-  ( R  Er  A  ->  dom  R  =  A )
5 errn 6929 . . 3  |-  ( R  Er  A  ->  ran  R  =  A )
64, 5xpeq12d 4905 . 2  |-  ( R  Er  A  ->  ( dom  R  X.  ran  R
)  =  ( A  X.  A ) )
73, 6sseqtrd 3386 1  |-  ( R  Er  A  ->  R  C_  ( A  X.  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    C_ wss 3322    X. cxp 4878   dom cdm 4880   ran crn 4881   Rel wrel 4885    Er wer 6904
This theorem is referenced by:  erex  6931  riiner  6979  efgval  15351
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-br 4215  df-opab 4269  df-xp 4886  df-rel 4887  df-cnv 4888  df-dm 4890  df-rn 4891  df-er 6907
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