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Theorem fssxp 5602
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 5594 . . 3  |-  ( F : A --> B  ->  Rel  F )
2 relssdmrn 5390 . . 3  |-  ( Rel 
F  ->  F  C_  ( dom  F  X.  ran  F
) )
31, 2syl 16 . 2  |-  ( F : A --> B  ->  F  C_  ( dom  F  X.  ran  F ) )
4 fdm 5595 . . . 4  |-  ( F : A --> B  ->  dom  F  =  A )
5 eqimss 3400 . . . 4  |-  ( dom 
F  =  A  ->  dom  F  C_  A )
64, 5syl 16 . . 3  |-  ( F : A --> B  ->  dom  F  C_  A )
7 frn 5597 . . 3  |-  ( F : A --> B  ->  ran  F  C_  B )
8 xpss12 4981 . . 3  |-  ( ( dom  F  C_  A  /\  ran  F  C_  B
)  ->  ( dom  F  X.  ran  F ) 
C_  ( A  X.  B ) )
96, 7, 8syl2anc 643 . 2  |-  ( F : A --> B  -> 
( dom  F  X.  ran  F )  C_  ( A  X.  B ) )
103, 9sstrd 3358 1  |-  ( F : A --> B  ->  F  C_  ( A  X.  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    C_ wss 3320    X. cxp 4876   dom cdm 4878   ran crn 4879   Rel wrel 4883   -->wf 5450
This theorem is referenced by:  fex2  5603  funssxp  5604  opelf  5606  fabexg  5624  dff2  5881  dff3  5882  f2ndf  6452  f1o2ndf1  6454  mapex  7024  uniixp  7085  hartogslem1  7511  wdom2d  7548  dfac12lem2  8024  infmap2  8098  axdc3lem  8330  tskcard  8656  dfle2  10740  ixxex  10927  imasvscafn  13762  imasvscaf  13764  fnmrc  13832  mrcfval  13833  isacs1i  13882  mreacs  13883  pjfval  16933  pjpm  16935  hausdiag  17677  isngp2  18644  volf  19425  fnct  24105  fndifnfp  26737  fgraphopab  27506
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  df-fun 5456  df-fn 5457  df-f 5458
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