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Theorem fssxp 5561
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 5553 . . 3  |-  ( F : A --> B  ->  Rel  F )
2 relssdmrn 5349 . . 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 5554 . . . 4  |-  ( F : A --> B  ->  dom  F  =  A )
5 eqimss 3360 . . . 4  |-  ( dom 
F  =  A  ->  dom  F  C_  A )
64, 5syl 16 . . 3  |-  ( F : A --> B  ->  dom  F  C_  A )
7 frn 5556 . . 3  |-  ( F : A --> B  ->  ran  F  C_  B )
8 xpss12 4940 . . 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 3318 1  |-  ( F : A --> B  ->  F  C_  ( A  X.  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    C_ wss 3280    X. cxp 4835   dom cdm 4837   ran crn 4838   Rel wrel 4842   -->wf 5409
This theorem is referenced by:  fex2  5562  funssxp  5563  opelf  5565  fabexg  5583  dff2  5840  dff3  5841  f2ndf  6411  f1o2ndf1  6413  mapex  6983  uniixp  7044  hartogslem1  7467  wdom2d  7504  dfac12lem2  7980  infmap2  8054  axdc3lem  8286  tskcard  8612  dfle2  10696  ixxex  10883  imasvscafn  13717  imasvscaf  13719  fnmrc  13787  mrcfval  13788  isacs1i  13837  mreacs  13838  pjfval  16888  pjpm  16890  hausdiag  17630  isngp2  18597  volf  19378  fnct  24058  fndifnfp  26627  fgraphopab  27397
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-br 4173  df-opab 4227  df-xp 4843  df-rel 4844  df-cnv 4845  df-dm 4847  df-rn 4848  df-fun 5415  df-fn 5416  df-f 5417
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