Theorem erssxp 6583

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 ⊆ (𝐴 × 𝐴))

Proof of Theorem erssxp

Step	Hyp	Ref	Expression
1		errel 6569	. . 3 ⊢ (𝑅 Er 𝐴 → Rel 𝑅)
2		relssdmrn 5167	. . 3 ⊢ (Rel 𝑅 → 𝑅 ⊆ (dom 𝑅 × ran 𝑅))
3	1, 2	syl 14	. 2 ⊢ (𝑅 Er 𝐴 → 𝑅 ⊆ (dom 𝑅 × ran 𝑅))
4		erdm 6570	. . 3 ⊢ (𝑅 Er 𝐴 → dom 𝑅 = 𝐴)
5		errn 6582	. . 3 ⊢ (𝑅 Er 𝐴 → ran 𝑅 = 𝐴)
6	4, 5	xpeq12d 4669	. 2 ⊢ (𝑅 Er 𝐴 → (dom 𝑅 × ran 𝑅) = (𝐴 × 𝐴))
7	3, 6	sseqtrd 3208	1 ⊢ (𝑅 Er 𝐴 → 𝑅 ⊆ (𝐴 × 𝐴))

Syntax hints:   → wi 4   ⊆ wss 3144   × cxp 4642  dom cdm 4644  ran crn 4645  Rel wrel 4649   Er wer 6557

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 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-br 4019  df-opab 4080  df-xp 4650  df-rel 4651  df-cnv 4652  df-dm 4654  df-rn 4655  df-er 6560

This theorem is referenced by:  erex  6584  riinerm  6635