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Theorem fssxp 5083
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 5074 . . 3  |-  ( F : A --> B  ->  Rel  F )
2 relssdmrn 4865 . . 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 5075 . . . 4  |-  ( F : A --> B  ->  dom  F  =  A )
5 eqimss 3052 . . . 4  |-  ( dom 
F  =  A  ->  dom  F  C_  A )
64, 5syl 14 . . 3  |-  ( F : A --> B  ->  dom  F  C_  A )
7 frn 5077 . . 3  |-  ( F : A --> B  ->  ran  F  C_  B )
8 xpss12 4467 . . 3  |-  ( ( dom  F  C_  A  /\  ran  F  C_  B
)  ->  ( dom  F  X.  ran  F ) 
C_  ( A  X.  B ) )
96, 7, 8syl2anc 403 . 2  |-  ( F : A --> B  -> 
( dom  F  X.  ran  F )  C_  ( A  X.  B ) )
103, 9sstrd 3010 1  |-  ( F : A --> B  ->  F  C_  ( A  X.  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1285    C_ wss 2974    X. cxp 4363   dom cdm 4365   ran crn 4366   Rel wrel 4370   -->wf 4922
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3898  ax-pow 3950  ax-pr 3966
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-br 3788  df-opab 3842  df-xp 4371  df-rel 4372  df-cnv 4373  df-dm 4375  df-rn 4376  df-fun 4928  df-fn 4929  df-f 4930
This theorem is referenced by:  fex2  5084  funssxp  5085  opelf  5087  fabexg  5102  dff2  5337  dff3im  5338  f2ndf  5872  f1o2ndf1  5874  tfrlemibfn  5971  tfr1onlembfn  5987  tfrcllembfn  6000  ixxex  8987
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