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Theorem homaf 13878
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 3775 . . . . . 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 3211 . . . . 5  |-  ( (  Hom  `  C ) `  x )  C_  _V
4 xpss12 4808 . . . . 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 4232 . . . . . 6  |-  { x }  e.  _V
7 fvex 5555 . . . . . 6  |-  ( (  Hom  `  C ) `  x )  e.  _V
86, 7xpex 4817 . . . . 5  |-  ( { x }  X.  (
(  Hom  `  C ) `
 x ) )  e.  _V
98elpw 3644 . . . 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 2296 . . 3  |-  ( x  e.  ( B  X.  B )  |->  ( { x }  X.  (
(  Hom  `  C ) `
 x ) ) )  =  ( x  e.  ( B  X.  B )  |->  ( { x }  X.  (
(  Hom  `  C ) `
 x ) ) )
1210, 11fmptd 5700 . 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 2296 . . . 4  |-  (  Hom  `  C )  =  (  Hom  `  C )
1713, 14, 15, 16homafval 13877 . . 3  |-  ( ph  ->  H  =  ( x  e.  ( B  X.  B )  |->  ( { x }  X.  (
(  Hom  `  C ) `
 x ) ) ) )
1817feq1d 5395 . 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 1632    e. wcel 1696   _Vcvv 2801    C_ wss 3165   ~Pcpw 3638   {csn 3653    e. cmpt 4093    X. cxp 4703   -->wf 5267   ` cfv 5271   Basecbs 13164    Hom chom 13235   Catccat 13582  Homachoma 13871
This theorem is referenced by:  homarcl2  13883  homarel  13884  arwhoma  13893
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-homa 13874
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