Theorem cnvssrndm 5191

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 5047	. . 3 ⊢ Rel ◡𝐴
2		relssdmrn 5190	. . 3 ⊢ (Rel ◡𝐴 → ◡𝐴 ⊆ (dom ◡𝐴 × ran ◡𝐴))
3	1, 2	ax-mp 5	. 2 ⊢ ◡𝐴 ⊆ (dom ◡𝐴 × ran ◡𝐴)
4		df-rn 4674	. . 3 ⊢ ran 𝐴 = dom ◡𝐴
5		dfdm4 4858	. . 3 ⊢ dom 𝐴 = ran ◡𝐴
6	4, 5	xpeq12i 4685	. 2 ⊢ (ran 𝐴 × dom 𝐴) = (dom ◡𝐴 × ran ◡𝐴)
7	3, 6	sseqtrri 3218	1 ⊢ ◡𝐴 ⊆ (ran 𝐴 × dom 𝐴)

Syntax hints:   ⊆ wss 3157   × cxp 4661  ^◡ccnv 4662  dom cdm 4663  ran crn 4664  Rel wrel 4668

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-br 4034  df-opab 4095  df-xp 4669  df-rel 4670  df-cnv 4671  df-dm 4673  df-rn 4674

This theorem is referenced by: (None)