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Theorem homaf 14140
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 3902 . . . . . 6  |-  ( x  e.  ( B  X.  B )  ->  { x }  C_  ( B  X.  B ) )
21adantl 453 . . . . 5  |-  ( (
ph  /\  x  e.  ( B  X.  B
) )  ->  { x }  C_  ( B  X.  B ) )
3 ssv 3328 . . . . 5  |-  ( (  Hom  `  C ) `  x )  C_  _V
4 xpss12 4940 . . . . 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 644 . . . 4  |-  ( (
ph  /\  x  e.  ( B  X.  B
) )  ->  ( { x }  X.  ( (  Hom  `  C
) `  x )
)  C_  ( ( B  X.  B )  X. 
_V ) )
6 snex 4365 . . . . . 6  |-  { x }  e.  _V
7 fvex 5701 . . . . . 6  |-  ( (  Hom  `  C ) `  x )  e.  _V
86, 7xpex 4949 . . . . 5  |-  ( { x }  X.  (
(  Hom  `  C ) `
 x ) )  e.  _V
98elpw 3765 . . . 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 204 . . 3  |-  ( (
ph  /\  x  e.  ( B  X.  B
) )  ->  ( { x }  X.  ( (  Hom  `  C
) `  x )
)  e.  ~P (
( B  X.  B
)  X.  _V )
)
11 eqid 2404 . . 3  |-  ( x  e.  ( B  X.  B )  |->  ( { x }  X.  (
(  Hom  `  C ) `
 x ) ) )  =  ( x  e.  ( B  X.  B )  |->  ( { x }  X.  (
(  Hom  `  C ) `
 x ) ) )
1210, 11fmptd 5852 . 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 2404 . . . 4  |-  (  Hom  `  C )  =  (  Hom  `  C )
1713, 14, 15, 16homafval 14139 . . 3  |-  ( ph  ->  H  =  ( x  e.  ( B  X.  B )  |->  ( { x }  X.  (
(  Hom  `  C ) `
 x ) ) ) )
1817feq1d 5539 . 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 224 1  |-  ( ph  ->  H : ( B  X.  B ) --> ~P ( ( B  X.  B )  X.  _V ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   _Vcvv 2916    C_ wss 3280   ~Pcpw 3759   {csn 3774    e. cmpt 4226    X. cxp 4835   -->wf 5409   ` cfv 5413   Basecbs 13424    Hom chom 13495   Catccat 13844  Homachoma 14133
This theorem is referenced by:  homarcl2  14145  homarel  14146  arwhoma  14155
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-homa 14136
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