Theorem erssxp 8648

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 8634	. . 3 ⊢ (𝑅 Er 𝐴 → Rel 𝑅)
2		relssdmrn 6217	. . 3 ⊢ (Rel 𝑅 → 𝑅 ⊆ (dom 𝑅 × ran 𝑅))
3	1, 2	syl 17	. 2 ⊢ (𝑅 Er 𝐴 → 𝑅 ⊆ (dom 𝑅 × ran 𝑅))
4		erdm 8635	. . 3 ⊢ (𝑅 Er 𝐴 → dom 𝑅 = 𝐴)
5		errn 8647	. . 3 ⊢ (𝑅 Er 𝐴 → ran 𝑅 = 𝐴)
6	4, 5	xpeq12d 5650	. 2 ⊢ (𝑅 Er 𝐴 → (dom 𝑅 × ran 𝑅) = (𝐴 × 𝐴))
7	3, 6	sseqtrd 3972	1 ⊢ (𝑅 Er 𝐴 → 𝑅 ⊆ (𝐴 × 𝐴))

Syntax hints:   → wi 4   ⊆ wss 3903   × cxp 5617  dom cdm 5619  ran crn 5620  Rel wrel 5624   Er wer 8622

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pr 5371

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-dif 3906  df-un 3908  df-ss 3920  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-br 5093  df-opab 5155  df-xp 5625  df-rel 5626  df-cnv 5627  df-dm 5629  df-rn 5630  df-er 8625

This theorem is referenced by:  erex  8649  riiner  8717  efgval  19596  qtophaus  33809