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Theorem homaf 13862
Description: Functionality of the disjointified hom-set function. (Contributed by Mario Carneiro, 11-Jan-2017.)
Hypotheses
Ref Expression
homarcl.h  |-  H  =  (Homa
`  C )
homafval.b  |-  B  =  ( Base `  C
)
homafval.c  |-  ( ph  ->  C  e.  Cat )
Assertion
Ref Expression
homaf  |-  ( ph  ->  H : ( B  X.  B ) --> ~P ( ( B  X.  B )  X.  _V ) )

Proof of Theorem homaf
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 snssi 3759 . . . . . 6  |-  ( x  e.  ( B  X.  B )  ->  { x }  C_  ( B  X.  B ) )
21adantl 452 . . . . 5  |-  ( (
ph  /\  x  e.  ( B  X.  B
) )  ->  { x }  C_  ( B  X.  B ) )
3 ssv 3198 . . . . 5  |-  ( (  Hom  `  C ) `  x )  C_  _V
4 xpss12 4792 . . . . 5  |-  ( ( { x }  C_  ( B  X.  B
)  /\  ( (  Hom  `  C ) `  x )  C_  _V )  ->  ( { x }  X.  ( (  Hom  `  C ) `  x
) )  C_  (
( B  X.  B
)  X.  _V )
)
52, 3, 4sylancl 643 . . . 4  |-  ( (
ph  /\  x  e.  ( B  X.  B
) )  ->  ( { x }  X.  ( (  Hom  `  C
) `  x )
)  C_  ( ( B  X.  B )  X. 
_V ) )
6 snex 4216 . . . . . 6  |-  { x }  e.  _V
7 fvex 5539 . . . . . 6  |-  ( (  Hom  `  C ) `  x )  e.  _V
86, 7xpex 4801 . . . . 5  |-  ( { x }  X.  (
(  Hom  `  C ) `
 x ) )  e.  _V
98elpw 3631 . . . 4  |-  ( ( { x }  X.  ( (  Hom  `  C
) `  x )
)  e.  ~P (
( B  X.  B
)  X.  _V )  <->  ( { x }  X.  ( (  Hom  `  C
) `  x )
)  C_  ( ( B  X.  B )  X. 
_V ) )
105, 9sylibr 203 . . 3  |-  ( (
ph  /\  x  e.  ( B  X.  B
) )  ->  ( { x }  X.  ( (  Hom  `  C
) `  x )
)  e.  ~P (
( B  X.  B
)  X.  _V )
)
11 eqid 2283 . . 3  |-  ( x  e.  ( B  X.  B )  |->  ( { x }  X.  (
(  Hom  `  C ) `
 x ) ) )  =  ( x  e.  ( B  X.  B )  |->  ( { x }  X.  (
(  Hom  `  C ) `
 x ) ) )
1210, 11fmptd 5684 . 2  |-  ( ph  ->  ( x  e.  ( B  X.  B ) 
|->  ( { x }  X.  ( (  Hom  `  C
) `  x )
) ) : ( B  X.  B ) --> ~P ( ( B  X.  B )  X. 
_V ) )
13 homarcl.h . . . 4  |-  H  =  (Homa
`  C )
14 homafval.b . . . 4  |-  B  =  ( Base `  C
)
15 homafval.c . . . 4  |-  ( ph  ->  C  e.  Cat )
16 eqid 2283 . . . 4  |-  (  Hom  `  C )  =  (  Hom  `  C )
1713, 14, 15, 16homafval 13861 . . 3  |-  ( ph  ->  H  =  ( x  e.  ( B  X.  B )  |->  ( { x }  X.  (
(  Hom  `  C ) `
 x ) ) ) )
1817feq1d 5379 . 2  |-  ( ph  ->  ( H : ( B  X.  B ) --> ~P ( ( B  X.  B )  X. 
_V )  <->  ( x  e.  ( B  X.  B
)  |->  ( { x }  X.  ( (  Hom  `  C ) `  x
) ) ) : ( B  X.  B
) --> ~P ( ( B  X.  B )  X.  _V ) ) )
1912, 18mpbird 223 1  |-  ( ph  ->  H : ( B  X.  B ) --> ~P ( ( B  X.  B )  X.  _V ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788    C_ wss 3152   ~Pcpw 3625   {csn 3640    e. cmpt 4077    X. cxp 4687   -->wf 5251   ` cfv 5255   Basecbs 13148    Hom chom 13219   Catccat 13566  Homachoma 13855
This theorem is referenced by:  homarcl2  13867  homarel  13868  arwhoma  13877
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-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
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-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-homa 13858
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