Theorem cnvssrndm 6224

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 6058	. . 3 ⊢ Rel ◡𝐴
2		relssdmrn 6222	. . 3 ⊢ (Rel ◡𝐴 → ◡𝐴 ⊆ (dom ◡𝐴 × ran ◡𝐴))
3	1, 2	ax-mp 5	. 2 ⊢ ◡𝐴 ⊆ (dom ◡𝐴 × ran ◡𝐴)
4		df-rn 5631	. . 3 ⊢ ran 𝐴 = dom ◡𝐴
5		dfdm4 5839	. . 3 ⊢ dom 𝐴 = ran ◡𝐴
6	4, 5	xpeq12i 5648	. 2 ⊢ (ran 𝐴 × dom 𝐴) = (dom ◡𝐴 × ran ◡𝐴)
7	3, 6	sseqtrri 3966	1 ⊢ ◡𝐴 ⊆ (ran 𝐴 × dom 𝐴)

Syntax hints:   ⊆ wss 3885   × cxp 5618  ^◡ccnv 5619  dom cdm 5620  ran crn 5621  Rel wrel 5625

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2707  ax-sep 5220  ax-pr 5364

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2714  df-cleq 2727  df-clel 2810  df-ral 3050  df-rex 3060  df-rab 3388  df-v 3429  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4264  df-if 4457  df-sn 4558  df-pr 4560  df-op 4564  df-br 5075  df-opab 5137  df-xp 5626  df-rel 5627  df-cnv 5628  df-dm 5630  df-rn 5631

This theorem is referenced by:  wuncnv  10642  fcnvgreu  32733  trclubgNEW  44033