MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cnvssrndm Unicode version

Theorem cnvssrndm 5210
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 5067 . . 3  |-  Rel  `' A
2 relssdmrn 5209 . . 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 4716 . . 3  |-  ran  A  =  dom  `' A
5 dfdm4 4888 . . 3  |-  dom  A  =  ran  `' A
64, 5xpeq12i 4727 . 2  |-  ( ran 
A  X.  dom  A
)  =  ( dom  `' A  X.  ran  `' A )
73, 6sseqtr4i 3224 1  |-  `' A  C_  ( ran  A  X.  dom  A )
Colors of variables: wff set class
Syntax hints:    C_ wss 3165    X. cxp 4703   `'ccnv 4704   dom cdm 4705   ran crn 4706   Rel wrel 4710
This theorem is referenced by:  wuncnv  8368
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-xp 4711  df-rel 4712  df-cnv 4713  df-dm 4715  df-rn 4716
  Copyright terms: Public domain W3C validator