Theorem erssxp 8704

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 8690	. . 3 ⊢ (𝑅 Er 𝐴 → Rel 𝑅)
2		relssdmrn 6258	. . 3 ⊢ (Rel 𝑅 → 𝑅 ⊆ (dom 𝑅 × ran 𝑅))
3	1, 2	syl 17	. 2 ⊢ (𝑅 Er 𝐴 → 𝑅 ⊆ (dom 𝑅 × ran 𝑅))
4		erdm 8691	. . 3 ⊢ (𝑅 Er 𝐴 → dom 𝑅 = 𝐴)
5		errn 8703	. . 3 ⊢ (𝑅 Er 𝐴 → ran 𝑅 = 𝐴)
6	4, 5	xpeq12d 5680	. 2 ⊢ (𝑅 Er 𝐴 → (dom 𝑅 × ran 𝑅) = (𝐴 × 𝐴))
7	3, 6	sseqtrd 3974	1 ⊢ (𝑅 Er 𝐴 → 𝑅 ⊆ (𝐴 × 𝐴))

Syntax hints:   → wi 4   ⊆ wss 3906   × cxp 5647  dom cdm 5649  ran crn 5650  Rel wrel 5654   Er wer 8677

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-ext 2736  ax-sep 5248  ax-pr 5392

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-ral 3079  df-rex 3089  df-rab 3417  df-v 3458  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5103  df-opab 5165  df-xp 5655  df-rel 5656  df-cnv 5657  df-dm 5659  df-rn 5660  df-er 8680

This theorem is referenced by:  erex  8705  riiner  8774  efgval  19759  qtophaus  34135