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Theorem homaf 14897
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 4016 . . . . . 6  |-  ( x  e.  ( B  X.  B )  ->  { x }  C_  ( B  X.  B ) )
21adantl 466 . . . . 5  |-  ( (
ph  /\  x  e.  ( B  X.  B
) )  ->  { x }  C_  ( B  X.  B ) )
3 ssv 3375 . . . . 5  |-  ( ( Hom  `  C ) `  x )  C_  _V
4 xpss12 4944 . . . . 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 662 . . . 4  |-  ( (
ph  /\  x  e.  ( B  X.  B
) )  ->  ( { x }  X.  ( ( Hom  `  C
) `  x )
)  C_  ( ( B  X.  B )  X. 
_V ) )
6 snex 4532 . . . . . 6  |-  { x }  e.  _V
7 fvex 5700 . . . . . 6  |-  ( ( Hom  `  C ) `  x )  e.  _V
86, 7xpex 6507 . . . . 5  |-  ( { x }  X.  (
( Hom  `  C ) `
 x ) )  e.  _V
98elpw 3865 . . . 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 212 . . 3  |-  ( (
ph  /\  x  e.  ( B  X.  B
) )  ->  ( { x }  X.  ( ( Hom  `  C
) `  x )
)  e.  ~P (
( B  X.  B
)  X.  _V )
)
11 eqid 2442 . . 3  |-  ( x  e.  ( B  X.  B )  |->  ( { x }  X.  (
( Hom  `  C ) `
 x ) ) )  =  ( x  e.  ( B  X.  B )  |->  ( { x }  X.  (
( Hom  `  C ) `
 x ) ) )
1210, 11fmptd 5866 . 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 2442 . . . 4  |-  ( Hom  `  C )  =  ( Hom  `  C )
1713, 14, 15, 16homafval 14896 . . 3  |-  ( ph  ->  H  =  ( x  e.  ( B  X.  B )  |->  ( { x }  X.  (
( Hom  `  C ) `
 x ) ) ) )
1817feq1d 5545 . 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 232 1  |-  ( ph  ->  H : ( B  X.  B ) --> ~P ( ( B  X.  B )  X.  _V ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   _Vcvv 2971    C_ wss 3327   ~Pcpw 3859   {csn 3876    e. cmpt 4349    X. cxp 4837   -->wf 5413   ` cfv 5417   Basecbs 14173   Hom chom 14248   Catccat 14601  Homachoma 14890
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4402  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rex 2720  df-reu 2721  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-op 3883  df-uni 4091  df-iun 4172  df-br 4292  df-opab 4350  df-mpt 4351  df-id 4635  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-homa 14893
This theorem is referenced by:  homarcl2  14902  homarel  14903  arwhoma  14912
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