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Theorem fssxp 5285
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 5272 . . 3  |-  ( F : A --> B  ->  Rel  F )
2 relssdmrn 5054 . . 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 5273 . . . 4  |-  ( F : A --> B  ->  dom  F  =  A )
5 eqimss 3146 . . . 4  |-  ( dom 
F  =  A  ->  dom  F  C_  A )
64, 5syl 14 . . 3  |-  ( F : A --> B  ->  dom  F  C_  A )
7 frn 5276 . . 3  |-  ( F : A --> B  ->  ran  F  C_  B )
8 xpss12 4641 . . 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 3102 1  |-  ( F : A --> B  ->  F  C_  ( A  X.  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1331    C_ wss 3066    X. cxp 4532   dom cdm 4534   ran crn 4535   Rel wrel 4539   -->wf 5114
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-br 3925  df-opab 3985  df-xp 4540  df-rel 4541  df-cnv 4542  df-dm 4544  df-rn 4545  df-fun 5120  df-fn 5121  df-f 5122
This theorem is referenced by:  fex2  5286  funssxp  5287  opelf  5289  fabexg  5305  dff2  5557  dff3im  5558  f2ndf  6116  f1o2ndf1  6118  tfrlemibfn  6218  tfr1onlembfn  6234  tfrcllembfn  6247  mapex  6541  uniixp  6608  ixxex  9675
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