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Theorem cnvssrndm 6110
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 5955 . . 3 Rel 𝐴
2 relssdmrn 6109 . . 3 (Rel 𝐴𝐴 ⊆ (dom 𝐴 × ran 𝐴))
31, 2ax-mp 5 . 2 𝐴 ⊆ (dom 𝐴 × ran 𝐴)
4 df-rn 5554 . . 3 ran 𝐴 = dom 𝐴
5 dfdm4 5752 . . 3 dom 𝐴 = ran 𝐴
64, 5xpeq12i 5571 . 2 (ran 𝐴 × dom 𝐴) = (dom 𝐴 × ran 𝐴)
73, 6sseqtrri 3990 1 𝐴 ⊆ (ran 𝐴 × dom 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wss 3919   × cxp 5541  ccnv 5542  dom cdm 5543  ran crn 5544  Rel wrel 5548
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5190  ax-nul 5197  ax-pr 5318
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-v 3482  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-br 5054  df-opab 5116  df-xp 5549  df-rel 5550  df-cnv 5551  df-dm 5553  df-rn 5554
This theorem is referenced by:  wuncnv  10146  fcnvgreu  30424  trclubgNEW  40174
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