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Theorem funssxp 5367
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 5228 . . . . . 6  |-  ( Fun 
F  <->  F  Fn  dom  F )
21biimpi 119 . . . . 5  |-  ( Fun 
F  ->  F  Fn  dom  F )
3 rnss 4841 . . . . . 6  |-  ( F 
C_  ( A  X.  B )  ->  ran  F 
C_  ran  ( A  X.  B ) )
4 rnxpss 5042 . . . . . 6  |-  ran  ( A  X.  B )  C_  B
53, 4sstrdi 3159 . . . . 5  |-  ( F 
C_  ( A  X.  B )  ->  ran  F 
C_  B )
62, 5anim12i 336 . . . 4  |-  ( ( Fun  F  /\  F  C_  ( A  X.  B
) )  ->  ( F  Fn  dom  F  /\  ran  F  C_  B )
)
7 df-f 5202 . . . 4  |-  ( F : dom  F --> B  <->  ( F  Fn  dom  F  /\  ran  F 
C_  B ) )
86, 7sylibr 133 . . 3  |-  ( ( Fun  F  /\  F  C_  ( A  X.  B
) )  ->  F : dom  F --> B )
9 dmss 4810 . . . . 5  |-  ( F 
C_  ( A  X.  B )  ->  dom  F 
C_  dom  ( A  X.  B ) )
10 dmxpss 5041 . . . . 5  |-  dom  ( A  X.  B )  C_  A
119, 10sstrdi 3159 . . . 4  |-  ( F 
C_  ( A  X.  B )  ->  dom  F 
C_  A )
1211adantl 275 . . 3  |-  ( ( Fun  F  /\  F  C_  ( A  X.  B
) )  ->  dom  F 
C_  A )
138, 12jca 304 . 2  |-  ( ( Fun  F  /\  F  C_  ( A  X.  B
) )  ->  ( F : dom  F --> B  /\  dom  F  C_  A )
)
14 ffun 5350 . . . 4  |-  ( F : dom  F --> B  ->  Fun  F )
1514adantr 274 . . 3  |-  ( ( F : dom  F --> B  /\  dom  F  C_  A )  ->  Fun  F )
16 fssxp 5365 . . . 4  |-  ( F : dom  F --> B  ->  F  C_  ( dom  F  X.  B ) )
17 xpss1 4721 . . . 4  |-  ( dom 
F  C_  A  ->  ( dom  F  X.  B
)  C_  ( A  X.  B ) )
1816, 17sylan9ss 3160 . . 3  |-  ( ( F : dom  F --> B  /\  dom  F  C_  A )  ->  F  C_  ( A  X.  B
) )
1915, 18jca 304 . 2  |-  ( ( F : dom  F --> B  /\  dom  F  C_  A )  ->  ( Fun  F  /\  F  C_  ( A  X.  B
) ) )
2013, 19impbii 125 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 103    <-> wb 104    C_ wss 3121    X. cxp 4609   dom cdm 4611   ran crn 4612   Fun wfun 5192    Fn wfn 5193   -->wf 5194
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-xp 4617  df-rel 4618  df-cnv 4619  df-dm 4621  df-rn 4622  df-fun 5200  df-fn 5201  df-f 5202
This theorem is referenced by:  elpm2g  6643  casef  7065
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