Theorem erssxp 6830

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 6816	. . 3 ⊢ (𝑅 Er 𝐴 → Rel 𝑅)
2		relssdmrn 5308	. . 3 ⊢ (Rel 𝑅 → 𝑅 ⊆ (dom 𝑅 × ran 𝑅))
3	1, 2	syl 14	. 2 ⊢ (𝑅 Er 𝐴 → 𝑅 ⊆ (dom 𝑅 × ran 𝑅))
4		erdm 6817	. . 3 ⊢ (𝑅 Er 𝐴 → dom 𝑅 = 𝐴)
5		errn 6829	. . 3 ⊢ (𝑅 Er 𝐴 → ran 𝑅 = 𝐴)
6	4, 5	xpeq12d 4799	. 2 ⊢ (𝑅 Er 𝐴 → (dom 𝑅 × ran 𝑅) = (𝐴 × 𝐴))
7	3, 6	sseqtrd 3286	1 ⊢ (𝑅 Er 𝐴 → 𝑅 ⊆ (𝐴 × 𝐴))

This proof depends on syntax axioms:   → wi 4   ⊆ wss 3220   × cxp 4772  dom cdm 4774  ran crn 4775  Rel wrel 4779   Er wer 6804

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-br 4131  df-opab 4193  df-xp 4780  df-rel 4781  df-cnv 4782  df-dm 4784  df-rn 4785  df-er 6807

This theorem is used by:  erex  6831  riinerm  6882