Theorem erssxp 6524

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 6510	. . 3 |- ( R Er A -> Rel R )
2		relssdmrn 5124	. . 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 6511	. . 3 |- ( R Er A -> dom R = A )
5		errn 6523	. . 3 |- ( R Er A -> ran R = A )
6	4, 5	xpeq12d 4629	. 2 |- ( R Er A -> ( dom R X. ran R ) = ( A X. A ) )
7	3, 6	sseqtrd 3180	1 |- ( R Er A -> R C_ ( A X. A ) )

Syntax hints:   -> wi 4   C_ wss 3116   X. cxp 4602   dom cdm 4604   ran crn 4605   Rel wrel 4609   Er wer 6498

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-xp 4610  df-rel 4611  df-cnv 4612  df-dm 4614  df-rn 4615  df-er 6501

This theorem is referenced by:  erex  6525  riinerm  6574