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Theorem cnvssrndm 5391
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  C_  ( ran  A  X.  dom  A )

Proof of Theorem cnvssrndm
StepHypRef Expression
1 relcnv 5242 . . 3  |-  Rel  `' A
2 relssdmrn 5390 . . 3  |-  ( Rel  `' A  ->  `' A  C_  ( dom  `' A  X.  ran  `' A ) )
31, 2ax-mp 8 . 2  |-  `' A  C_  ( dom  `' A  X.  ran  `' A )
4 df-rn 4889 . . 3  |-  ran  A  =  dom  `' A
5 dfdm4 5063 . . 3  |-  dom  A  =  ran  `' A
64, 5xpeq12i 4900 . 2  |-  ( ran 
A  X.  dom  A
)  =  ( dom  `' A  X.  ran  `' A )
73, 6sseqtr4i 3381 1  |-  `' A  C_  ( ran  A  X.  dom  A )
Colors of variables: wff set class
Syntax hints:    C_ wss 3320    X. cxp 4876   `'ccnv 4877   dom cdm 4878   ran crn 4879   Rel wrel 4883
This theorem is referenced by:  wuncnv  8605
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-br 4213  df-opab 4267  df-xp 4884  df-rel 4885  df-cnv 4886  df-dm 4888  df-rn 4889
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