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Theorem homaf 14177
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 3934 . . . . . 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 3360 . . . . 5  |-  ( (  Hom  `  C ) `  x )  C_  _V
4 xpss12 4973 . . . . 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 4397 . . . . . 6  |-  { x }  e.  _V
7 fvex 5734 . . . . . 6  |-  ( (  Hom  `  C ) `  x )  e.  _V
86, 7xpex 4982 . . . . 5  |-  ( { x }  X.  (
(  Hom  `  C ) `
 x ) )  e.  _V
98elpw 3797 . . . 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 2435 . . 3  |-  ( x  e.  ( B  X.  B )  |->  ( { x }  X.  (
(  Hom  `  C ) `
 x ) ) )  =  ( x  e.  ( B  X.  B )  |->  ( { x }  X.  (
(  Hom  `  C ) `
 x ) ) )
1210, 11fmptd 5885 . 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 2435 . . . 4  |-  (  Hom  `  C )  =  (  Hom  `  C )
1713, 14, 15, 16homafval 14176 . . 3  |-  ( ph  ->  H  =  ( x  e.  ( B  X.  B )  |->  ( { x }  X.  (
(  Hom  `  C ) `
 x ) ) ) )
1817feq1d 5572 . 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 1652    e. wcel 1725   _Vcvv 2948    C_ wss 3312   ~Pcpw 3791   {csn 3806    e. cmpt 4258    X. cxp 4868   -->wf 5442   ` cfv 5446   Basecbs 13461    Hom chom 13532   Catccat 13881  Homachoma 14170
This theorem is referenced by:  homarcl2  14182  homarel  14183  arwhoma  14192
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-homa 14173
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