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Theorem cnvssrndm 5574
 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
StepHypRef Expression
1 relcnv 5422 . . 3 Rel 𝐴
2 relssdmrn 5573 . . 3 (Rel 𝐴𝐴 ⊆ (dom 𝐴 × ran 𝐴))
31, 2ax-mp 5 . 2 𝐴 ⊆ (dom 𝐴 × ran 𝐴)
4 df-rn 5049 . . 3 ran 𝐴 = dom 𝐴
5 dfdm4 5238 . . 3 dom 𝐴 = ran 𝐴
64, 5xpeq12i 5061 . 2 (ran 𝐴 × dom 𝐴) = (dom 𝐴 × ran 𝐴)
73, 6sseqtr4i 3601 1 𝐴 ⊆ (ran 𝐴 × dom 𝐴)
 Colors of variables: wff setvar class Syntax hints:   ⊆ wss 3540   × cxp 5036  ◡ccnv 5037  dom cdm 5038  ran crn 5039  Rel wrel 5043 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-cnv 5046  df-dm 5048  df-rn 5049 This theorem is referenced by:  wuncnv  9431  fcnvgreu  28855  trclubgNEW  36944
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