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Theorem cnvssrndm 6260
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 6093 . . 3 Rel 𝐴
2 relssdmrn 6257 . . 3 (Rel 𝐴𝐴 ⊆ (dom 𝐴 × ran 𝐴))
31, 2ax-mp 5 . 2 𝐴 ⊆ (dom 𝐴 × ran 𝐴)
4 df-rn 5677 . . 3 ran 𝐴 = dom 𝐴
5 dfdm4 5885 . . 3 dom 𝐴 = ran 𝐴
64, 5xpeq12i 5694 . 2 (ran 𝐴 × dom 𝐴) = (dom 𝐴 × ran 𝐴)
73, 6sseqtrri 4011 1 𝐴 ⊆ (ran 𝐴 × dom 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wss 3940   × cxp 5664  ccnv 5665  dom cdm 5666  ran crn 5667  Rel wrel 5671
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-br 5139  df-opab 5201  df-xp 5672  df-rel 5673  df-cnv 5674  df-dm 5676  df-rn 5677
This theorem is referenced by:  wuncnv  10720  fcnvgreu  32333  trclubgNEW  42824
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