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Theorem funssxp 5682
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 5558 . . . . . 6  |-  ( Fun 
F  <->  F  Fn  dom  F )
21biimpi 194 . . . . 5  |-  ( Fun 
F  ->  F  Fn  dom  F )
3 rnss 5179 . . . . . 6  |-  ( F 
C_  ( A  X.  B )  ->  ran  F 
C_  ran  ( A  X.  B ) )
4 rnxpss 5381 . . . . . 6  |-  ran  ( A  X.  B )  C_  B
53, 4syl6ss 3479 . . . . 5  |-  ( F 
C_  ( A  X.  B )  ->  ran  F 
C_  B )
62, 5anim12i 566 . . . 4  |-  ( ( Fun  F  /\  F  C_  ( A  X.  B
) )  ->  ( F  Fn  dom  F  /\  ran  F  C_  B )
)
7 df-f 5533 . . . 4  |-  ( F : dom  F --> B  <->  ( F  Fn  dom  F  /\  ran  F 
C_  B ) )
86, 7sylibr 212 . . 3  |-  ( ( Fun  F  /\  F  C_  ( A  X.  B
) )  ->  F : dom  F --> B )
9 dmss 5150 . . . . 5  |-  ( F 
C_  ( A  X.  B )  ->  dom  F 
C_  dom  ( A  X.  B ) )
10 dmxpss 5380 . . . . 5  |-  dom  ( A  X.  B )  C_  A
119, 10syl6ss 3479 . . . 4  |-  ( F 
C_  ( A  X.  B )  ->  dom  F 
C_  A )
1211adantl 466 . . 3  |-  ( ( Fun  F  /\  F  C_  ( A  X.  B
) )  ->  dom  F 
C_  A )
138, 12jca 532 . 2  |-  ( ( Fun  F  /\  F  C_  ( A  X.  B
) )  ->  ( F : dom  F --> B  /\  dom  F  C_  A )
)
14 ffun 5672 . . . 4  |-  ( F : dom  F --> B  ->  Fun  F )
1514adantr 465 . . 3  |-  ( ( F : dom  F --> B  /\  dom  F  C_  A )  ->  Fun  F )
16 fssxp 5681 . . . 4  |-  ( F : dom  F --> B  ->  F  C_  ( dom  F  X.  B ) )
17 xpss1 5059 . . . 4  |-  ( dom 
F  C_  A  ->  ( dom  F  X.  B
)  C_  ( A  X.  B ) )
1816, 17sylan9ss 3480 . . 3  |-  ( ( F : dom  F --> B  /\  dom  F  C_  A )  ->  F  C_  ( A  X.  B
) )
1915, 18jca 532 . 2  |-  ( ( F : dom  F --> B  /\  dom  F  C_  A )  ->  ( Fun  F  /\  F  C_  ( A  X.  B
) ) )
2013, 19impbii 188 1  |-  ( ( Fun  F  /\  F  C_  ( A  X.  B
) )  <->  ( F : dom  F --> B  /\  dom  F  C_  A )
)
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    C_ wss 3439    X. cxp 4949   dom cdm 4951   ran crn 4952   Fun wfun 5523    Fn wfn 5524   -->wf 5525
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pr 4642
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-br 4404  df-opab 4462  df-xp 4957  df-rel 4958  df-cnv 4959  df-dm 4961  df-rn 4962  df-fun 5531  df-fn 5532  df-f 5533
This theorem is referenced by:  elpm2g  7342  volf  21154
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