Theorem erssxp 6610

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 6596	. . 3 |- ( R Er A -> Rel R )
2		relssdmrn 5186	. . 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 6597	. . 3 |- ( R Er A -> dom R = A )
5		errn 6609	. . 3 |- ( R Er A -> ran R = A )
6	4, 5	xpeq12d 4684	. 2 |- ( R Er A -> ( dom R X. ran R ) = ( A X. A ) )
7	3, 6	sseqtrd 3217	1 |- ( R Er A -> R C_ ( A X. A ) )

Syntax hints:   -> wi 4   C_ wss 3153   X. cxp 4657   dom cdm 4659   ran crn 4660   Rel wrel 4664   Er wer 6584

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 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-br 4030  df-opab 4091  df-xp 4665  df-rel 4666  df-cnv 4667  df-dm 4669  df-rn 4670  df-er 6587

This theorem is referenced by:  erex  6611  riinerm  6662