Theorem erssxp 6642

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 6628	. . 3 ⊢ (𝑅 Er 𝐴 → Rel 𝑅)
2		relssdmrn 5202	. . 3 ⊢ (Rel 𝑅 → 𝑅 ⊆ (dom 𝑅 × ran 𝑅))
3	1, 2	syl 14	. 2 ⊢ (𝑅 Er 𝐴 → 𝑅 ⊆ (dom 𝑅 × ran 𝑅))
4		erdm 6629	. . 3 ⊢ (𝑅 Er 𝐴 → dom 𝑅 = 𝐴)
5		errn 6641	. . 3 ⊢ (𝑅 Er 𝐴 → ran 𝑅 = 𝐴)
6	4, 5	xpeq12d 4699	. 2 ⊢ (𝑅 Er 𝐴 → (dom 𝑅 × ran 𝑅) = (𝐴 × 𝐴))
7	3, 6	sseqtrd 3230	1 ⊢ (𝑅 Er 𝐴 → 𝑅 ⊆ (𝐴 × 𝐴))

Syntax hints:   → wi 4   ⊆ wss 3165   × cxp 4672  dom cdm 4674  ran crn 4675  Rel wrel 4679   Er wer 6616

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-v 2773  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-br 4044  df-opab 4105  df-xp 4680  df-rel 4681  df-cnv 4682  df-dm 4684  df-rn 4685  df-er 6619

This theorem is referenced by:  erex  6643  riinerm  6694