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Theorem funssxp 5088
Description: Two ways of specifying a partial function from  A to  B. (Contributed by NM, 13-Nov-2007.)
Assertion
Ref Expression
funssxp  |-  ( ( Fun  F  /\  F  C_  ( A  X.  B
) )  <->  ( F : dom  F --> B  /\  dom  F  C_  A )
)

Proof of Theorem funssxp
StepHypRef Expression
1 funfn 4959 . . . . . 6  |-  ( Fun 
F  <->  F  Fn  dom  F )
21biimpi 117 . . . . 5  |-  ( Fun 
F  ->  F  Fn  dom  F )
3 rnss 4592 . . . . . 6  |-  ( F 
C_  ( A  X.  B )  ->  ran  F 
C_  ran  ( A  X.  B ) )
4 rnxpss 4782 . . . . . 6  |-  ran  ( A  X.  B )  C_  B
53, 4syl6ss 2985 . . . . 5  |-  ( F 
C_  ( A  X.  B )  ->  ran  F 
C_  B )
62, 5anim12i 325 . . . 4  |-  ( ( Fun  F  /\  F  C_  ( A  X.  B
) )  ->  ( F  Fn  dom  F  /\  ran  F  C_  B )
)
7 df-f 4934 . . . 4  |-  ( F : dom  F --> B  <->  ( F  Fn  dom  F  /\  ran  F 
C_  B ) )
86, 7sylibr 141 . . 3  |-  ( ( Fun  F  /\  F  C_  ( A  X.  B
) )  ->  F : dom  F --> B )
9 dmss 4562 . . . . 5  |-  ( F 
C_  ( A  X.  B )  ->  dom  F 
C_  dom  ( A  X.  B ) )
10 dmxpss 4781 . . . . 5  |-  dom  ( A  X.  B )  C_  A
119, 10syl6ss 2985 . . . 4  |-  ( F 
C_  ( A  X.  B )  ->  dom  F 
C_  A )
1211adantl 266 . . 3  |-  ( ( Fun  F  /\  F  C_  ( A  X.  B
) )  ->  dom  F 
C_  A )
138, 12jca 294 . 2  |-  ( ( Fun  F  /\  F  C_  ( A  X.  B
) )  ->  ( F : dom  F --> B  /\  dom  F  C_  A )
)
14 ffun 5076 . . . 4  |-  ( F : dom  F --> B  ->  Fun  F )
1514adantr 265 . . 3  |-  ( ( F : dom  F --> B  /\  dom  F  C_  A )  ->  Fun  F )
16 fssxp 5086 . . . 4  |-  ( F : dom  F --> B  ->  F  C_  ( dom  F  X.  B ) )
17 xpss1 4476 . . . 4  |-  ( dom 
F  C_  A  ->  ( dom  F  X.  B
)  C_  ( A  X.  B ) )
1816, 17sylan9ss 2986 . . 3  |-  ( ( F : dom  F --> B  /\  dom  F  C_  A )  ->  F  C_  ( A  X.  B
) )
1915, 18jca 294 . 2  |-  ( ( F : dom  F --> B  /\  dom  F  C_  A )  ->  ( Fun  F  /\  F  C_  ( A  X.  B
) ) )
2013, 19impbii 121 1  |-  ( ( Fun  F  /\  F  C_  ( A  X.  B
) )  <->  ( F : dom  F --> B  /\  dom  F  C_  A )
)
Colors of variables: wff set class
Syntax hints:    /\ wa 101    <-> wb 102    C_ wss 2945    X. cxp 4371   dom cdm 4373   ran crn 4374   Fun wfun 4924    Fn wfn 4925   -->wf 4926
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-br 3793  df-opab 3847  df-xp 4379  df-rel 4380  df-cnv 4381  df-dm 4383  df-rn 4384  df-fun 4932  df-fn 4933  df-f 4934
This theorem is referenced by: (None)
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