MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cnvssrndm Structured version   Visualization version   GIF version

Theorem cnvssrndm 6267
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 6100 . . 3 Rel 𝐴
2 relssdmrn 6264 . . 3 (Rel 𝐴𝐴 ⊆ (dom 𝐴 × ran 𝐴))
31, 2ax-mp 5 . 2 𝐴 ⊆ (dom 𝐴 × ran 𝐴)
4 df-rn 5686 . . 3 ran 𝐴 = dom 𝐴
5 dfdm4 5893 . . 3 dom 𝐴 = ran 𝐴
64, 5xpeq12i 5703 . 2 (ran 𝐴 × dom 𝐴) = (dom 𝐴 × ran 𝐴)
73, 6sseqtrri 4018 1 𝐴 ⊆ (ran 𝐴 × dom 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wss 3947   × cxp 5673  ccnv 5674  dom cdm 5675  ran crn 5676  Rel wrel 5680
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-br 5148  df-opab 5210  df-xp 5681  df-rel 5682  df-cnv 5683  df-dm 5685  df-rn 5686
This theorem is referenced by:  wuncnv  10721  fcnvgreu  31876  trclubgNEW  42302
  Copyright terms: Public domain W3C validator