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Theorem cnvssrndm 5372
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 5222 . . 3  |-  Rel  `' A
2 relssdmrn 5371 . . 3  |-  ( Rel  `' A  ->  `' A  C_  ( dom  `' A  X.  ran  `' A ) )
31, 2ax-mp 5 . 2  |-  `' A  C_  ( dom  `' A  X.  ran  `' A )
4 df-rn 4860 . . 3  |-  ran  A  =  dom  `' A
5 dfdm4 5042 . . 3  |-  dom  A  =  ran  `' A
64, 5xpeq12i 4871 . 2  |-  ( ran 
A  X.  dom  A
)  =  ( dom  `' A  X.  ran  `' A )
73, 6sseqtr4i 3497 1  |-  `' A  C_  ( ran  A  X.  dom  A )
Colors of variables: wff setvar class
Syntax hints:    C_ wss 3436    X. cxp 4847   `'ccnv 4848   dom cdm 4849   ran crn 4850   Rel wrel 4854
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-sep 4543  ax-nul 4551  ax-pr 4656
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-ral 2780  df-rex 2781  df-rab 2784  df-v 3083  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3910  df-sn 3997  df-pr 3999  df-op 4003  df-br 4421  df-opab 4480  df-xp 4855  df-rel 4856  df-cnv 4857  df-dm 4859  df-rn 4860
This theorem is referenced by:  wuncnv  9155  fcnvgreu  28264
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