Theorem erssxp 6452

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

Step	Hyp	Ref	Expression
1		errel 6438	. . 3 |- ( R Er A -> Rel R )
2		relssdmrn 5059	. . 3 |- ( Rel R -> R C_ ( dom R X. ran R ) )
3	1, 2	syl 14	. 2 |- ( R Er A -> R C_ ( dom R X. ran R ) )
4		erdm 6439	. . 3 |- ( R Er A -> dom R = A )
5		errn 6451	. . 3 |- ( R Er A -> ran R = A )
6	4, 5	xpeq12d 4564	. 2 |- ( R Er A -> ( dom R X. ran R ) = ( A X. A ) )
7	3, 6	sseqtrd 3135	1 |- ( R Er A -> R C_ ( A X. A ) )

Syntax hints:   -> wi 4   C_ wss 3071   X. cxp 4537   dom cdm 4539   ran crn 4540   Rel wrel 4544   Er wer 6426

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 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-opab 3990  df-xp 4545  df-rel 4546  df-cnv 4547  df-dm 4549  df-rn 4550  df-er 6429

This theorem is referenced by:  erex  6453  riinerm  6502