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Theorem fssxp 5260
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 5247 . . 3  |-  ( F : A --> B  ->  Rel  F )
2 relssdmrn 5029 . . 3  |-  ( Rel 
F  ->  F  C_  ( dom  F  X.  ran  F
) )
31, 2syl 14 . 2  |-  ( F : A --> B  ->  F  C_  ( dom  F  X.  ran  F ) )
4 fdm 5248 . . . 4  |-  ( F : A --> B  ->  dom  F  =  A )
5 eqimss 3121 . . . 4  |-  ( dom 
F  =  A  ->  dom  F  C_  A )
64, 5syl 14 . . 3  |-  ( F : A --> B  ->  dom  F  C_  A )
7 frn 5251 . . 3  |-  ( F : A --> B  ->  ran  F  C_  B )
8 xpss12 4616 . . 3  |-  ( ( dom  F  C_  A  /\  ran  F  C_  B
)  ->  ( dom  F  X.  ran  F ) 
C_  ( A  X.  B ) )
96, 7, 8syl2anc 408 . 2  |-  ( F : A --> B  -> 
( dom  F  X.  ran  F )  C_  ( A  X.  B ) )
103, 9sstrd 3077 1  |-  ( F : A --> B  ->  F  C_  ( A  X.  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1316    C_ wss 3041    X. cxp 4507   dom cdm 4509   ran crn 4510   Rel wrel 4514   -->wf 5089
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-br 3900  df-opab 3960  df-xp 4515  df-rel 4516  df-cnv 4517  df-dm 4519  df-rn 4520  df-fun 5095  df-fn 5096  df-f 5097
This theorem is referenced by:  fex2  5261  funssxp  5262  opelf  5264  fabexg  5280  dff2  5532  dff3im  5533  f2ndf  6091  f1o2ndf1  6093  tfrlemibfn  6193  tfr1onlembfn  6209  tfrcllembfn  6222  mapex  6516  uniixp  6583  ixxex  9650
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