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Theorem funssxp 5003
Description: Two ways of specifying a partial function from to . (Contributed by NM, 13-Nov-2007.)
Assertion
Ref Expression
funssxp  Fun  F  F  C_  X.  F : dom  F -->  dom  F  C_

Proof of Theorem funssxp
StepHypRef Expression
1 funfn 4874 . . . . . 6  Fun 
F  F  Fn  dom  F
21biimpi 113 . . . . 5  Fun 
F  F  Fn  dom  F
3 rnss 4507 . . . . . 6  F 
C_  X.  ran  F  C_ 
ran  X.
4 rnxpss 4697 . . . . . 6  ran  X.  C_
53, 4syl6ss 2951 . . . . 5  F 
C_  X.  ran  F  C_
62, 5anim12i 321 . . . 4  Fun  F  F  C_  X.  F  Fn  dom  F  ran  F  C_
7 df-f 4849 . . . 4  F : dom  F -->  F  Fn  dom  F  ran  F  C_
86, 7sylibr 137 . . 3  Fun  F  F  C_  X.  F : dom  F -->
9 dmss 4477 . . . . 5  F 
C_  X.  dom  F  C_ 
dom  X.
10 dmxpss 4696 . . . . 5  dom  X.  C_
119, 10syl6ss 2951 . . . 4  F 
C_  X.  dom  F  C_
1211adantl 262 . . 3  Fun  F  F  C_  X.  dom  F  C_
138, 12jca 290 . 2  Fun  F  F  C_  X.  F : dom  F -->  dom  F  C_
14 ffun 4991 . . . 4  F : dom  F -->  Fun 
F
1514adantr 261 . . 3  F : dom  F -->  dom  F  C_  Fun  F
16 fssxp 5001 . . . 4  F : dom  F -->  F 
C_  dom  F  X.
17 xpss1 4391 . . . 4  dom 
F  C_  dom  F  X.  C_  X.
1816, 17sylan9ss 2952 . . 3  F : dom  F -->  dom  F  C_  F  C_  X.
1915, 18jca 290 . 2  F : dom  F -->  dom  F  C_  Fun  F  F  C_  X.
2013, 19impbii 117 1  Fun  F  F  C_  X.  F : dom  F -->  dom  F  C_
Colors of variables: wff set class
Syntax hints:   wa 97   wb 98    C_ wss 2911    X. cxp 4286   dom cdm 4288   ran crn 4289   Fun wfun 4839    Fn wfn 4840   -->wf 4841
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-cnv 4296  df-dm 4298  df-rn 4299  df-fun 4847  df-fn 4848  df-f 4849
This theorem is referenced by: (None)
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