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Theorem fssxp 5416
Description: A mapping is a class of ordered pairs. (Contributed by NM, 3-Aug-1994.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
Assertion
Ref Expression
fssxp  |-  ( F : A --> B  ->  F  C_  ( A  X.  B ) )

Proof of Theorem fssxp
StepHypRef Expression
1 frel 5408 . . 3  |-  ( F : A --> B  ->  Rel  F )
2 relssdmrn 5209 . . 3  |-  ( Rel 
F  ->  F  C_  ( dom  F  X.  ran  F
) )
31, 2syl 15 . 2  |-  ( F : A --> B  ->  F  C_  ( dom  F  X.  ran  F ) )
4 fdm 5409 . . . 4  |-  ( F : A --> B  ->  dom  F  =  A )
5 eqimss 3243 . . . 4  |-  ( dom 
F  =  A  ->  dom  F  C_  A )
64, 5syl 15 . . 3  |-  ( F : A --> B  ->  dom  F  C_  A )
7 frn 5411 . . 3  |-  ( F : A --> B  ->  ran  F  C_  B )
8 xpss12 4808 . . 3  |-  ( ( dom  F  C_  A  /\  ran  F  C_  B
)  ->  ( dom  F  X.  ran  F ) 
C_  ( A  X.  B ) )
96, 7, 8syl2anc 642 . 2  |-  ( F : A --> B  -> 
( dom  F  X.  ran  F )  C_  ( A  X.  B ) )
103, 9sstrd 3202 1  |-  ( F : A --> B  ->  F  C_  ( A  X.  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    C_ wss 3165    X. cxp 4703   dom cdm 4705   ran crn 4706   Rel wrel 4710   -->wf 5267
This theorem is referenced by:  fex2  5417  funssxp  5418  opelf  5420  fabexg  5438  dff2  5688  dff3  5689  mapex  6794  uniixp  6855  hartogslem1  7273  wdom2d  7310  dfac12lem2  7786  infmap2  7860  axdc3lem  8092  tskcard  8419  dfle2  10497  ixxex  10683  imasvscafn  13455  imasvscaf  13457  fnmrc  13525  mrcfval  13526  isacs1i  13575  mreacs  13576  pjfval  16622  pjpm  16624  hausdiag  17355  isngp2  18135  volf  18904  fnct  23356  scprefat  25174  1alg  25825  fnctartar  26010  fnctartar2  26011  fndifnfp  26859  fgraphopab  27632
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  df-fun 5273  df-fn 5274  df-f 5275
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