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Theorem fssxp 5400
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 5392 . . 3  |-  ( F : A --> B  ->  Rel  F )
2 relssdmrn 5193 . . 3  |-  ( Rel 
F  ->  F  C_  ( dom  F  X.  ran  F
) )
31, 2syl 15 . 2  |-  ( F : A --> B  ->  F  C_  ( dom  F  X.  ran  F ) )
4 fdm 5393 . . . 4  |-  ( F : A --> B  ->  dom  F  =  A )
5 eqimss 3230 . . . 4  |-  ( dom 
F  =  A  ->  dom  F  C_  A )
64, 5syl 15 . . 3  |-  ( F : A --> B  ->  dom  F  C_  A )
7 frn 5395 . . 3  |-  ( F : A --> B  ->  ran  F  C_  B )
8 xpss12 4792 . . 3  |-  ( ( dom  F  C_  A  /\  ran  F  C_  B
)  ->  ( dom  F  X.  ran  F ) 
C_  ( A  X.  B ) )
96, 7, 8syl2anc 642 . 2  |-  ( F : A --> B  -> 
( dom  F  X.  ran  F )  C_  ( A  X.  B ) )
103, 9sstrd 3189 1  |-  ( F : A --> B  ->  F  C_  ( A  X.  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    C_ wss 3152    X. cxp 4687   dom cdm 4689   ran crn 4690   Rel wrel 4694   -->wf 5251
This theorem is referenced by:  fex2  5401  funssxp  5402  opelf  5404  fabexg  5422  dff2  5672  dff3  5673  mapex  6778  uniixp  6839  hartogslem1  7257  wdom2d  7294  dfac12lem2  7770  infmap2  7844  axdc3lem  8076  tskcard  8403  dfle2  10481  ixxex  10667  imasvscafn  13439  imasvscaf  13441  fnmrc  13509  mrcfval  13510  isacs1i  13559  mreacs  13560  pjfval  16606  pjpm  16608  hausdiag  17339  isngp2  18119  volf  18888  fnct  23341  scprefat  25071  1alg  25722  fnctartar  25907  fnctartar2  25908  fndifnfp  26756  fgraphopab  27529
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-xp 4695  df-rel 4696  df-cnv 4697  df-dm 4699  df-rn 4700  df-fun 5257  df-fn 5258  df-f 5259
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