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Theorem cnvssrndm 4892
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 4753 . . 3  |-  Rel  `' A
2 relssdmrn 4891 . . 3  |-  ( Rel  `' A  ->  `' A  C_  ( dom  `' A  X.  ran  `' A ) )
31, 2ax-mp 7 . 2  |-  `' A  C_  ( dom  `' A  X.  ran  `' A )
4 df-rn 4402 . . 3  |-  ran  A  =  dom  `' A
5 dfdm4 4575 . . 3  |-  dom  A  =  ran  `' A
64, 5xpeq12i 4413 . 2  |-  ( ran 
A  X.  dom  A
)  =  ( dom  `' A  X.  ran  `' A )
73, 6sseqtr4i 3041 1  |-  `' A  C_  ( ran  A  X.  dom  A )
Colors of variables: wff set class
Syntax hints:    C_ wss 2982    X. cxp 4389   `'ccnv 4390   dom cdm 4391   ran crn 4392   Rel wrel 4396
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2612  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-br 3806  df-opab 3860  df-xp 4397  df-rel 4398  df-cnv 4399  df-dm 4401  df-rn 4402
This theorem is referenced by: (None)
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