Theorem cnvssrndm 5152

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 5008	. . 3 ⊢ Rel ◡𝐴
2		relssdmrn 5151	. . 3 ⊢ (Rel ◡𝐴 → ◡𝐴 ⊆ (dom ◡𝐴 × ran ◡𝐴))
3	1, 2	ax-mp 5	. 2 ⊢ ◡𝐴 ⊆ (dom ◡𝐴 × ran ◡𝐴)
4		df-rn 4639	. . 3 ⊢ ran 𝐴 = dom ◡𝐴
5		dfdm4 4821	. . 3 ⊢ dom 𝐴 = ran ◡𝐴
6	4, 5	xpeq12i 4650	. 2 ⊢ (ran 𝐴 × dom 𝐴) = (dom ◡𝐴 × ran ◡𝐴)
7	3, 6	sseqtrri 3192	1 ⊢ ◡𝐴 ⊆ (ran 𝐴 × dom 𝐴)

Syntax hints:   ⊆ wss 3131   × cxp 4626  ^◡ccnv 4627  dom cdm 4628  ran crn 4629  Rel wrel 4633

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-br 4006  df-opab 4067  df-xp 4634  df-rel 4635  df-cnv 4636  df-dm 4638  df-rn 4639

This theorem is referenced by: (None)