Theorem cnvssrndm 5188

Description: The converse is a subset of the cartesian product of range and domain. (Contributed by Mario Carneiro, 2-Jan-2017.)

Assertion

Ref	Expression
cnvssrndm	⊢ ◡𝐴 ⊆ (ran 𝐴 × dom 𝐴)

Proof of Theorem cnvssrndm

Step	Hyp	Ref	Expression
1		relcnv 5044	. . 3 ⊢ Rel ◡𝐴
2		relssdmrn 5187	. . 3 ⊢ (Rel ◡𝐴 → ◡𝐴 ⊆ (dom ◡𝐴 × ran ◡𝐴))
3	1, 2	ax-mp 5	. 2 ⊢ ◡𝐴 ⊆ (dom ◡𝐴 × ran ◡𝐴)
4		df-rn 4671	. . 3 ⊢ ran 𝐴 = dom ◡𝐴
5		dfdm4 4855	. . 3 ⊢ dom 𝐴 = ran ◡𝐴
6	4, 5	xpeq12i 4682	. 2 ⊢ (ran 𝐴 × dom 𝐴) = (dom ◡𝐴 × ran ◡𝐴)
7	3, 6	sseqtrri 3215	1 ⊢ ◡𝐴 ⊆ (ran 𝐴 × dom 𝐴)

Syntax hints:   ⊆ wss 3154   × cxp 4658  ^◡ccnv 4659  dom cdm 4660  ran crn 4661  Rel wrel 4665

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 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-br 4031  df-opab 4092  df-xp 4666  df-rel 4667  df-cnv 4668  df-dm 4670  df-rn 4671

This theorem is referenced by: (None)